zotero/storage/B3434XQ8/.zotero-ft-cache

What is the Electron?
Edited by Volodimir Simulik
Apeiron
Montreal


Published by C. Roy Keys Inc. 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com
© C. Roy Keys Inc. 2005
First Published 2005
Library and Archives Canada Cataloguing in Publication
What is the electron? / edited by Volodimir Simulik.
Includes bibliographical references. ISBN 0-9732911-2-5
1. Electrons. I. Simulik, Volodimir, 1957
QC793.5.E62W48 2005 539.7'2112 C2005-902252-3
Cover design by François Reeves. Cover graphic by Gordon Willliams.


Table of Contents
Preface ................................................................................................. i
Jaime Keller – A Comprehensive Theory of the Electron from START.................................................................................... 1
Hidezumi Terazawa – The Electron in the Unified Composite Model of All Fundamental Particles and Forces .................. 29
Thomas E. Phipps, Jr. – Prospects for the Point Electron ................ 43
Martin Rivas – The Spinning Electron ............................................. 59
Claude Daviau – Relativistic Wave Equations, Clifford Algebras and Orthogonal Gauge Groups.............................. 83
H. Sallhofer – What is the Electron? .............................................. 101
Volodimir Simulik – The Electron as a System of Classical Electromagnetic and Scalar Fields...................................... 105
Malcolm H. Mac Gregor – What Causes the Electron to Weigh?................................................................................ 129
Paolo Lanciani and Roberto Mignani – The Electron in a (3+3)-Dimensional Space-Time ......................................... 155
Vesselin Petkov – Can Studying the Inertial and Gravitational Properties of the Electron Provide us with an Insight into its Nature?.................................................................... 169
Horace R. Drew – The Electron as a Periodic Waveform of Spin 1⁄2 Symmetry ............................................................... 181
Fabio Cardone, Alessio Marrani and Roberto Mignani – A Geometrical Meaning of Electron Mass from Breakdown of Lorentz Invariance ...................................... 195
Paramahamsa Tewari – On the Space-Vortex Structure of the Electron............................................................................... 207
Milo Wolff and Geoff Haselhurst – Solving Nature’s Mysteries: Structure of the Electron and Origin of Natural Laws....................................................................... 227


William Gaede – Light: Neither Particle nor Transverse Wave..... 251
Françoise Tibika-Apfelbaum – A Matter of Life... or Hypothesis on the Role of Electron Waves in Creating “Order out of Chaos” .......................................................... 269
Adolphe Martin – The Electron as an Extended Structure in Cosmonic Gas..................................................................... 273


What is the Electron? i edited by Volodimir Simulik (Montreal: Apeiron 2005)
Preface
The electron is the first elementary particle, from both the physical and the historical point of view. It is the door to the microworld, to the physics of elementary particles and phenomena. This book is about electron models. The year 1997 marked the centenary of the discovery of the electron as a particle by J.J. Thomson. We have already passed the centenary of Planck’s great discovery and the beginning of quantum physics; 2001 marked the 75th anniversary of Schrödinger’s equation and the beginning of quantum mechanics, while the year 2003 was the 75th anniversary of the Dirac equation and Dirac’s model of the electron. Today the most widely used theoretical approaches to the physics of the electron and atom are quantum mechanical and field theoretical models based on the non-relativistic Schrödinger and the relativistic Dirac equations and their probabilistic interpretation. This is the basis of modern quantum field theory. More than 75 years is a long time for a physical theory! This theory is the basis for all contemporary calculations of physical phenomena. After 75 years most physical theories tend to be supplanted by new theories, or to be modified. The theory’s successes, as well as its difficulties, are now evident to specialists. There is no proof of the uniqueness of the quantum field theory approach to the model of the electron and atom. Are other approaches possible? Quantum field theory may be sufficient to describe the electron, but is it necessary? This theory and its mathematics are very complicated; can we now propose a simpler construction? Is the electron an extended structure, a compound object made up of sub-particles, or is it a point-like elementary particle, which does not consist of any sub-particles? What is the limit of application of modern classical physics (based either on the corpuscular or wave model) in the description of the electron? These and many other questions remain without definitive answers, while experiments on quantum entanglement have given rise to new discussion and debate. New high-precision experimental data, e.g., on the electric and magnetic dipole moments of the electron, may prove decisive. This book, What is the electron?, brings together papers by a number of authors. The main purpose of the book is to present original papers containing new ideas about the electron. What is the electron? presents different points of view on the electron, both within the framework of quantum theory and from competing approaches. Original modern models and hypotheses, based on new principles, are well represented. A comparison of different viewpoints (sometimes orthogonal) will aid further development of the physics of the electron. More than ten different models of the electron are presented here. More than twenty models are discussed briefly. Thus, the book gives a complete pic


ii Preface
ture of contemporary theoretical thinking (traditional and new) about the physics of the electron. It must be stressed that the vast majority of the authors do not appeal to quantum field theory, quantum mechanics or the probabilistic Copenhagen interpretation. The approaches adopted by these authors consist in using “lighter” mathematics and a “lighter” interpretation than in quantum theory. Some of them are sound approaches from the methodological point of view. The editor will not presume to judge the models or the authors. We will not venture to say which model is better, and why. The reasons are simple. (i) Readers can reach their own conclusions themselves. (ii) Investigation of the electron is by no means finished. (iii) My own point of view is presented in my contribution to the book. So I want my paper to be on an equal footing with other new models of the electron presented here. The general analysis of the electron models presented here shows that they can be classified as follows: corpuscular and wave, classical and quantum, point and extended, structureless and with structure. The reader can compare and ponder all these approaches! I would like to thank the authors for their contributions. It is my hope that this volume will prove worthwhile for readers, and encourage them to pursue further investigation of electron models.
Volodimir Simulik Senior Research Associate Institute of Electron Physics Ukrainian National Academy of Science. Uzhgorod, Ukraine


What is the Electron? 1 edited by Volodimir Simulik (Montreal: Apeiron 2005)
A Comprehensive Theory of the Electron from START
Jaime Keller Departamento de Física y Química Teórica Facultad de Química Universidad Nacional Autónoma de México AP 70-528, 04510, México D.F., MEXICO E-mail: keller@servidor.unam.mx and keller@cms.tuwien.ac.at
Space and Time are primitive concepts in science, used to describe material objects in relation with other material objects and the evolution of those relations. Mathematical description of those relations results in an observer’s geometric frame of reference. To describe the object’s behaviour, we add one more geometric element: the Action attributed to the system of objects. A fundamental concept is that of action carriers. The resulting Theory has a deductive character. A comprehensive (mass, charge, weak charge, spin, magnetic moment) theory of the electron is presented from this point of view. The main emphasis is given to the mathematical structures needed and the epistemological issues of the theory.
PACS number(s): 01.55.+b, 31.15.Ew, 71.10.-w, 71.15.Mb
Keywords: Space-Time-Action, Electron, Neutrino, Magnetic Interaction, Weak Interaction, START.
1. Introduction: space, time and material objects mathematical structures
This paper contributes to the construction of a deductive theory of matter, starting from first principles and using a single mathematical tool, geometric analysis. We present a comprehensive theory, where the analysis is centered in the theory of the electron. It represents a logical continuation of the material presented in the volume The Theory of the Electron, A theory of matter from START and a series of publications [1-5]. We recast here our fundamental philosophical and methodological remark. The theory of the electron developed in the above mentioned book is based on two main theoretical considerations: the nature of a scientific theory and the elements used to describe nature. The basic purpose of the theory presented here is a description of what can be observed, inferred, related and predicted within the fundamental limitations of experimental and theoretical science. We do not go beyond these limitations in any sense, nor seek to derive


2 Jaime Keller
fundamental concepts from model structures which might be supposed to be more fundamental. We use three basic elements of physical objects and phenomena: time, space, and action density. The first element, a one-dimensional manifold time, an evolution parameter, (a primitive concept, universally accepted) is defined by its mathematical properties. The concept of space, frame of reference, is defined, using the same considerations, through its mathematical description; this requires a three dimensional manifold in agreement with our anthropological apprehension of nature. The third element of physical nature considered here is given the unfortunate name of density of action, and describes the existence of physical objects, assuming action is a one dimensional manifold joined to the previous four in a geometrical unity. We have refrained from giving this concept a new name because we want to emphasize that we are presenting new conceptual and mathematical structures (Principia Geometrica Physicae). In our presentation an action density field is introduced into the space-time frame of reference to describe matter through the properties of this action density distribution. The space-time-matter concept is tautological: it is a set of nonseparable concepts in nature. Geometry is introduced through the use of a quadratic form to give a quadratic space structure to the variables:
2 2 22 2
2 2 2 22 2
2 2 2 22 2 2
Quadratic Forms from Pythagoras to the XXI Century
Quadratic form Dim diff Op Group
Statics
( ) 3 D Galileo
Kinetics
( ) ( ) 4 D , Poincaré
Dynamics
( ) 5 D , ST
lxyz t
s ct x y z D D
S ct x y z w K K
⎛⎞
⎜⎟
⎝⎠
/.
= + + − ∇,∇
= − ++ −
= − ++ − −
+
()
(0)
(0)
22 (0) (0)
ART
,
dc hE
wa a
aμ
μ
=κ , κ = = = ∑
"
When distributions of action in space-time are made to correspond to physical objects, we conclude that, as time evolves, the permanence of these objects is related to a set of symmetry constraints on that action distribution. The presentation used is then both a mathematical and an epistemological approach to the study of matter, and of physics itself.
1.1 Epistemological approach
The procedure followed in this article is: 1. To define a frame of reference to describe physical objects as a distribution field (carrier), the geometric space-time-action frame allows the definition of velocities and of energy-momentum as derivatives. 2. To define fields of carriers through a set of properties.


A comprehensive theory of the electron from START 3
3. To describe interaction as the possibility of exchange of energymomentum among the carriers (of sets of properties). 4. To find the equations for the interaction fields. 5. To find the sources of the interaction fields. 6. To determine the physical properties of a field of sources in such a form that those fields can be used as carriers. 7. To find the equations for the (source) carriers. 8. To find the observable properties of those carriers and identify them as the observed electron and its elementary particle partner, the neutrino.
1.2 Position and localization
Once the sub-frame space-time is defined there is a fundamental difference between position and localization. Position refers to a mathematical point x in space (which in general can be described by an anchored vector). The fixing point is called Coordinate Origin and a Poincaré coordinate transformation includes a change of this reference point. Localization refers to the possibility of assigning a restricted, continuous, set of position points ( ) 0
ρ x ≠ to a physical object (or phenomena). Localized objects are those for which a domain of position points can be assigned, the size of the domain being defined as the size of the physical object (or phenomenon). Non-localized objects correspond to those for which the domain of explicitly considered position points is larger than some assumed size of the object (or phenomenon).
1.3 Mass, charge, action, space and time
In our theory action, as a fundamental variable, is distributed among a set of carrier of action fields. An action density w(x,t) , action w per unit spacetime hypervolume 0 1 2 3
Δx Δx Δx Δx at point (x,t) with x0 = ct , is the fundamental concept defining space (parameterized by x), time (parameterized by t), and action density (parameterized by a scalar analytical function w(x,t) , as primitive concepts from which all other physical quantities will be derived or at least related directly or indirectly. The different forms of distributing the action among these carriers define the carriers themselves. This is fundamental in the practical use of the four principles below. For an elementary carrier n we will define ( , ) ( , )
n nn
w x t = f ρ x t . With constant in space fn . Within our fundamental formulation we will have to define properties of the fields we call carriers. A carrier will have physical significance through its set of properties. The density ρ of an elementary carrier field can be defined through a set of scalar constants, such that the integral of the product of these constants, and the density gives the experimentally attributed value of a property for that carrier. We will use an example: a carrier field identified with an electron will have a density ( ,t)
ρ x , and if the property is Q we will define Q q( ,t)d q ( ,t)d
= =ρ
∫∫
x x x x for all t, which determines that Q is a constant property (in space and time) for that field. The set of properties {Q} characterizes a carrier field and in turn establishes the conditions for a density field to correspond to an acceptable carrier.


4 Jaime Keller
The concept of charge appears in the theory first of all from the necessity to define the objects which exchange action (charges are always relative properties) in order to give a formal meaning to the principle that action will be exchanged in integer units of the Planck constant. In this context for an electronlike carrier both mass and electric charge belong to the generic name of ‘charges’. This program can obviously not be achieved if the formulation is not suitable to deduce of the theory of elementary particles, giving a geometric meaning to this theory. The definition of w is as a finite analytical action density and, to agree with standard formulations, the energy density E w / t
=∂ ∂
 and the momentum density / i
pi = ∂w ∂x are the fundamental rates of change of the primitive concept of action (considering a unit time-like interval Δx0 = 1 ).
In our full geometrization scheme a vectorial representation X x eμ
μ
= for {ct, x, y, z; xμ,μ= 0,1, 2,3} is used, and from the space-time gradient of w we recover the positive semi-definite energy-momentum expression
22 2 2 2 2
/ ,,
x yz
E c p p p P P p eμ
μ
− − − = = (1)
as well as the space-time 2 2 2 2 2 2
ds = c dt − dx − dy − dz . Action change
dK = P ⋅ dX is introduced through quadratic terms 2
dK (see appendix)
{}
2
2 2 2 2 222 2 2 2 2 2 2
0( / ) ,
xyz
dS − ds = − dK = −κ E c c dt − p dx − p dy − p dz (2)
creating a unified geometrical quadratic form 2
dS . The dK vector, the directional in space-time change of action, is a new theoretical quantity formally defined by (2). Notice that the generalization 2 2
ds ⇒ dS also corresponds to a (generally curved) generalization of the space-time metric
( ) ( )( )( )
2 22 2 2 2 22 2 22 2 22 2 0 000
1/ 1 1 1
xyz
dS = −κ E c c dt − −κ p dx − −κ p dy − −κ p dz
1.4 Hypotheses and principles of START
The set of hypotheses and principles which are explicitly included in our theory are called START [3]:
Physics is the science which describes the basic phenomena of Nature within the procedures of the Scientific Method.
We consider that the mathematization of the anthropocentric primary concepts of space, time and the existence of physical objects (action carriers), is a suitable point of departure for creating intellectual structures which describe Nature.
We introduce a set of principles: Relativity, Existence, Quantization and Choice as the operational procedure, and a set of 3 mathematical postulates to give these principles a formal, useful, structure.
We have derived in this and previous papers some of the fundamental structures of Physics: General Relativity, Density Functional Theory, Newtonian Gravitation and the Maxwell formulation of Electromagnetism. A fundamental common concept is the definition of energy (action) carriers. Most of the relations presented here are known, our procedure derives these structures and theories from START.


A comprehensive theory of the electron from START 5
1.5 Energy, momentum and interaction fields
There are in the theory two different forms of studying contribution to energy and momentum: the quantities defined in the paragraphs above and, second, quantities that will be called relative energy or relative momentum.
Principle of Space-Time-Action Relativity. In a space-time-action manifold an unstructured observer cannot determine his own state of motion; he can only determine the relative motion of other bodies in relation to himself and among the other bodies themselves. Light in the space-time-action manifold is assigned the “speed” c. An observer of a “system of bodies” will describe first each body as in motion relative to the observer with the concept: motion originated momentum (p), and, second, the motion of that body in relation to the rest of the system with the concept: interaction originated momentum (Δp=eA). The interaction originated momentum is the result of a non-unique description procedure, this freedom of definition will mathematically appear as a “gauge freedom” in the formulations below. The total momentum to enter in the descriptions for bodies in interaction is p + Δp.
1.6 Action carriers in START
Consider a set of scalar field “carriers” in such a form that the total action density in space-time is the sum of the action attributed to the carriers. Some properties arise from the START geometry itself, others from the description of a physical system as a time evolving energy distribution. In stationary systems, for a given observer, an elementary carrier field c is defined to have an energy density εcρc (x) with εc being a constant in space, and an integer number of carriers Nc of type c. The density ( , )
ρc x t obeys (x, ) x
cc V
ρ t d =N
∫ in the sys
tem’s volume V . We make a sharp distinction between action density and Lagrangian density. The Lagrangian contains, in general, prescriptions (and Lagrange multipliers) for the description of the system. Both the action density function w(X ) and the splitting among carrier fields will be considered analytically well-behaved functions. A description is
introduced when we treat the energy E( t) of a system as a sum of the different carrier types {c} such that ( ) ( )
c c
E t =∑E t
  , a sum of constants ( )
Ec t
 in space for a given observer.
1.7 Carriers and physical bodies
The charges are to be defined in our theory from a geometrical analysis of the distribution w( X ) when momentum is described in two ways: the amount which is related to the rate of change of action with respect to relative position, and the amount, per unit charge, which is pairwise shared, adding to zero, among the carriers. The rates of change of relative energy and momentum are called forces. A carrier for which a current of charges can be defined is by definition a body. A body corresponds to our hitherto undefined concept of matter.


6 Jaime Keller
Our study below will show that we cannot define an elementary body unless other properties, in addition to charge, are given to the carrier.
In our presentation the word “particle” is systematically avoided as for many authors it refers to a “point” body, with no spatial dimensions. The word body, on the other hand, conveys the idea of spatial distribution. Point-like distributions can only be introduced as a practical tool for handling a distribution confined to a region of space small in relation to the total system’s volume.
1.7.1 Maxwell equations from START
In our formalism [3,4] the Maxwell equations in their standard textbook form are analytical properties of the third derivatives of the action density attributed to a test carrier (with ‘electric’ charge) as induced by a collection of interacting carriers. The energy per carrier can be considered the derivative of a scalar field, but the momentum for interacting carriers cannot be solely considered the gradient of a scalar field. In this particular case, assume that we describe a set of carriers as interacting by partitioning an amount of energy (the interaction energy E ( )
eX
 ) among them, allowing the partitioning to be described as the sum of the overall momentum ( ( ) / i )ei
∂we X ∂x plus the momentum e,i e
i Rp
Δ
induced by interactions among the carriers. These interaction moment fields might then have a non-null rotational part. Consider, in the reference frame of a given observer, the induced action density (arising from the interaction), denoted by ( )
∂we X , per unit charge
( ⇒ p.u.ch) of a test carrier at space-time point X xμe
μ
= . Here the Greek indices 0,1, 2,3
μ= and x0 = ct whereas the space vectors = e = e ,
ii ii
qq q
0
e = , = 1, 2,3
ii
e e i are written in bold face letters, and we use the standard definitions of “dot” and “cross” products. From it define the related energy density ( )
Ee X and the total (external plus induced) momentum density pe , per unit charge of the test carrier, as
,,
() ()
( ) , e e,
ii
ee e e ei R ei
i
wX wX
X pp p
tx
∂ ⎛∂ ⎞
= = = +Δ
⎜⎟
∂∂
⎝⎠
E {def. 1} (3)
also, by definition, the electric field strength E as the force (p.u.ch) corresponding to these terms
,
() p
e () ,
ei i
ee e
i
p
XX
xt t
∂
⎛⎞
∂∂
= + =∇ +
⎜⎟
∂∂ ∂
⎝⎠
E
E E (4)
with time dependence
22
23 ,, 2
()
() ()
e 2 e e.
()
ei R ei
ii i
ee ii
pp
X aX
t tx tt tx t t
⎛⎞
∂ ∂Δ
∂∂
∂= + = +
⎜⎟
⎜⎟
∂ ∂∂ ∂∂ ∂∂ ∂ ∂
⎝⎠
E
E
By definition of interacting carriers, we have added in {def. 1} the term , ei
ΔR pe i as the effect of the conservation of interaction transverse moment between the fields representing the rest of the carriers with that sort of charges. This is by definition the origin, in START, of a magnetic field intensity


A comprehensive theory of the electron from START 7
Bk
= Bke that will appear as the curl of the momentum (p.u.ch) of an interac
tion field acting on a carrier of type b. The axial vector
,( )
B e e p,
ei j i
e
i
pX
x
⎛∂ ⎞
= × =∇×
⎜⎟
∂
⎝⎠ with time dependence
2
,( )
B e e.
ei j i i
pX
t tx
⎛⎞
∂
∂= ×
⎜⎟
⎜⎟
∂ ∂∂
⎝⎠
Otherwise the space variation of E , including the interaction transverse moment, ∇E = ∇ ⋅ E + ∇ × E, will also include a transversal (rotational) term
2
,( ) B
ee
ej i j i
pX
xt t
⎛⎞
∂∂
∇× = × =−
⎜⎟
⎜⎟
∂∂ ∂
⎝⎠
E {2nd Maxwell Equation} (5)
relation which is the direct derivation in START of this well known Maxwell equation. The scalar term ∇ ⋅ E being a divergence of a vector field should be defined to be proportional to a source density
2 2
0
()
1 ( ),
e e
ii i
wX wX
xxt t
ρ
ε
⎛⎞
∂∂
∇⋅ = = = ∇
⎜⎟
∂∂∂ ∂
⎝⎠
∑
E {1st Maxwell Equation}
and will be given full physical meaning below. For the space variation of B we have
∇B = ∇ ⋅ B + ∇ × B .
The first term vanishes identically in our theory because it corresponds to the divergence of the curl of a vector field ∇ ⋅ B = 0 , {3rd Maxwell Equation} while the last term, using U V W V (U W ) (U V )W
×× = ⋅ − ⋅
22 00
B ( ( )) p J ,
ee
wX t
μ ε∂
⎛⎞
∇× =∇ ∇ −∇ = +
⎜⎟
∂
⎝⎠
E {4th Maxwell Equation}
The additional dimensional constant μ0 is needed to transform from time units (used in the conceptual definition of a current 2
0
( ( )) /
e
J = ∇ ∇ a X μ ) into distance units. The units of 0 0
ε μ are of 2 2
T / D or inverse velocity squared, in fact (see below) 2
ε0μ0 c−
= corresponding to have used above twice the derivative with respect to t and not to x0 = ct . The (4th Maxwell Equation), defining J, is related to the analog of the (1st Maxwell Equation) and the analog of the (2nd Maxwell Equation), also to a Lorentz transformation of the (1st Maxwell Equation). The Maxwell equations can be formulated in 4-D form ( eμ
μ
,= ∂ )
0 00
1 1 e , J , , 0,1, 2,3, ) :
μ
= ∂ + ∇ = ∂ + ∂ = μ = ρ μ= =

,
t t ii
e e J J x ct
cc
00
11 1
F E cB , F J F , F J.
cc
ρ
εε
∂
⎛⎞
= + ∇= + − =
⎜ ⎟∂
⎝⎠

,
t


8 Jaime Keller
The here derived Maxwell equations are formally equivalent to the original Maxwell equations, then they are: first local equations and second linear in the sources (ρand J). Both the (4th Maxwell Equation), defining J, related to a Lorentz transformation of the (1st Maxwell Equation) defining ρ, can immediately be integrated using geometric analysis techniques, the standard approach being of fundamental conceptual consequences in START. The space divergence of a non-solenoidal vector field like E is immediately interpreted as its ‘source’ using the standard geometric theorem that the volume integral of a divergence ∇ ⋅ E equals the surface integral of the normal (to the surface) component of the vector field n ⋅ E . Consider:
()
22
00
2 0
4 1 (r n)
( E) ( ) 4 ,
rr
E , rn .
4
πρ π
εε
πε
⋅
′′ ′
∇⋅ = = = =
= = ⋅=
∫∫ ∫
VV S
dV r r dr Q E(r) dS r E r
r
Q
E(r) r
r rr
That is: the inverse square law of the Newtonian and Coulomb forces are geometrical consequences of the definition of interaction among charged carriers. Nevertheless this is not a derivation of the value(s) of the (Newtonian and) Coulomb constant(s) G and ε0 . For a small ( l ≤ r ) current source at the origin of coordinates: (in the sphere 2 2
r ( , ) r 0, (r ) (r ) 1
t ct t ct
θφ ⋅ = = = )
2
22 0 00 2
( B) (r ( , ) n) 4 r ,
4 J ( ) r 4 r B ( )r r ,
4
t ct
VS
ct ct t t
V
dV B(r) dS r fB(r)
M
r r dr M r fB(r) B r r f
θφ π
μ
πμ δ μ π π
∇× = × =
′ ′ ′= = ⇒ = =
∫∫
∫
and its Amperian inverse square law is also a geometrical consequence of the definition of transverse interaction among charged carriers.
1.7.2 Beyond Newtonian gravity
The analysis above depends only on the assumption of the decomposition of the action and of the energy momentum into contributions per carrier. The analysis above can applied to gravitation considering the mass M = E / c2 . The Newtonian gravitational potential equation per unit test mass m
() ,
M
Vr Gr
= − that is 2
E,
M
Gr
=−
the usual relations in the textbook formulation of Newtonian gravity. The constant ( )
0
1/ 4 g
G = πε . If we define 2 ( ) ( )
00 1
gg
c μ ε = then ( ) 2
0 4 /.
μ g = πG c
In this approach to gravitation there is no quantization properly, there being no exchange of action, only a description of the sharing of energy between a source carrier and a test carrier. We include the transverse momentum in the interaction between sources of the gravitational field:


A comprehensive theory of the electron from START 9
2 ( )( ) B
E,
g
jg
ij gi
p Xe e
xt t
∂∂
∇× = × =−
∂ ∂ ∂ (6)
()
2 () 2 () () () 0 0 22
EE
41
B () p J J
gg
g gg g ge e g g
G
wX t c c t
π
μ ε∂ ∂
⎛⎞
∇× =∇ ∇ −∇ = + +
⎜⎟
∂∂
⎝⎠
1.8 Formal definition of carrier fields
We follow our presentation in [4] (Keller and Weinberger). A carrier-domain B is a connected open set whose elements can be put into bijective correspondence with the points of a region (domain in some instances) B of an Euclidian point space E. B is referred to as a configuration of B; the point in B to which a given element of B corresponds is said to be “occupied” by that element. If X denotes a representative element of B and x the position relative to an origin 0 of the point x occupied by X in B, the preceding statement implies the existence of a function θ: B → B0 , ( B0 , stands for the totality of the positions relative to 0 of the points of B) and its inverse Θ: B0 → B such that x ( X ), X= (x)
=θ Θ (7)
In a motion of a carrier-domain the configuration changes with time x = φ(X,t), X = Φ( x ,t) (8) In a motion of B a typical element X occupies a succession of points which together form a curve in E. This curve is called the path of X and is given parametrically by equation (8). The rate of change v of x in relation to t is called the velocity of the element X, (our definitions run parallel to those of an extended body in continuum mechanics; see for example Spencer 1980 [8]). The velocity and the acceleration of X can be defined as the rates of change with time of position and velocity respectively as X traverses its path. “Kinematics” is this study of motion per se, regardless of the description in terms of physical forces causing it. In space-time a body is a bundle of paths. Equations (8) depict a motion of a carrier-domain as a sequence of correspondences between elements of B and points identified by their positions relative to a selected origin 0. At each X a scalar quantity is given, called carrier density ρ(X), such that if x = φ(X,t) then ρ(X) → ρ( x ,t) defines a scalar field called local carrier density. As already mentioned, a carrier will have physical significance through its set of properties. We used charges as example. The set of properties {Q} characterizes a carrier field and in turn establishes the conditions for a density field to correspond to an acceptable carrier.
1.9 Carriers in interaction
In B the carrier has existence only, whereas in B the carrier c has a distribution characterized by the density ( , )
ρc x t . There is no restriction in defining a reference space BR where the carrier exists in the points x with constant density ρ0
occupying a volume V0 such that (0)
ρ V0 = 1 . These two quantities are unob


10 Jaime Keller
servable as far as any “observation” requires an “interaction,” only then the distribution acquires meaningful space dependence as a function, by definition, of an external interaction V(x,t) , which will be defined below. Here it is important to state that as a result of this interaction, and of the properties attributed to the carrier, the density evolves into a current: (0) (x, )
V
ρ ⇒ Gjc t . The density is characterized by the properties of the carrier and the self-organization of the carrier, which adapts to the external interactions.
1.10 Composite, decomposable, elementary, average and average description of carriers
There are several ways to analyze the density. Each allows a physical interpretation. For example: • A composite carrier is defined as one for which the density
( , ) ( , ),
c
C Cc c
ρ t =∑A ρ t
x x (9)
with the definition of each of the ( , )
ρc x t being also meaningful as a description of a carrier. • Similarly a non-decomposable carrier is defined as one for which (9) applies but for which the meaning of each of the ( , )
ρc x t cannot be defined without reference to the global ( , )
ρC x t .
• An (non-decomposable) elementary carrier is one for which a single ( ,)
ρc x t is all it is needed; in this case we emphasize the discrete nature of an elementary carrier, but we do not assume a point-like or any internal structure for them. • An average carrier is defined as one for which its density can be described as ( 1,
c cn A
WA
=
=Σ )
1,
1
(x, ) (x, ),
c
A Ac cn
t At
W
ρρ
=
= ∑ (10)
with the definition of each of the ( , )
ρc x t being meaningful as a description of a carrier itself. • Similarly an average description of a carrier can be defined either as a space average over carrier descriptions as in (10) or as a time average
of a description, or sum of descriptions (
0
0
1,
1 ()
tt
c
c n tt
W A t dt
r
= +τ
==
= ∑ ∫ , the
choice W = 1 presents less manipulation difficulty)
0
0
1,
11
( ) () ( ) .
tt
c
c n tt
A t ,t dt
Wr
τ
ρρ
=+
==
=∑ ∫
x x (11)
This paper is centered on the definition of the elementary carriers and their correspondence with the fields describing the elementary particles, in particular the electron and its partner particle, the neutrino.


A comprehensive theory of the electron from START 11
1.11 The density
For a physically acceptable carrier density: D1. (x, )
ρc t is a real function (x, )
ρc t ⊂ R . D2. The density 0 (x, )
≤ ρc t < ∞ in order to represent a finite amount of charges and of action. D3. The derivatives of the density (x, )
ct
−∞ < ∂μρ < +∞ in order to represent a finite amount of energy-momentum.
Theorem 1 If Ψ(x,t) is an analytical quadratic integrable complex or
multivector function, conditions D1, D2 and D3 are fulfilled identically if
2
( ,) ( ,)
cc
ρ x t = Ψ x t . Here f 2 means the real quadratic form of any more general function f, even if f itself is not necessarily a real function and we define: if
f 2 f+f
= then f 2
∂μ = ( f ) f f ( f )
μμ
++
∂ +∂.
Condition D1 is fulfilled by the definition 2
( ,) ( ,)
cc
ρ x t = Ψ x t , D2 by the
requirement of quadratic integrability, D3 by the definition f 2
∂μ = ( ) ()
ff f f
μμ
++
∂ + ∂ and the analytical properties of Ψ(x,t) . It is seen that the conditions D1, D2, D3 and ( , )
cc
ρ t d =N
∫ x x correspond to the Ψ(x,t) being
quadratic integrable Hilbert functions.
1.12 Wave function quantum mechanics and density functional theory from START
We proceed now to establish the basic theoretical aspects of the study of carriers, which result in a stationary state Wave Function Quantum Mechanics and Density Functional Theory of the carriers. • The total energy of the system is a functional of the density, which can be defined in two steps. The first is to establish that there is a ground, least action, minimum energy, state of the system, which defines the carriers themselves:
() () () 00 0
(x) x (x) x ,
NN N
E = E d = ρ εd = Nε
∫ ∫ (12)
()
0 (x) x
ρN d = N
∫ (N = number of carriers), (13)
where the density of energy ( )
0 ()
E N x at a given space point x has been factorized as the product of the energy ε per carrier and the carrier density ( )
0
N
ρ . This by itself is the definition of elementary carriers of a given type: they are indistinguishable, equivalent, and the energy of the carrier is a constant in space, for all points of the distribution and, in a given system, the same for all elementary carriers of the given type. • The constant defining the energy per carrier is a real functional of the carrier density and of the auxiliary function Ψ(x) .
()
0 (x), (x)
N
ε ε⎡ρ ⎤
=Ψ
⎣ ⎦ . (14)
Because the reference energy has to be freely defined, this constant may be positive, negative, or null. The functional may, in some cases, become a local density functional (LDF).


12 Jaime Keller
The energy density, assuming indistinguishable (independent or interacting) carriers of a given type is now subject to the needs or desires of the observer describing the system. This defines independent carriers from interacting carriers, in that this energy appears as a property of the carrier in the system (a pseudo-carrier in condensed matter physics language), different from an isolated carrier.
Physics studies both the system in itself and, mainly, its response to external excitations. In the simplest approximation the necessary description is that of the possible stationary states of the system. • The study of different excitation energies of the system hν is now equivalent to the Heisenberg approach to studying a physical system through its excitation spectra, which was properly termed quantum mechanics due to the direct use of Planck’s constant h. • Density functional theory describes the self-organization of the carrier system with density ( )
ρ x in the presence of some external potential.
1.12.1 The density as the basic variable
It is convenient to define the action in a form that distinguishes the part corresponding to the self-organization of the distribution and the part that corresponds to the ‘external’ influences on the distribution. The volume (in space) of integration is considered large enough for the ‘kinetic’ energy to be internal; there should be no need to change the integration domain as a function of time. If the external influence is represented by the external potential V( X ) we can write for the total (invariant) action
[ ( )] xV( ) ( ) ,
I
A dt E ρ X d X ρ X
⎡⎤
=+
⎣⎦
∫∫
 (15)
where the functional [ ( )]
EI ρ X
 corresponds to the energy of the distribution of
carriers ρ(X ) . This functional EI has the interesting property that at a given time
V( )
()
EI X
X
δ
δρ = −

. (16)
This is a basic relation in Action-DFT as far as there is an intrinsic definition of the external potential. This shows the tautological nature of the concept of carriers, once they are defined, by [ ( )]
EI ρ X
 , the external potential is defined through the density of the carriers themselves. The tautological cycle is closed when given V(X ) and ρ( X ) the kinetic energy and the interaction terms define [ ( )]
EI ρ X
 . Reminder: in practice more general forms of V( X ) should also be acceptable. From the definitions above we can extend the description to consider a set {b} of types of carriers, each carrier type with density ρb . In this case for each b the ‘external potential’ depends in all types b b
′≠ .


A comprehensive theory of the electron from START 13
1.12.2 Introducing gauge freedom for the description of the action
The density ρ( X ) at space-time point X is required to be gauge invariant, whereas the description of the energy (action) is gauge dependent. This is achieved by constructing the energy density as the product of an average energy per carrier εwith the two conjugated quantities Ψ( X ) and Ψ† (X ) such that †
ρ( X ) = Ψ ( X )Ψ( X ) is gauge invariant. Here we define an auxiliary quantity: a gauge phase φ( X ) , similar to that proposed by Klein and by Fock as early as 1926 [10]:
0( ) ( )
() () () ,
ia X i X
geom
X AX X e P
φ
ρ −+
+↑
Ψ = (17)
where we are restricted (even if φ( X ) can be very general [3] and can represent electroweak, color and gravitational interactions), by definition, to ( 0 ( ) ( )) ,
aX X
t
φε
∂ −=
∂
= (18)
showing the gauge freedom of the description of the energy (action) associated with the carrier. We have then recovered the equivalent to the HohenbergKohn Theorems [11] and, with our definition of ( )
geom ρ X below, the HartreeFock or the Kohn-Sham minimization procedures [12] from
{}
()
δ E[ p] −ε ρ(x)dx − N = 0,
∫
 (19)
allowing the direct self-consistent determination of ( )
ρ x and ε(see [5]).
1.13 Least action amplitude functions in START
We can now follow the START definitions and the Schrödinger procedure to obtain the stationary action states of the elementary carriers system. 1. Let the Schrödinger (1926) definition of action W(x,t) in terms of an auxiliary function ( ,t)
Ψ x be
†
W(x,t) = K ln Ψ(x,t) = −K ln Ψ (x,t) , (20)
that is: action is considered a sum of terms. The action W(x,t) is required to correspond to the stationary states of the system to be described, if ensured through a variational optimization procedure. 2. Let the carrier density ρbe the real quantity defined above
†
ρ(x,t) = Ψ (x,t)Ψ(x,t), (21)
where , ρ(x,t) , Ψ and Ψ† are: unique-valued, continuous and twicedifferentiable and obey the additional condition ρ(x, t) space boundary = 0 . 3. Let the canonically conjugated variables be X = (x,t) and
†
,W = iK,ln Ψ = −iK,ln Ψ , with eμ
μ
,= ∂ the space-time gradient operator.
4. Let the local energy description be ( E0 is not a density)
†
2 2 2 2 2 22 00
†
K ( )( ) c E (Pc) (E ) (m c ) ,
Ψ Ψ =− = =
ΨΨ

, , (22)


14 Jaime Keller
(in the case where an interaction, through a gauge, is assumed to exist
2 2 22 0
(E − V ) − (Pc − eA) = (m c ) ) with the Euler-Lagrange (density of energy and constrain) function
2 † 2 22 † 0
J = K (,Ψ ) ⋅ (,Ψ)c − (m c ) Ψ Ψ, (23)
and perform the variational search for the extremum energy E (minimum of action for a stationary state system) δJ = 0 to obtain from the standard variational approach the condition ( 2 2
K == )
2 † 2 2 † 2† 0
K [Ψ (, Ψ) + (, Ψ )Ψ] = m c Ψ Ψ, (24)
and then the equation for the auxiliary function Ψ (the SchrödingerKlein-Gordon-like Equation (SKG)) is
2 222 2 2 22 0
2 2 2 2 c (m c ) 0.
t xyz
⎡⎤
⎛⎞
⎛⎞
∂ ∂∂∂
− + + − Ψ=
⎢⎥
⎜⎟
⎜⎟
⎜⎟
∂ ∂∂∂
⎢⎥
⎝⎠
⎝⎠
⎣⎦
= (25)
We must emphasize that in the relativistic (and in the non-relativistic) case we obtain, through the Schrödinger optimization procedure, the Ψ (or Ψ† ) function which minimizes the action of the system. A geometric factorization of the operator in the SKG equation transforms it into a Dirac-like equation. The gauge potentials are to be added.
1.13.1 General case of the auxiliary amplitude function Ψ
The auxiliary amplitude function Ψ describing the (set of) carrier(s) is constructed from sums of per carrier c, contributions ψc (sets of sums also). The space-time distribution ψc of a carrier and its intrinsic properties is given by the (four factors) functions:
()
(1) ( , ) ( , ) i t kx
geom
c xt Rxte P
ω
ψ ρ −−
+↑
= (26)
the first factor, the geometric square root geomρ , describes the per carrier local density, the second, the multivector transformation ( , )
R x t , the carrier local properties, the third i( t kx)
e− ω − the observer-relative carrier local motion and the last, the P+↑ , is a projector describing the reference sign of the mass and the reference direction of the spin.
1.13.2 First order equation as a factorization-projection
Consider (here again 2 2
0
k = (m c / =) , i2 = −1)
2
22 22
1 k 0,
c tψ ψ ψ
∂ −∇ + =
∂ (27)
and propose the factorization of the operator in the Dirac sense ( eμ
μ
,= ∂ )
( ik)( ik) ( ik)( ik) ,
μμ μμ
,+ ,− ψ = γ ∂ + γ ∂ − ψ (28)
defining the projected function
( ik) ,
μ μ
Ψ = γ ∂ − ψ ( ik) 0
μ μ
γ ∂ + Ψ = (29)
which obeys, by construction, the well known Dirac equation, showing that the auxiliary function Ψ, optimized to obtain the least action, is a geometric function (using a representation (eμ) μ
γ = γ of the geometry).


A comprehensive theory of the electron from START 15
For a massless carrier field k = 0 , this factorization is not unique, as
2†
( m ) (D mi)(D mi),
μ μ
∂ ∂ + = + − (30)
requires that
−D†m + mD = 0 and D†D μ 2 ,
μ
= ∂ ∂ =, (31)
therefore we can have a set of choices, either:
1. any value of m and D† = D (the standard Dirac operator D0 =, );
2. or when m = 0 the possibility D† ≠ D also becomes acceptable.
The basic requirement † †
D D DD μ
μ
= = ∂ ∂ limits the choices of D. Here they will be written in the Lorentz invariant form. The ( f )
Γμ are a generalization (irreducible or reducible representation) of the Dirac γμ matrices. The limitation is so strong that the only possible choice, within the algebra, is when the chirality generator iγ5 , which has the same action on all γμ , that is 5 5
ii
μμ
γ γ = −γ γ , is used (see Keller [3]). We define the differential
() 5
1cos( ) sin( ) ,
22
dd d
nt i nt
μ μ μμ
ππ
γ
⎧⎫
∂= + + + ∂
⎨⎬
⎩ ⎭ (32)
with n and tdμ integers, a choice which results in the simplest multi-vector. Here, to take the electron as a reference, we use n = 1. Then, in a particular frame we have the ‘diagonal’ structure:
()
5
if are even,
if are odd.
d d d
nt
i nt
μμ μ
μμ
γ
⎧∂ +
⎪
∂ =⎨ ∂ +
⎪⎩
(33)
The vectors, which can be represented by the standard γμ matrices, correspond to an irreducible representation of the Clifford algebra 1,3
C useful for writing the wave equations of the fundamental family of leptons and quarks ( , , ,{ ,
RL L L l
e eν u d
− − ; color}} of elementary particles. The electron requires a combination of two massless fields ( , )
RL
e ee
− −−
= for the standard phenomenology of electroweak-color interactions. The case of the neutrino presented here is the simplest of these structures.
1.14 Interaction fields and charges
Consider the particular case of an initial situation without electromagnetic phenomena being present E = 0, J = 0, B = 0 and 0
ρ= , and that in the process of creating a pair of interacting carriers with electric charges Q , an initial pulse of current 0
J(r,t) = Qv(r)δ(t ) is assumed to have been generated. This induces an electric field for 0
t > t from the Maxwell Equations derived above:
0 00 0 0
E J(r, ) / / v(r) ( ) / ,
∂t = − t ε + ∇ × B ε μ = Q δ t ε (34)
00
E(r,t;t > t ) = −Qv(r) /ε then ρ(r,t) = −Q∇ ⋅ v(r), (35)
and
0 00
B(r, ; ) E(r, ; ) v(r) / ,
∂t t t > t = −∇ × t t > t = Q∇ × ε (36)
showing that this virtual mechanism (in our process to establish a partitioning of energy and momentum among charged carriers) requires the actual alloca


16 Jaime Keller
tion of physical properties, to the collection of created carriers, since the divergence of the current pulse creates a charge and the rotational of the current pulse of velocity field v(r) creates a magnetic dipole. We can see that the definitions are a circular procedure: sources create fields or fields generate the concept of sources. In START the charge Q corresponds to the rotations (a new type of “spin”) in the planes generated by the (space axis)-(action axis) basis vectors. Notice that a charged source with a circular current generates an electric and a magnetic field, the case of the electron, and the carrier also shows the presence of the spin associated to the solenoidal current. Within the postulates above the action of circular currents will have to be quantized in terms of Planck’s = , a pair of currents in terms of = / 2 each. The emitted, excess energy-momentumangular momentum (electromagnetic wave), is itself (from the Maxwell Equations) an action carrier traveling at the speed 2
00
c (ε μ )
− = . We see that the concept of electromagnetic (light) wave is basic to the study of action density and its distribution in space-time, and the quantization condition is also fundamental in this case.
2. The experimental electron
The matter fields enter into the theory as charge-current distribution densities jμ → (ρ, J). The currents J (for example those generated by the electron field) can, in general, be decomposed into their solenoidal jsol and irrotational j parts. In Dirac’s theory the solenoidal parts analyzed via the Gordon decomposition contain two components: one which is intrinsically solenoidal; and a second which is solenoidal only with reference to the boundary conditions and the observer’s frame of reference. Then the electron sources of the electromagnetic fields, in units of the electron charge e , are described in fact by a set of seven basic quantities:
,,
,, , , ,
i j k i sol j sol
ρ j j j j j and k,sol
j (37)
We have already reminded the reader that an electron cannot exist without its electromagnetic fields, that is, it exists with an electrostatic field generated by the electron’s charge, an intrinsic magnetic field generated by its intrinsic solenoidal current and an additional electromagnetic field generated by the, extrinsic, electric current. A satisfactory theory considers physical entities as constituted by whatever is observable. The intrinsic solenoidal current of the electron implicates not only a magnetic moment but also an angular momentum 1
() ,
2
S= S x =
∫ = (38)
then in (39) above jsol could also be replaced by an angular moment field S ( x)
G
. Dirac’s theory shows that the magnitude of S (x)
G
is S(x) S (x),
= ρ (39)


A comprehensive theory of the electron from START 17
then only the direction of S (x)
G
is independent of ρ(x) but not its magnitude. This is one of the most important features of the geometrical content of the electron theory. It says that, even if the analysis of an electron distribution shows some solenoidal current, there is a curl of the distribution at every point and, as is well known in vector analysis, the overall intrinsic solenoidal current is the result of the application of Gauss’ theorem to the ensemble: every point of the distribution contains the same amount of intrinsic angular momentum per unit density.
There is no indication whatsoever of a structure giving rise to spin, and in fact a spin field 1
12 12
S ψγ ψ* ρψγ ψ−
= = is one of the most fundamental quantities of the standard theory. In all experiments performed up to date an electron appears as a distribution of charge, currents and electromagnetic (electroweak, in fact) fields. Problems arise from the attempt to rationalize the experimental facts starting from a point particle idea as the basis for the interpretation of experiment or for the interpretation of the results of the now standard quantum mechanical calculations. Experiment shows that there is no internal structure of the electron, but the experiment does not disagree with the existence of distribution. The ‘interpretation’ of the distribution is a not a question of quantum mechanics, nor of the electron theory. That is, there is no experiment resolving the electron ‘cloud’ into instantaneous positions of a ‘point’ particle, nor at the same time is there any evidence at all of a possible excitation of internal structures of an electron. We could speak in terms of electromagnetic quantities alone. The densities, which we commonly refer to the sources, can be substituted by electromagnetic quantities through the integral form of the Maxwell equations. For example, to relate E and ∇ ⋅ E
0 1 12 21
2
0 12 12
E( )
1
E( ) ,
4
rr
r dV
rr
ε πε
∇⋅
=∫
GG
G (40)
or to relate ∇ × H and ∇ ⋅ E for time independent E, ∇ × H = ( E)v,
∇ ⋅ (41)
and we can even think of the electromagnetic potentials Aμ as quantities related to the sources in special forms
20
0
D.
A
A tε
∂ ∇⋅
∇ +∇⋅ =
∂
G
(42)
We can then assume that besides the field intensities E and H we have a vector distribution (reminder 0
J = ε (ρ+ J)
)
∇ ⋅ E → ρ, (43)
( E)v J,
∇ ⋅ → (44)
the energy-momentum related to this vector being
00
E = γm ε ∇ ⋅ E/e, (45)


18 Jaime Keller
00
P γm ε ( E)v′/e.
= ∇ ⋅ (46)
Here m0 appears as a parameter providing the correct dimensions and v′ corresponds to the relative velocity between the inertial system where ∇ ⋅ E has been computed and that of the observer. Remember that relativistically E and H cannot be separated, nor do they have a unique formulation; in fact, they can always be expressed as Lorentz transformations and duality rotation of a reference bi-vector 12
H =ψγ ψ* .
2.1 Spin, magnetic moment and mass
We now give a meaning to the proposed amplitude function ψc in the free particle approximation. The rest mass parameter m0 of the carrier will be directly related to the amplitude term of the non-dispersive wave packet. This is a consequence of the fact that a non-dispersive wave packet, ψ, is a solution of the equation
0,
,ψ = where
2 22 22
1.
ct
∂
=∇ = ∂
, (47)
Then a non-dispersive wave for a carrier of mass m travelling in the +x direction with velocity v relative to the observer takes the form [13, 14, 15])
00
(sin / ) exp[ ( ) ,
ψ geom ρ k r k r i ωt kx P+↑
= − (48)
where
1 22
22 0 0 22
2 22 0
()
m /, ,
1( / )
m/, m/, ( /) ,
x vt
k c r yx
vc
c k v k wc k
ω
⎧⎫
−
= = ++
⎨⎬
−
⎩⎭
= = =−
=
==
(49)
with (x,t)
ρ = constant representing the time average of a steady state. That (48) is a solution of (47) follows by simple substitution. It is also one form of the standard spherically symmetrical solution of (47) after it has been subjected to a Lorentz transformation. Then a solution of (47) takes the form (we leave out the reference projector P+↑ in this part of the discussion; note R > 0 )
exp[ ]
ψ = geom ρR iS . (50)
Inserting this ψ into (47) and then separating real and imaginary parts, the following two equations are obtained:
2 2 2
1
2 ( ) 0,
S
RR S c t
⎧⎫
∂
⎪⎪
⎛⎞
− ∇− =
⎨⎬
⎜⎟
⎝∂ ⎠
⎪⎪
⎩⎭
, (51)
2
1
2 2 0.
RS
R S RS c t t
⎧ ∂ ∂⎫
⎛ ⎞⎛ ⎞
+ ∇∇ − =
⎨⎬
⎜ ⎟⎜ ⎟
∂∂
⎝ ⎠⎝ ⎠
⎩⎭
, (52)
If one takes the exp[iS] to be the de Broglie wave, so that S = ωt − kx , (52) now leads directly to the result that


A comprehensive theory of the electron from START 19
22 0 2
2 .
mc
R
R ==
, (53)
It is well known that equation (48) represents the superposition of two spherically symmetrical waves, one converging and one diverging, both having phase velocity c [15], and if the waves are electromagnetic waves, this combination constitutes a phase-locked cavity similar to that proposed by Jennison [16], who has also shown that such cavities have many of the inertial properties of particles. The structure of the field can be associated with the electron. We now compute the spin of the distribution (48). The momentum P of the field is
††
[ ( ) ] h.c.,
4i
μ μ
P = = ψ ∇ψ +ψ αα ∇ ψ + (54)
†† †
[ ( ) ],
P = 2=i ψ ∇ψ − ∇ψ ψ + =4 ∇ ×ψ σψ (55)
and, for k = 0 corresponding to a particle at rest,
22
2 22
sin ˆ ˆ
(x, ) ( 2 2 ),
42
k kr
t yx xy
r kr
ρπ
⎛ ⎞⎛ ⎞
∂
= −+
⎜ ⎟⎜ ⎟
∂
⎝ ⎠⎝ ⎠
P = (56)
which represents a circular flow of the field in the plane ˆ ˆ
x ∧ y . The angular momentum is given by
† †3 †3
J [ ( )] ,
22
x dx dx
= i × ψ ∇ψ − ∇ψ ψ + ψ σψ
∫∫
= = (57)
and again the second term, spin, will be the relevant quantity. If we assume R to be normalized, then the integral of the spin part would be trivially of magnitude = / 2 . As for a de Broglie wave packet k0 = m0c / = , then the same prefactor R(r) that generates the mass generates the spin of the total wave. This appears to be the real origin of the structural parts discussed above. Notice that
2
4πr R(r) = 0 as r → 0. The prefactor R(r) provides, additionally, a connection with the standard model of elementary particles given that
sin
() ,
ikr ikr
kr e e
R r kr ikr ikr
−
= = − (58)
and it corresponds to a standing spherical wave: eikr / kr is an outgoing spherical wave and ikr /
e kr
− an incoming spherical wave. Given a spin direction they will have opposite helicities, and the standing spherical wave will be the realization of the well-known sum of a left-handed and a right handed wave.
2.1.1 Conserved electromagnetic quantities
The integrated quantities
33
E = E(x,t)(dx) , P = P(x,t)(dx) ,
∫ ∫ and 3
M = (P × x)(dx)
∫ (59)
electromagnetic energy-momentum and angular momentum for a steady cur
rent J are time independent: 0 0 0
dE / dx = dP / dx = dM / dx = 0.


20 Jaime Keller
2.2 The basic structural relationship between charge, magnetic moment, spin and mass
A crucial argument of the present paper is that once we have defined a field of sources for an electromagnetic field, which contains a static electric and a static magnetic part, and we have shown that this field carries a spin 1/ 2= , the field configurations correspond to a charged “particle” with spin. The charge of this particle is e and its spin is / 2
z
M = = . In this case we have (see Appendix)
22
ge = = or 2 e ,
g ee α
==
= (60)
where α = e2 / =c = 7.29735308 3
10 1/137
−
× ≈ is the fine structure constant (above in units where c = 1). The relation e/g is a fundamental dimensionless structural constant of the formulation of the theory. We now argue that the usual relationship between a magnetic moment μand a spin s is
m
μ= e s (61)
and, as for the carrier field, we have defined that the magnetic moment is μ0 the carrier should be attributed a mass m0 . This being a structural relationship, which should be obeyed at each point of the distribution with carrier density ρ. The space integral of ρ for one carrier is 1, and then the relation (61) is obeyed as a relation between physical constants at each and every point of the carrier distribution, its space integral being 0 0
μ , e, m and s. The radius r0 is implicitly contained in the definitions.
2.3 Action and energy involved in the interaction
The logical cycle of the interaction structure would close when the energy and action related to these logical and mathematical structures are determined. From the definition of the divergence of the interaction fields as the sources
0
1,
FJ
β
αβ α
ε
∂ = − (62)
which allows the calculations of the energy given off by the source itself, the energy of the interaction field. For this purpose, consider a variation of the four-vector potential Aμ and the scalar product of these Aμ
δ with the source carrier current to obtain, from (62) after integration in a volume Ω of four-dimensional space
0
F 1 J A d 0,
βα
αβ α δ
ε
⎛⎞
∂ + Ω=
⎜⎟
⎝⎠
∫ (63)
this quantity refers to the action related to the source and also to the field. Notice that where Jα ≠ 0 the integrand vanishes by definition. An integration by
parts, using a boundary condition Aμ
δ (boundary) = 0 and the antisymmetry of the F F
μν νμ
= − gives
( ) [ ( ) ( )] ( ) ..
F Ad F A F A d F A d
β α β α βα βα αβ αβ αβ αβ
∂ δ Ω= ∂ δ − ∂ δ Ω= − ∂ δ Ω
∫ ∫ ∫ (64)
because Aα 0
δ = on the boundary.


A comprehensive theory of the electron from START 21
() () ()
( ) () ( )
1
2
1 1 1,
2 24
F Ad F A F A d
F A A d F F d FF d
βα βα αβ αβ αβ βα
β α α β αβ αβ αβ αβ αβ
δ δδ
δ δδ
⎡⎤
− ∂ Ω= − ∂ − ∂ Ω
⎣⎦
⎡⎤
= − ∂ −∂ Ω= Ω= Ω
⎣⎦
∫∫
∫ ∫∫
(65)
to obtain finally
0
1 1 0.
4F F J A d
αβ α αβ α
δε
⎛⎞
+ Ω=
⎜⎟
⎝⎠
∫ (66)
The field energy density is
0
11
() .
4
EX F F J A
αβ α αβ α
ε
=+
 (67)
We have followed the Huang and Lin [9] analysis in his equivalent work to obtain the Lagrangian of the electromagnetic field. We should remember that in our approach carriers do not interact with themselves, and the quantity in (67) should be taken to be zero if J (x,t) 0
α ≠.
3. The many-electron problem
The N electron problem (fermions) should solve the set of equations † in
N one electron the N electrons system
ρ= ρ = Ψ Ψ .
The statistics are the Fermi-Dirac statistics and require: • The density for the N equivalent fermion carriers system is to be con
structed as a sum of M independent alternative contributions
() ( )( )
i 1,M N i i
ρ x,t = Σ = ≥ ρ x,t, s .
• There should be at least one linearly independent function (pseudocarrier amplitude) contributing to the density for each of the N equivalent carriers in the system. A linear transformation would then give one different spin-orbital (SO or “state”) per electron, the usual argument.
()
1
22
1
1
()
1
MN ii c ci i c c i i
M
ii i i i i cc c c c c
i
t a ts a b N
aa a a a a N
ψφ α
′′
≥
=
=
, = ,, =
=− ≤ ; =
∑
∑
xx
The total amplitude function should be a sum of single (pseudo-) carrier amplitude functions ( )
ψc x,t, s , such that the exchange among two carriers of the space–time–spin (x,t, s) descriptions.
†
( ) and ( )
cc cc cc
ωψ t s π ′ψ t s
′ ′
Ψ= , , Ψ= , ,
∑∑
xx
This defines Ψ as a vector, linear form, expressed in the basis {ωc} . The Ψ are defined and the products ordered to obtain
∫ Ψ 2dx = ∫ ΨΨdx = N


22 Jaime Keller
( ) ( ) c c c c c c cc
ii
i i ii
φ t s φ t s d δ ωω ω ω alsoπ ω δ
′ ′ ′′ ′′ ′
∫ x, , x, , x = ; = − ; =
The 1st condition is double: first the orthonormality among the φi functions (requiring them to be eigenfunctions of the same differential equation operator) to fulfill the condition of making linear independent combinations ψc and sec
ond the Grassmann character of the i
ac coefficients, to make the local density per carrier corresponds to the sum of the squares
( )2
i
ci i
a φ x,t, s ,
and the 2nd condition, equivalent to the Pauli principle, defines the ωc as Grassmann variables and the c
π as their Grassmann conjugates. The (stationary state) Hamiltonian for a many-electron atom (or molecule or solid) may be written in the form
2
ˆ ˆ () 11
ˆˆ
ˆˆ
() () , 1
ω π ω ω ππ
ωπ
⎡⎤
⎛⎞
⎢⎥
⎜⎟
⎢⎥
⎜⎟
⎢⎥
⎜⎟
⎢⎥
⎜⎟
⎢⎥
⎜⎟
⎢⎥
⎝⎠
⎢⎥
⎣⎦
∗
′ ′′
= +∫ / ,
∑ ′∑
= =+
⎡⎤
= + =Ψ Ψ
∑⎣ ⎦
=∫
NN
core
c c c c cc c c
H x dx e r
H j ij
c cc
N core interaction
cc
H x x E H dx
HH
c
where the core Hamiltonian for the electron c, consists of the kinetic-energy and nuclear attraction terms for electron c with coordinates i. Hamiltonian () ()
core interaction
xx
H + H is a one-electron operator, even if 1 cc
rij ′
/ , being dependent on the inter-electron distance, is a two-electron operator; ωc and c
π act here as projection operators. The energy of the N -electron system is given by E and we should determine both Ψ and E There is a total density function ( )
ρN x which should be integrable in a final volume, and everywhere in that volume should be a finite and non negative function, corresponding to a many electron function ({ } 1 )
Ψ N xn ; n = ,..., N
where 2
()
NN
ρ x =Ψ 1 1
() ()
N MN ca
ca
ρx ρx
≥
==
==
∑ ∑ . This Hermitian square can
be described as both a sum of ( )
ρc x = 2
ψc or as a sum of SO contributions
2
()
a aa
ρ x = bφ .
Third, in the case of the many electron (fermion) system we are studying all N electrons (fermions) are equivalent. This requires that the density itself
is a sum 2
() ()
N N electron
ρ x = Ψ = N ρ x , and each ( )
ρelectron x should be generated by equivalent contributions. That is ( ) 1
ρelectron x = /N ( )
a
aρ x
∑.
As density appears as a sum of densities, then the wave function should both be the square root of the total density and also provide the square root of each one of the contributions to the total density. For this we require the use of geometric (multivector analysis) techniques. In fact the problem is similar to that of finding the linear form (geometric square root) d = ae1 + be2 + ce3 + ...
JG
which corresponds to the quadratic form d 2 = a2 + b2 + c2 + ... .
3.1 Configuration space and real space
A basic concept in the study of a many-electron system (N interacting fermions) is, from the considerations above, the simultaneous, repeated, use of


A comprehensive theory of the electron from START 23
real space (the space of the observer) for each one of the fermions of the system: configuration space. Then, if x represents a point in real space, it is customary to represent by { }
X = xa ;a = 1,..., N the set of points in the configuration space X for N fermions.
Here and in the rest of our presentation we use a geometric notation
{}
a ; a 1, , N; n m m n
= ω = ωω = −ω ω
∑
X x ... and the projection operators a
π such that a b ab
π ω = δ selecting the part of the configuration space which corresponds to electron a : a a
π X = x . This allows a clear formal definition of the electrons involved in each part of the calculation. Our geometric procedure introduces the statistics of the fermion system from the beginning because the interchange of two electrons in a given expression will change the sign of the corresponding terms.
3.2 The energy calculation
In correspondence with our formal definition of configuration space the total electronic energy operator or Hamiltonian is
22 2 2
1
ˆ
22
KKW n m m n
n mn e n nm
Ze e
H mx x
ω ω ππ
≠
⎞
⎛∇
= − −+ ⎟
⎜⎜ ⎟
⎝⎠
∑∑
= , (H) (68)
and the wave function KKW
Ψ N is , with ij
and
i j ji i j
αα = −αα ααδ
1
N
KKW Nn
n ψω
=
Ψ = ∑ where ( )
1 (WF)
MN i ii i
ψ i bαφ x
≥
=
= ∑ (69)
Here the electron, or pair of electrons, under consideration is explicitly selected. Note that a double set of Grassmann numbers { }
n; i
ω α has been introduced; this has an analytical analogue in the HF method, where in a determinant the exchange of columns or of rows changes the sign of the determinant. The exchange terms arise from the definition of the wave function (WF) when used in (H). In (H) the core Hamiltonian for the electron n with coordinates i consists of the kinetic-energy term and the nuclear attraction local potential. In the calculation of ψthe effective Hamiltonian is ( ) ( )
core interaction
H x + H x where the second term is a one-electron operator, even if the electron repulsion, being dependent on the inter-electron distance, is a two-electron (i for n, j for m) operator. The resulting exchange and correlation potential is the same for all components of ψ. Orthonormality and equivalence are used
1
†
1
1 ˆ ()
1ˆ
( ) () ( )
N
cc
core core c c
N
cc
core
dd dd c
d
d
cd
d
E cd
H
N
cd
H N
ωπ
π ψ ω π ωψ
′
′
′ ′
=
=
⎧⎫
⎪⎪
⎨⎬
⎪⎪
⎩⎭
⎧⎫
⎪⎪
⎨⎬
⎪⎪
⎩⎭
=Ψ Ψ
=
∫∑
∫∑
∑∑
x
x xx
2
1
1
†
1 1 11 1 1 11 1 11
1 11 1
()
ˆ
( ) (1) ( ) ( ) ( ) ()
( ) ( ) (formal definition)
i i
core
core core i i
core
a
N
Ed d
H
N
d
ρ
ψψ ε ρ ρ
ερ
==
=
⎡⎤
⎢⎥
⎢⎥
⎣⎦
∫ ∫∑
∫
x
x xx x xx x
x xx


24 Jaime Keller
For the electron-electron interaction (e-e), because the equivalence of the N electrons and using the expansion of the ψ, we obtain
2
12 12
(1) (2) (2) (1)
2
int i i j j k k l l ij k l
Ne
E dd
x
φ α φ α φ α φα
∗∗ ∗ ∗
= ∑ ∑ ∑ ∑ x x.
Considering the property i j ij
αα δ
∗ = , there are 3 types of e-e terms: I) j = k and i = l which gives
2
22
1 2 1 1 1 11 1 12
1 (2) (1) ( ) ( )
2
ji
I
ji i ji
e
a da d V d
x
ρ
ρρ
≠
⎧⎫
⎡⎤
⎪⎪
⎢⎥
⎪⎪
⎢⎥
⎨⎬
⎢⎥
⎪⎪
⎢⎥
⎪⎪
⎣⎦
⎩⎭
∫ ∫ =∫
∑∑ x x x x x
II) j l i
= ≠ and i = k (one interchange i j j i
a a = −a a is needed!)
2 22
11 2 11 1
12
1 11 1
1
(2) (2) (1) (1) ( ) ( ) 2
( ) ( ) (the arises from spins orthonormal)
j
i
j
i
sj i s ji ij ij i
s II s
e
dd
aa
x
V d ij
δ φ φ φφ ρ ρ
ρδ
∗∗
,≠
⎧⎫
⎪⎪
⎨⎬
⎪⎪
⎩⎭
−/
= ≠,
⎡⎤
⎢⎥
⎣⎦
∫ ∫∑
∫
x x xx
x xx
III) Null terms, where ( i ≠ l and i ≠ k ) or ( j ≠ l and j ≠ k ). The total electron-electron interaction energy:
[] ( )
1 1 11 1 1 11 1
, () () () ()() 2
inter inter I II
N
E V V dN d
ρ Ψ = ∫ x + x ρ x x = ∫ε x ρ x x
Then we have two different contributions which will also contribute to the formal interpretation of the Pauli Exclusion Principle: a given electron is not interacting with itself and there is an “exchange” term for fermions, where
from i j j i
a a = −a a a negative sign appears. Those terms are related, and similar in structure, to the integrals related to “exchange-correlation” in the HF+CI sense. Finally the total energy of N equivalent electrons is
[] { }
1 1 11 1
1 1 11 1
() () ()
( ) ( ) constant , ( )
core inter
core inter
EN d
E N dN
ρε ερ
ερ ε ε ρ ερ ρ ε
=+
=+ = = =
⎡⎤ ⎡⎤ ⎡⎤
⎣⎦ ⎣⎦ ⎣⎦
∫
∫
x x xx
x x xx
In principle it should be written [ ]
E = E ρ, Ψ . The variational procedure is to be carried with respect to the ψ’s. The basic set of equations for our KKW method, presented in comparison with HF and HF+CI, is as follows:


A comprehensive theory of the electron from START 25
KKW HF (HF)+CI
HlΨ = NεΨ l
00
SC SC HF
HΨ = ε Ψ l SC SC
CI CI
H Ψ = NεΨ
2
1
1
N n
n
nm mn MN ii
ia
ωψ ωω ωω
ψφ
⎧⎪ =
⎪⎪⎪⎨⎪⎪ ≥
⎪⎪ =
⎩
Ψ=
=−
=
∑
∑
1 0
det
SC
i
N
ii
φ
φφ
⎡⎢⎣ ⎤⎥⎦
!
⎡⎢⎣ ⎤⎥⎦
Ψ=
≡
{}
0
SC CI B B B
B
Nc
Bi i
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎝⎠
⎧⎨⎩ ′ ⎫⎬⎭
Ψ = Ψ+ Ψ
=→
∑
K n KW
H ψ = εψ n
()
HF SC SC i i ii
H φ = εφ l 0
SC SC B D BD
H εδ
⎡⎤
⎢⎥
⎢⎥
⎣⎦
Ψ Ψ− =
4
n
()
2
1
22
() ()
MN i
i
i ii
i
KKW i ii
i
ii
i ii
i j ij
i j ji
xx
b
dx bi
H
ab
ρρ
ρφ
ρ
φ εφ ε ρρ ε
α
αα δ
αα αα
≥
⎧⎪ =
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
=
=
=
=
/=
=
=
=−
∑
∫
∑
2
HF HF i i
SC
HF ii
ρρ
ρφ
=
=
∑
2
2
2
2
11
HF B B B
SC SC B D BD
SC BB
B
B
Dc
Nc
DN
ρρ ρ
δ
ρ
⎛⎞
⎜⎟
⎜⎟
⎜⎟
⎝⎠
=+
ΨΨ =
=Ψ
=/ +
=
∑
∑
Row 1 presents the basic equation, row 2 the structure of the wave function, row 3 the resulting equation after the variation, energy minimization procedure, row 4 the definitions for the total density in each method and some auxiliary conditions.
Appendix: Definitions and Notation
We use the term space to denote the 3-D space of our perception of the distribution of physical objects in Nature and for its mathematical representation as an R3 manifold with a quadratic form. Its points are denoted by the letter x and represented as a vectorial quantity x ie
i
= x . We use the traditional indices i = 1, 2,3. We use the term time to denote the 1-D space of our perception of the evolution of physical phenomena in Nature and for its mathematical representation as an R1 manifold with a quadratic form. The normal-face letter t denotes its points. We use the term space-time to denote the 4-D Minkowski space of our perception of the physical world in the sense of relativity theory, and for its mathematical representation as an R4 manifold with a quadratic form: ds2 g dx dx ,
μν μν
= ( , 0,1, 2,3).
μν = Its points are denoted by the Normal-face letter X and represented as a vectorial quantity X X μe
μ
= . We use the traditional indices μ= 0,1, 2,3 . The vectors eμ in the geometry of space-time generate the ST
G 16 dimensional space-time geometry of multivectors. The basis vectors 0 1 2 3
{e , e , e , e } , with 2 2 2 2
e0 = −e1 = −e2 = −e3 = 1 and the definition property ee ee
μν ν μ
= − generate a Clifford group 1,3
Cl . We also use the notation 0 0 e ( 1, 2,3)
j jj
e = e e = j = and e5 = e0e1e2e3 = e0123. A special property of the


26 Jaime Keller
pseudo-scalar (and also hypervolume and inverse hypervolume) in space-time e5 is that 5 5
ee e e
μμ
= − (from e e e e ,
μ ν = − ν μ μ≠ν) and then it has the same commuting properties with the generating vectors of ST
G as generating vectors have among themselves. A vector e4 can be used to introduce an additional basis vector, giving one more dimension (action). We thereby obtain the five dimensional carrier space spanned by the basic vectors , 0,1, 2,3, 4
eu u = (identified as ,
eu e u
⇒ μ = μ and e4 ) with metric diag( 1, 1, 1, 1, 1)
guv = + − − − − . This is used to construct a geometrical framework for the description of physical processes: a unified space-time-action geometry STA
G , mathematically a vector space with a quadratic form. An auxiliary element j anti-commutes with all e : e j je
μμ μ
= − and j2 = +1. Multivector Representation. The base space R5 corresponds to the real variables set { 0 } { 0 1 2 3 4}
u
ct, x, y, z,κα ↔ x ;u = , , , , , that is, time, 3-D space and action (in units of distance introducing the universal speed of light in vacuum c and the system under observation dependent, using the Compton wavelength λ for a system with energy 2
mc : κ0 = λ/h = 1/mc ). Time is usually an independent evolution coordinate. Action is distributed in space, then we consider the functions ( ) ( ) ( )
x t , y t , z t and w(t, x, y, z) =κ0α(t, x, y, z) . The nested vectors
0
01234 5
0123 4
123 3
u u
i
i ii
i
dS dx e u D
ds dx e D
d dx i e e D
μ
μ μ
μμ
= ; = ,, , , −
= ; = ,, , −
= ;=,, ; = −
∑ ∑ ∑
xe e
are members of a Clifford algebra generated by the definition of a quadratic form
()
()
2
2
2
† 012 3 4
11111
,
u START u v u uv
START uv u v v u
uu
dS dx e g dx dx
dS
g diag e e e e
e e e e e e e e e ee
μ μν
⎛⎞
⎜⎟
⎜⎟
⎝⎠
≡= = ,
= ,− ,− ,− ,− , = − = =− =
∑∑
This 5-D geometry has two types of rotations: space rotations associated with angular momentum (in particular spin 12 = and intrinsic magnetic moment) and “rotations” in the action-space planes, with degeneracy 3 and intrinsic value / 2mc
℘= , associated with the electric charge of the field.
Observable objects are extended in space described by an action density α in space-time. Then a) defining 2
m(x,t)c = ( )
εtotal x,t , b) the inverse of the space-time volume 0 1 2 3
e e e e /+x+y+z+t , c) the space-time d’Alembertian operator μ e μ
μ
, = Σ ∂ (for a given observer with time vector e0 the operator , has
the property ( ) ( )
01 1
t t ii
e , = c ∂ + ∇ = c ∂ + e ∂ ), d) along b b e
μμ
μ
= Σ the directional change operator is db μ db μ
μ
= Σ ∂ (apply for 0 1 2 3
b = cte , xe , ye , ze to obtain the sum of directed changes of w) to obtain:
( 0)
40 40 0
2 2 ()
() ()
1
( ) ( ) m t mct
m tc t m tc t
te te e e e
mc
x yzt x yzt xyzt
κα κ , /
,,
, = , = = =x
xx
ax x +
++
++++ ++++ ++++


A comprehensive theory of the electron from START 27
()
()
0 4
() ()
() ()
()
m t m ct w t
te e e w te
xyzt xyzt
edw w t dx e e
μ
μμ
μ
,/ ,
,= = =,
⎡⎤
= ∂,
⎣⎦
∑
xx
ax x
x
+
++++ ++++
( ) ( )( )
() () ()
2†
2 222
2 22
2
1 () 1 1 1
00 01 0 2 03
dS dS
dS
κ p cdt κ p dx κ p dy κ p dz
⎛⎞
⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞
⎜⎟
⎛ ⎞ ⎛⎞ ⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟
⎜⎟
⎜ ⎟ ⎜⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟
⎜⎟
⎝ ⎠ ⎝⎠ ⎝ ⎠ ⎝ ⎠
⎜ ⎟⎜ ⎟⎜ ⎟
⎜⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠
⎝⎠
=
= − − − +− +−
here ( )
pt
μ = ∂μα x, is a momentum density. Notice that w(x,t) is the distance equivalent to a reduced action density, this makes the approach universal for all systems. We use the term action a to denote the 1-D space of our perception of the objects of physical phenomena in Nature and for its mathematical representation as an R1 manifold with a quadratic form 2
da .
e) We use the term space-time-action to denote the 5-D space of our perception of physical phenomena in Nature and for its mathematical representation as an R5 manifold with a quadratic form
2 2 22 22
0 0,
( , 0,1, 2,3, 4), ( , 0,1, 2,3).
AB AB
dS ds da g dx dx da g dx dx
AB v
μν μν
κκ
μ
=− = − =
= = (70)
Its points are represented by the set 0
( X ,κ a), 0 0
κ =1/ m c . f) We use the term description to denote the partitioning of the total action (or energy-momentum) into carriers c. We use the term theoretical structure for a set of defining mathematical considerations.
Acknowledgement
J.K. is a member of the SNI (CoNaCyT, México). The author is grateful to Mrs. Irma Vigil de Aragón for technical support.
References
[1] Keller J. Advances in Applied Clifford Algebra, 9(2), 309-395 (1999). [2] Keller J. Rev. Soc. Quim. Mex., 44(1), 22-28 (2000).
[3] Keller J. The Theory of the Electron; A Theory of Matter from START, Foundations of Physics Series 117. Dordrecht: Kluwer Academic Publishers (2001). [4] Keller J. Unification of Electrodynamics and Gravity from START, Annales de la Fondation Louis de Broglie, 27(S), 359-410 (2002); Keller J. Advances in Applied Clifford Algebras, 11(S2), 183204 (2001). Keller J. A Theory of the Neutrino from START, Electromagnetic Phenomena, 3,1(9) 122-139 (2003). Keller J. and Weinberger P. A Formal definition of Carriers, Advances in Applied Clifford Algebras. 12(1), 39-62 (2002).
[5] Keller J., Keller A., Flores J.A. La Búsqueda de una Ecuación para la (Raíz Cuadrada de la) Densidad Electrónica, Acta Chimica Theoretica Latina, XVIII (4), 175-186 (1990). Flores, J.A. and Keller, J., Differential Equations for the Square Root of the Electronic Density in Symmetry Constrained Density Functional Theory, Phys. Rev. A, 45 (9), 6259--6262 (1992). [6] de Broglie L. Annales de Physique, 3, 22, (1925). [7] Schrödinger E. Annalen der Physik, 79, 361, 489, 734; 81, 109 (1926). [8] Spencer A. J. M. Continuum Mechanics, New York: Longman (1980). [9] Huang L.S. and. Ling C. L. Am. J. Phys., 70 (7), 471-743 (2002). [10] Fock V. A. Z. Phys., 39: 226 (1926).


28 Jaime Keller
[11] Hohenberg P. and Kohn W. Phys. Rev. B, 136, 864-867 (1964). [12] Kohn W. and Sham, L.J. Phys. Rev. A., 140, 1133-1138 (1965). [13] Mackinnon L. Lett. Nuovo Cimento, 31, 37 (1981). [14] Mackinnon L. Found. Phys., 8, 157 (1978).
[15] de Broglie, L. C. R. Acad, Sci., 180, 498 (1925); Théorie Générale des Particules à Spin. Paris: Gauthier-Villars (1943); Ondes Électromagnetiques et Photons. Paris: Gauthier-Villars (1968). [16] Jennison, R. C. J. Phys. A: Math, Nucl. Gen., 11, 1525 (1978).


What is the Electron? 29 edited by Volodimir Simulik (Montreal: Apeiron 2005)
The Electron in the Unified Composite Model of All Fundamental Particles and Forces
Hidezumi Terazawa Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki, 305-0801, Japan E-mail address: terazawa@post.kek.jp
The electron, one of the most fundamental particles in nature, is described in detail in the unified composite model of all fundamental particles and forces, a candidate for the most fundamental theory in physics.
I. Introduction
In 1897, J.J. Thomson discovered the electron, one of the most fundamental particles in nature. For more than a century since then, the electron has played a key role in physics as well as science and technology. What is the electron? From the remarkable progress in experimental and theoretical physics in the twentieth century, it has become well known that matter consists of atoms, an atom consists of a nucleus and electrons, a nucleus consists of nucleons (protons or neutrons) and a nucleon consists of quarks. There exist at least twentyfour fundamental fermions, the six flavours of leptons including the electron and the eighteen (six flavours and three colors) of quarks. In addition, there exist at least twelve gauge bosons including the photon, the three weak bosons, and the color-octet of gluons. The quarks have the strong interaction with the gluons while both the quarks and leptons have the electroweak interactions with the photon or the weak bosons. In addition, all these fundamental particles have the gravitational interaction with themselves (or through the graviton). Furthermore, it has also become clear that the strong and electroweak forces of these fundamental particles fit the standard model in which the strong interaction can be described by quantum chromodynamics, the Yang-Mills gauge theory of color SU(3) , while the electroweak interactions can be described by the unified gauge theory of weak-isospin SU(2) × hypercharge U(1) . The latter theory assumes the existence of additional fundamental particles, the Higgs scalars, which should be found in the near future. Since there exist so many fundamental particles in nature and so many parameters in the standard model of fundamental forces, it is now hard to believe that all these particles are fundamental, and it is rather natural to assume that the standard model is not the


30 Hidezumi Terazawa
most fundamental theory, but what can be derived as an effective theory at low energies from a more (and probably the most) fundamental theory in physics. In this paper, I shall describe the electron in detail in the unified composite model of all fundamental particles and forces, a candidate for the most fundamental theory in physics. This paper is organized as follows: In Section II, I will introduce the unified composite model of all fundamental particles and forces, in which not only all quarks and leptons, including the electron, but also all gauge bosons including the photon, the weak bosons and even the graviton as well as the Higgs scalars are taken as composite states of subquarks, the more (and most) fundamental particles. In Section III, I will explain all properties of the electron such as the electric charge, the intrinsic spin angular momentum and the mass in the unified composite model. In Section IV, I will describe all interactions of the electron such as the electroweak and gravitational interactions as effective interactions at low energies (or at long distances) in the unified composite model. Finally, the last Section will be devoted to conclusions and further discussion. Throughout this paper, the natural unit system of ( h/2 ) c 1
= ≡ π = = where 34
h [ 6.62606876 10− Js]
≅ × is the Planck constant and c (= 299792458 m/s) is the speed of light in vacuum should be understood for simplicity unless otherwise stated. Also, note that electric charges should be understood as in units of electron charge 19
e [ 1.602176462 10− C]
≅ × unless otherwise stated.
II. Unified composite model of all fundamental particles and forces
The unified composite model of all fundamental particles and forces consists of an iso-doublet of spinor subquarks with charges ±1⁄2, w1 and w2 (called “wakems” standing for weak and electromagnetic) [1] and a Pati-Salam colorquartet of scalar subquarks with charges +1⁄2 and 16
− , C0 and Ci (i = 1, 2,3) (called “chroms” standing for colors) [2]. The spinor and scalar subquarks with the same charge +1⁄2, w1 and C0 , may form a fundamental multiplet of N = 1 supersymmetry [3]. Also, all the six subquarks, wi (i = 1, 2) and Cα (α= 0,1, 2,3) , may have “sub colors,” the additional degrees of freedom [4], and belong to a fundamental representation of sub color symmetry. Although the sub color symmetry is unknown, a simplest and most likely candidate for it is SU(4) . Therefore, for simplicity, all the subquarks are assumed to be quartet in sub color SU(4) . Also, although the confining force is unknown, a simplest and most likely candidate for it is the one described by quantum subchromodynamics (QSCD), the Yang-Mills gauge theory of sub color SU(4) [4]. Note that the subquark charges satisfy not only the Nishijima-GellMann rule of ( ) / 2
w
Q = I + B − L but also the “anomaly-free condition” of 0
wC
ΣQ = ΣQ = .
In the unified composite model, we expect at least 36 (= 6 × 6) composite states of a subquark (a) and an antisubquark ( a* or a) which are sub color


The electron in the unified composite model 31
singlet. They include: 1) 16 = (4 × 2 × 2) spinor states corresponding to one generation of quarks and leptons and their antiparticles of
** * * 01 0 2 1 2
,, , ,
e ii ii
v =C w e=C w u =C w d =C w
and their hermitian conjugates (i = 1, 2,3) ; 2) 4 (= 2 × 2) vector states corresponding to the photon and weak bosons of
W w2w1;
+=
11 2 2 0 0
, , , ,;
ii
γ Z =ww w w C C CC
W w1w2 ;
−=
or 4 ( 2 2)
= × scalar states corresponding to the Higgs scalars of
11 21 1 2 2 2
( ) ( ) / ( ) ( ) ( , 1, 2)
φij ⎡ w w w w w w w w ⎤ i j
==
⎣ ⎦;
and 3) 16 = 4 × 4 vector states corresponding to a) the gluons, “leptogluon” and “barygluon” of
( ) 0 00 9
2 ; ; ( , 1, 2,3)
a i a j ii
ij
G =C λ C G =C C G =CC i j= ,
where ( 1, 2,3,...,8)
λa a = is the Gell-Mann’s matrix of SU(3) and b) the “vector leptoquarks” of
i 0i
X =C C
and the hermitian conjugates (i = 1, 2,3) or 16 = 4 × 4 scalar states corresponding to the “scalar gluons,” “scalar leptogluon,” “scalar barygluon” and “scalar leptoquarks” of ( , 0,1, 2,3).
Φαβ = CαCβ α β =
Quarks and leptons with the same quantum numbers but in different generations can be taken as dynamically different composite states of the same constituents. In addition to these “meson-like composite states” of a subquark and an antisubquark, there may also exist “baryon-like composite states” of 4 subquarks, which are sub color-singlet.
III. Quantum numbers and electron mass
In the unified composite model the electron of charge –1 and spin 1⁄2 is taken as a composite S-wave ground state of the spinor subquark w2 of charge –1⁄2 and spin 1⁄2 and the scalar antisubquark C0 of charge –1⁄2 and spin 0. The quantum numbers of the electron come from those of subquarks, the constituents of the electron. In order to explain the mass of the electron, we must consider all the masses of quarks and leptons together, since the electron is not the only isolated member but one of the at least twenty-four fundamental fermions, the quarks and leptons. By taking the first generation of quarks and leptons as almost Nambu-Goldstone fermions [5] due to spontaneous breakdown of approximate supersymmetry between a wakem and a chrom, and the second generation of them as quasi Nambu-Goldstone fermions [6], the superpartners of the Nambu-Goldstone bosons due to spontaneous breakdown of approximate


32 Hidezumi Terazawa
global symmetry, we have not only explained the hierarchy of quark and lepton masses, , , ,
e u c td s b
m m mm m mm m m
μτ
<< << << << << << but also ob
tained the square-root sum rules for quark and lepton masses [7],
1 1 1 11 1 1
2 2 2 22 2 2
and ,
e du e sd
mm m m mmm
μ
= − − = − and the simple relations among quark and lepton masses [8], 2 3 3 2 3 2
and ,
e us t dc b
mm m mm m mm m
τμ
==
all of which are remarkably well satisfied by the experimental values and estimates. By solving a set of these two sum rules and two relations [9], given the inputs of me = 0.511 MeV, mμ= 105.7 MeV, mu = 4.5 ± 1.4 MeV, mc = 1.35 ± 0.05 GeV and mb = 5.3 ± 0.1 GeV [10], we can obtain the following predictions: mτ = 1520 MeV (1776.99 + 0.29 /− 0.26 MeV) ,
8.0 1.9 MeV (5 to 8.5 MeV)
md = ± ,
154 8 Mev (80 to 155 Mev)
ms = ± ,
187 78 GeV (174.3 5.1 or 178.1+10.4/ 8.3 GeV)
mt = ± ± − ,
where the values in the parentheses denote either the experimental data or the phenomenological estimates [10], to which our predicted values should be compared. Furthermore, if we solve a set of these two sum rules and these two relations, and the other two sum rules for the W boson mass mW and the Higgs scalar mass ( mH ) derived in the unified composite model of the Nambu-JonaLasinio type [1],
( )12
2 ,
3,
W ql
m= m
( )12
42 ,,
2,
H ql ql
m = Σm Σm
where , ' s
mq l are the quark and lepton masses and < > denotes the average value for all the quarks and leptons, we can predict not only the four quark and/or lepton masses such as , ,
d st
m m m , and mτ as above but also the Higgs scalar and weak boson masses as 2 366 156GeV,
Ht
m ≅ m= ±
12
(3/ 8) 112 24GeV
Wt
m ≅ m= ± ,
which should be compared to the experimental value of mW = 80.423 0.039 GeV
± [10]. What is left for future theoretical investigations is to try to complete the ambitious program for explaining all the quark and lepton masses by deriving more sum rules and/or relations among them and by solving a complete set of the sum rules and relations. To this end, my private concern is to see whether
one can take the remarkable agreement between my prediction of
( )12
32 3 t dcb us
m = m m m m m ≅ 180 GeV and the experimental data as an evidence for the unified composite model. Recently, I have been more puzzled by the “new Nambu empirical quark-mass formula” of
0
2n
M= M


The electron in the unified composite model 33
with his assignment of n = 0,1,5,8,10,15 for u, d, s, c, b, t [11], which makes my relation of 3 2 3 2
ust tt b
m m m = m m m hold exactly. More recently, I have been even more puzzled by the relations of
22
and
ub s dt c
mm ≅m m m ≅m
suggested by Davidson, Schwartz and Wali (D-S-W) [12], which can coexist with my relation and which are exactly satisfied by the Nambu assignment. If we add the D-S-W relations to a set of our two sum rules, our two relations and our sum rules for mW and if we solve a set of these seven equations by taking the experimental values of 0.511 MeV
me = , mμ = 105.7 MeV and mW = 80.4 GeV as inputs, we can find the quark and lepton mass spectrum of mτ = 1520 MeV (1776.99+0.29/ − 0.26 MeV),
3.8MeV (1.5 to 4.5MeV),
mu =
7.2MeV (5 to 8.5MeV)
md = ,
150MeV (80 to 155MeV)
ms =
0.97GeV (1.0 to 1.4GeV)
mc = ,
5.9GeV (4.0 to 4.5GeV)
mb = ,
131GeV (174.3 5.1GeV or 178.1+10.4/ 8.3GeV)
mt = ± − ,
where an agreement between the calculated values and the experimental data or the phenomenological estimates seems reasonable. This result may be taken as one of the most elaborate theoretical works in elementary particle physics.
IV. Interactions and coupling constants of the electron
In the unified composite model, the unified gauge theory of Glashow-SalamWeinberg for electroweak interactions of the composite quarks and leptons [13] is not taken as the most fundamental theory, but as an effective theory at low energies which can be derived from the more (and, probably, most) fundamental theory of quantum subchromodynamics for confining forces of elementary subquarks [4]. It is an elementary exercise to derive the GeorgiGlashow relations [14],
2 22 3
(sin ) ( ) / 3/ 8
θw = Σ I ΣQ = and
22 2 3
( ) ( ) / ( / 2) 1
a
f g =Σ I Σλ = ,
for the weak-mixing angle θw , the gluon and weak-boson coupling constants (f
and g), the third component of the isospin (I), the charge (Q) and the color-spin
()
/2
λa of subquarks without depending on the assumption of grand unifica
tion of strong and electroweak interactions. The experimental value [10] is
[ ]2
sin ( ) 0.23113 0.00015
wz
θ M = ± . The disagreement between the value of 3/8 predicted in the subquark model and the experimental value might be excused by insisting that the predicted value is viable as the running value renormalized à la Georgi, Quinn and Weinberg [15] at extremely high energies (as high as 15
10 GeV , given the “desert hypothesis.”


34 Hidezumi Terazawa
The CKM quark-mixing matrix V [16] is given by the expectation value of the subquark current between the up and down quark states as [17]
Vus = u w1w2 d .
By using the algebra of subquark currents [18], the unitarity of quark-mixing matrix V †V = VV † = 1 has been demonstrated although the superficial nonunitarity of V as a possible evidence for the substructure of quarks has also been discussed by myself [19]. In the first order perturbation of isospin breaking, we have derived the relations of , ,...,
us cd cb ts
V = −V V = −V which agree well with the experimental values of 0.219 0.226
Vus = − and 0.219 0.226
Vcd = − [10] and some other relations such as ( ) ( / ) 0.021
cb ts s b us
V = V ≅ m m V ≅ , which roughly agree with the latest experimental value of 0.038 0.044
Vcb = − [10]. In the second-order perturbation, the relations of ( / ) 0.0017
ub s c us cb
V ≅ m m V V ≅ and 0.0046
td us cb
V ≅V V ≅
have been predicted. The former relation agrees remarkably well with the latest experimental data of 0.0025 0.0048
Vub ≅ − [10]. The predictions for Vts and Vtd also agree fairly well with the experimental estimates from the assumed unitarity of V, 0.037 0.044
Vts ≅ − and 0.004 0.014
Vtd ≅ − [10]. In short, we have succeeded in predicting all the magnitudes of the CKM matrix elements except for a single element, say, Vus . On the contrary, the lepton-mixing has a different feature. In 1998, the Super-Kamiokande Collaboration [20] found an evidence for the neutrino oscillation [21] due to neutrino-mixing among three generations of
neutrinos ( )
,,
e μτ
ν ν ν in the atmospheric neutrinos. More recently, neutrinomixing has been confirmed not only by the K2K Collaboration [22] for longbase-line neutrino oscillation by neutrino beams from KEK to SuperKamiokande, but also by the SNO Collaboration [23] for solar neutrinos at the Sudbury Neutrino Observatory. They have concluded that the data are consistent with two-flavour νμ ↔ ντ oscillations with 2
(sin 2 ) 0.88
θμτ ≥ and
23
mμτ 2 10
−
Δ = × to 3 2
5 10 (eV)
−
× [20]. The neutrino oscillation indicates not only the non-vanishing mass of neutrinos but also the breakdown of lepton number conservation [24]. I have found a simple model of neutrino masses and mixings [25], whose predictions are consistent not only with such a large mixing and such a small mass-squared difference between νμ and ντ suggested by the Super-Kamiokande data but also with a small mixing
23 2
((sin ) 2 10 to 4 10
θeμ − −
= × × and a large mass-squared difference
22
0.3 to 2.2(eV)
meμ
Δ = between νe and νμ suggested by the LSND data [26] but not with the solar neutrino deficit [27]. However, the LSND data has not been confirmed by any other experiments [26, 28] but seems to contradict the latest result from the KamLAND Collaboration [29], which has excluded all oscillation solutions but the ‘Large Mixing Angle’ solution to the solar neutrino problem with a large mixing 2
[(sin 2 ) 0.86 to 1.00]
θeμ ≅ and a small
mass-squared difference 2 5 2
( 6.9 10 (eV) )
meμ
−
Δ ≅ × . Also note that the CHOOZ experiment [28] has given the constraints of 2
(sin ) 0.15
θeτ < and


The electron in the unified composite model 35
2 32
, 1 10 (eV)
ee
mμτ
−
Δ < × . Furthermore, the Heidelberg-Moscow group has recently reported the first evidence for neutrinoless double beta decay deducing the effective neutrino mass of 0.11 to 0.56 eV with a best value of 0.39 eV [30]. On the other hand, very lately, the determination of absolute neutrino masses from Z-bursts caused by ultrahigh energy neutrinos scattering on relic neutrinos has predicted the heaviest neutrino mass to be 2.75 1.28 / 0.98 eV
+−
for galactic halo and 0.26 + 0.20 /− 0.14 eV for extragalactic origin [31]. More lately, by comparing the power spectrum of fluctuations derived from the Two Degree Field Galaxy Redshift Survey with power spectra for models with four components: baryons, cold dark matter, massive neutrinos and a cosmological constant, an upper limit on the total neutrino mass of 1.8 eV has been obtained [32]. As it stands now, it seems difficult to make a simple model of neutrino masses and mixings which is consistent with all the experimental results since some experimental results contradict others. In the unified pregauge and pregeometric theory of all fundamental forces, the gauge-coupling and gravitational constants are related to each other through the most fundamental length scale of nature. A pregauge theory is a theory in which a gauge theory appears as an effective and approximate theory at low energies (lower than a cut-off λ1 ) from a more fundamental theory [33], while a pregeometric theory (or pregeometry) is a theory in which Einstein theory of general relativity for gravity appears as an effective and approximate theory at low energies (lower than a cut-off λ2 ) from a more fundamental theory [34]. Let us suppose that the cut-off in electrodynamics e.m
λ and the cut-off in geometrodynamics grav
λ are the same or at least related to each other as
e.m grav
λ ≈ λ . In most pregeometric theories of gravity in which Einstein-Hilbert action is induced as an effective and approximate action at long distances by quantum effects of matter fields, the Newtonian gravitational constant is naturally related to the ultra-violet cut-off as 2
grav
G λ−
≈ . If this is the case, these two equations lead to the relation
12
λe.m ≈ G .
This is the famous conjecture by Landau in 1955 [35]: there must be a natural ultra-violet cut-off at the Planck energy 12
G where gravity becomes strong. On the other hand, in most of pregauge theories of electomagnetism, in which the Maxwell action is induced as an effective and approximate action at long distances by quantum effects of charged particles, the fine-structure constant is naturally related to the ultra-violet cut-off as
22 .
1/ ln( / )
α≈ λe m M ,
where M is a parameter of mass dimension. If this is the case, these two relations lead to the relation of
2
α≈ 1/ ln(GM ) .
This is the so-called α-G relation first derived by us in 1977 in the unified pregauge and pregeometric theory of all fundamental forces [36]. Note, however, that in some pregauge and pregeometric theories these fundamental constants


36 Hidezumi Terazawa
are determined as 2 4
G ≈ M /λ and 4
α≈ (M /λ) so that the α-G relation becomes
2
α≈ GM .
Hereafter, I will concentrate on the α-G relation of the former type, leaving the α-G relation of the latter type for later discussion. For definiteness let us write the α-G relation as
2
α= 1/ Aln(1/ GM ) ,
where A is a constant parameter depending on a particular unified pregauge and pregeometric model of all fundamental forces. In our unified pregauge and pregeometric model of all elementary particle forces including gravity [36], for example, the constant is simply given by A = ΣQ2 / 3π, where ΣQ2 is the sum of squared charges over all fundamental fermions. For N generations of quarks and leptons, ΣQ2 = 8N / 3 , so that A = 8N / 9π . Also, the mass parameter is approximately given by 2 2
5 / 24
W
M ≅ Nm π for N generations, so that the α-G relation approximately becomes 2
9 / 8 ln(24 / 5 )
W
α≅ π N π NGm , where mW is the charged weak boson mass. Furthermore, we also know that for six generations of quarks and leptons (N = 6) the α-G relation of
( 2)
3 16 ln 4 / 5 W
α≅ π π Gm
is very well satisfied by the experimental values of α≅ 1/137 ,
12 19
G ≅ 1.22 ×10 GeV and 80GeV
mW ≅ . Therefore, from now on let us assume that there exist six generations of quarks and leptons, or three generations of quarks and leptons and their mirror particles, or that there exist three generations of quarks and leptons, their super-partners and more, so that A = 16 / 3π. We now suppose that the fundamental length scale 1/λ be time-varying with respect to the mass scale related to the mass parameter M. Then, we expect that both the fine-structure and gravitational constants α and G are timevarying [37] and their time-derivatives dα/ dt and dG / dt may satisfy the relation of
2
(dα/ dt)α = AdG / dt ,
which can be derived by differentiating the both hand sides of the α-G relation with respect to time. If instead 1/λ stays constant and if M varies, dG / dt must vanish but the time-derivatives dα/dt and dM/dt may satisfy the other relation of
2
(dα/ dt)α = 2A(dM / dt) / M .
For A = 16 / 3π, the above relations become
2
(dG / dt) / G = (3π/16)(dα/ dt) /α
and
2
(dM / dt)M = (3π/ 32)(dα/ dt) /α .
Now the first relation together with the latest result of
15
dα/ dt /α (2.25 0.56) 10− / yr
< > = ± × for redshift of 0.5 < z < 3.5 by Webb et al. [38] immediately leads to our prediction of


The electron in the unified composite model 37
12
(dG / dt) / G (0.181 0.045) 10− / yr
=±×
We find that this prediction is not only consistent with the most precise limit of
12
(dG / dt) / G ( 0.6 2.0) 10− / yr
= − ± × by Thorsett [39] but also feasible for future experimental test. If the α-G relation of the latter type holds instead of the one of the former type, it leads to either the relation (dα/ dt) /α= (dG / dt) / G
or (dα/ dt) /α= 2(dM / dt) / M ,
depending on whether the length scale 1/λ or the mass scale M varies while M or 1/λ stays constant. Then, the first relation together with the result of Webb et al. [38] immediately leads to another prediction of
15
(dG / dt)G (2.25 0.56) 10− / yr
=±× .
We find that this predicted value for (dG / dt)G seems too small to be feasible for experimental tests in the near future although it is consistent with the limit of Thorsett [39]. On the other hand either one of the second relations together with the result of Webb et al. [38] immediately leads to another prediction of
12
(dM / dt)M (0.081 0.023) 10− / yr
=±× ,
or
15
(dM / dt)M (1.13 0.28) 10− / yr
=±× .
However, I suspect that either one of these predicted values for (dM / dt)M is too small to be feasible for experimental tests in the near future, although a prediction for the possible time-varying particle masses seems extremely interesting at least theoretically. In concluding this Section, I would like to emphasize that the recent result of Webb et al. [38] suggesting a varying fine-structure constant may indicate not only a varying gravitational constant but also a varying cosmological constant [40], if our picture for varying constants of nature is right and future experiments to test our predictions for (dG / dt)G in this Section may check not only the α-G relation but also the unified pregauge and pregeometric theory of all fundamental forces. A few questions would still remain: What is the origin of the varying length scale 1/λ or of the varying mass scale M? Is it related to the mass field [41], the “quintessence” [42] or the Kaluza-Klein extra space in extra dimensions? Are no “constants” of nature constant? After all, it may be that nothing is constant or permanent, as emphasized by the Greek and Indian philosophers some two and a half millennia ago!
V. Conclusions and further discussion
I have explained almost all the properties of the electron including the charge, spin, mass, mixing-angle and coupling constants in the unified composite model of all fundamental particles and forces. There remain some other impor


38 Hidezumi Terazawa
tant properties such as the electric and magnetic moments and the possible nonvanishing size of the electron. First of all, the latest experimental value for the electron mass is [43] 0.0005485799092 0.0000000000004 u
me = ± ,
where u is the unified atomic mass unit [ = (931.494013 ± 0.000037) MeV =
27
(1.66053873 0.00000013)10 kg]
−
± , while that for the electron charge magnitude is [10]
19
(1.602176462 0.000000063) 10 C
(4.80320420 0.00000019) esu.
e−
=±
=±
The fine structure constant is given by [10]
2
α( ≡ e /4π=c) = 1/(137.03599976 ± 0.00000050) .
The experimental upper bound on the charge difference between the electron and the positron is [10]
8
4 10
ee
q qe −
+−
+ ≤× .
The experimental upper bound on the mass difference between the electron and the positron is [10]
9
8 10 |
e ee
m mm −
+−
− <× ,
which gives strong constraint on possible violation of CPT invariance. On the other hand, the current experimental constraint on the electric dipole moment of the electron is [10, 44]
26
(0.07 0.07) 10 e cm
de −
≤ ±× ,
which may allow a small violation of CP or T invariance in the electron sector. Furthermore, the experimental value for the electron magnetic moment is [10]
()
1.001159652187 0.000000000004 2
ee
μ= ± e m ,
which is consistent with the standard model. The experimental data on the difference between the electron and positron g-factor is [10]
12
( 0.5 2.1) 10
e e average
g gg −
+−
− = − ± × |,
which gives another strong constraint on possible violation of CPT invariance. In 1996, the CDF Collaboration at Tevatron [45] released their data on the inclusive jet differential cross section for jet transverse energies ET from 15 to 440 GeV with the significant excess over current predictions based on perturbative QCD calculations for 200
ET > GeV , which may indicate the presence of quark substructure at the compositeness energy scale λC of the order of 1.6T eV. This could be taken as an exciting and intriguing historical discovery of the substructure of quarks (and leptons), which had been long predicted, or as the first evidence for the composite model of quarks (and leptons), which had been proposed since the middle 1970s [1]. It might dramatically change not only so-called “common sense” in physics or science but also that in philosophy, which often states that quarks (and leptons) are the smallest and most fundamental forms (or particles) of matter in mother nature. Note that such a


The electron in the unified composite model 39
relatively low energy scale for λC of the order of 1 TeV had been anticipated theoretically [46] or by precise comparison between currently available experimental data and calculations in the composite model of quarks (and leptons) [47]. In 1997, the H1 and ZEUS Collaborations at HERA [48] reported their data on the deep inelastic e + p scattering with a significant excess of events over the expectation of the standard model of electroweak and strong interactions for high momentum-transfer squared 2 2
Q >15000 (GeV) , which might indicate new physics beyond the standard model. Although neither one of these indications have been confirmed by the other experiments and the significance of the HERA anomaly has decreased with higher statistics, not only the substructure of quarks and leptons as well as Higgs scalars and gauge bosons, but also the possible existence of leptoquarks has been extensively reinvestigated [49]. As it stands now, I must emphasize that both the CDF and HERA anomalies are still with us, and that the explanation of the latter anomaly either by the leptoquark with the mass between 280 GeV and 440 GeV, or by the excited electron with the mass between 300 GeV and 370 GeV [50] is still very viable. The current lower bound on the mass of the excited electron is *>223GeV
me [10] while that on the compositeness energy scale of the elec
tron is ( ) 8.3 TeV
λL+L eeee > and ( ) 10.3 TeV
λL−L eeee > [10], which means that
the size of the electron (1/λe ) is smaller than the order of 18
10 cm
−.
The possible substructure of fundamental fermions such as the electron was considered in some detail by McClure-Drell and Kroll [51] and by Low and myself [52] already in the middle of nineteen sixties, while that of quarks was pointed out by Wilson and others [53] in the early nineteen seventies. Also, the possible substructure of fundamental bosons such as the weak bosons was discussed in great detail by myself and others [54] in the mid-nineteen seventies. In conclusion, let me repeat what I said in my talks at the Paris Conference in 1982 [55] and at the Leipzig Conference in 1984 [56]. “It seems to me that it has taken and will take about a quarter century to go through one generation of physics: atomic physics in 1900-1925, nuclear physics in 19251950, hadron physics in 1950-1975, quark-lepton physics in 1975-2000, “subquark physics” in 2000-2025 and so on.” “I would like to emphasize that the idea of composite models of quarks and leptons (and also gauge bosons as well as Higgs scalars), which was proposed by us, theorists, in the mid-seventies, has just become a subject of experimental relevance in the mid-eighties.” A century has past since the discovery of the electron, the “first elementary particle,” and, hopefully, the compositeness of “elementary particles” will soon be found.
Acknowledgements
The author would like to thank Professor Volodimir Simulik, the editor of this book, entitled What is the electron? Modern structures, theories, hypotheses, for inviting him to publish the present paper in the book and for encouraging him until it was written. He also wishes to thank Mr. Yoshizumi Terazawa, his


40 Hidezumi Terazawa
brother, to whom he dedicates the present paper, for having introduced him to science and technology in his childhood.
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What is the Electron? 43 edited by VolodimirSimulik (Montreal: Apeiron 2005)
Prospects for the Point Electron
Thomas E. Phipps, Jr. 908 South Busey Avenue Urbana, Illinois 61801
Introduction: the relativity of physical size
Many ingenious models for the electron, perhaps the most stable and fundamental of known particles, have been devised. Some of these are discussed elsewhere in this book. Here we shall examine the advantages and prospects of the simplest model of all, the mathematical point. It might seem that this could be dismissed at the outset on general philosophical grounds; e.g., that a point is “infinitely small,” therefore (like the “infinitely large”) operationally indefinable and hence non-physical. This overlooks the fact that models are not to be confused with that which is modeled, but are to be judged by results rather than by inferred resemblances to truth. So, I shall not address the imponderable ontology of what the electron “is,” but only what a point-particle model of it might accomplish. As a point of my own philosophy, I claim that this sort of metaphorical approximation to reality is what the science of physics at best provides. When it pretends to do more, it trespasses on the territories of philosophy, religion, and faith. On the scale of Newtonian physics the point particle has done yeoman service as an approximation to everything from planets to bullets, and has given us a Newtonian principle of relativity of physical size. Dirac believed that the granularity of atoms brought an end to this Newtonian relativity. He wrote:(1) “So long as big and small are merely relative concepts, it is no help to explain the big in terms of the small. It is therefore necessary to modify classical ideas in such a way as to give an absolute meaning to size.” But if we examine not the words Dirac used but the parameters, we discover that his own most wildly successful and seminal theory of the electron describes not an extended particle but a mathematical point! Therefore he himself was the active agent in saving Newtonian relativity on all size scales. It is upon Dirac’s success in describing the point electron that we shall build here in seeking consciously to implement a universal Principle of Relativity of Physical Size. It will be evident that such a principle by no means implies the physical proposition that “size does not matter.” It merely implies the hopeful view that the point-particle approximation can be descriptively useful on all size scales... and, more cogently, that the form or the parameterization of physical descriptive equations is invariant; i.e., does not abruptly change at some threshold such as the “atomic” or “nuclear.” Thus, like another better-known relativity


44 Thomas E. Phipps, Jr.
principle, it implies mathematical form preservation or invariance. Stated in that modest way, it seems to make a good deal of sense, does it not? For if there were an abrupt change in parameterization, would this not have to reflect an (unobserved) abrupt change in the physics? How else than by form preservation are we to make useful inferences from known physics about the topography of unknown physical-descriptive territory? With such questions for clues, we should be able to do a bit of elementary detective work that would not strain Sherlock. Indeed, the implications of size relativity are the only real clue we have to guide us in bettering our pretend-knowledge (as embodied in the ludicrously over-hyped Standard Theory) of particles on nuclear and subnuclear size scales.
A rigorized formal correspondence
If size relativity is to be a useful guide in physical exploration, it must apply to the descriptive transition between the Newtonian and atomic descriptive realms. And it must embody rigorous form preservation. That is the first test that the principle must pass before its “universality” can be substantiated. Here an immediate breakdown of the principle occurs, if Dirac(1) is to be believed. Exercising equal parts of optimism and scepticism, let us set aside Dirac’s judgment and look closely at the best that can be done in formally aligning classical and quantum mechanics ... that is, in setting up a “formal Correspondence” between the two. We need to inquire: “Form preservation” under what transformation or group? At once we see that our best chance is to improve the existing formal Correspondence between the Hamilton-Jacobi (H-J) mechanics (of point particles) and the Schrödinger equation; for these two already bear to each other a remarkable formal resemblance. Regrettably, it may be that the reader’s education has been skimped in regard to the H-J formalism and classical canonical mechanics(2) in favour of something more trendy, such as string theory(3). We shall not attempt a proper tutorial, but merely provide a reminder that the classical equations of motion of n point particles take the H-J form,
S
Ht
∂
=−∂ , j
j
S
pq
∂
=∂ , j
j
S
PQ
∂
− = ∂ , j = 1, 2,",3n , (1a,b,c)
where ( )
,,
jj
H = H q p t is the Hamiltonian or energy function, and
()
,,
jj
S = S q Q t is a scalar known as Hamilton’s principal function. These equations completely describe the point-particle mechanics of the classical domain. Observe that there are two complementary sets of descriptive parame
ters, apart from time; namely, the so-called “old canonical variables” ( )
j, j
qp
and the “new canonical variables” ( )
j, j
Q P . The transformations between these two are termed “canonical.” Just as special relativity preserves form under coordinate transformations, we might expect the principle of size relativity to imply “form” preservation under canonical transformations. That is manifestly impossible if we accept the universal opinion that Schrödinger’s equations,


Prospects for the point electron 45
H it
∂
Ψ=− Ψ
∂
= ,j
j
p iq
∂
Ψ= Ψ
∂
= , Ψ = Ψ (qj ,t ) , (2a,b)
tell the complete story of mechanics in the quantum domain. For, observe that there are three differences between the H-J and Schrödinger equations: (a) The latter contain a wave function or operand Ψ ; hence the “old canonical vari
ables” have morphed into “dynamical variables” ( )
j, j
q p that are operators. (b) The S-function has disappeared and been formally replaced by = / i . (c) The
“new canonical variables” ( )
j, j
Q P have disappeared and been replaced by nothing. It is this latter abrupt disappearance that blocks all possibility of invariance under the formal Correspondence transformation—and that must be corrected if a size relativity principle, implying a rigorized Correspondence, is to be implemented. How can we formulate operator equations that combine all features of both (1) and (2), when applied in their appropriate physical domains? The following set of equations does exactly that:
ff
S
Ht
∂
Ψ =− Ψ
∂ , jf f
j
S
pq
∂
Ψ= Ψ
∂ , jf f
j
S
PQ
∂
−Ψ= Ψ
∂ . (3a,b,c)
In order to recover quantum mechanics from (3) it is necessary to postulate S = = / i , which is equivalent to the Heisenberg postulate in view of
k j j k f j j f jk f kk
jk f
p q q p Sq q S S
qq
i
δ
δ
⎡⎤
⎛⎞ ⎛⎞
∂∂
⎡⎤
− Ψ= − Ψ= Ψ
⎢⎥
⎜⎟ ⎜⎟
⎣ ⎦∂ ∂
⎢⎥
⎝⎠ ⎝⎠
⎣⎦
⎛⎞
→Ψ
⎜⎟
⎝⎠
= , (4)
where δjk is 1 if j = k , 0 otherwise. Here ( )
,,
jj
H=H q p t ,
()
, ,,
f f j jj
Ψ = Ψ q Q P t , and in general ( )
,,
jj
S = S q Q t , if we set aside the specialization S = = / i that describes atomic physics. More generally, we shall find it advantageous to assume
()
,,
jj
S sq Q t
= i= , (5)
where s is some real scalar function to be determined... in case we might want to generalize beyond the atomic case ( s → 1) to describe, e.g., point particles in nuclear states. The element of “innovation” here, Eq. (3c), is not really new. It is the restoration of an operator analog of (1c)—as is obviously essential in order to avoid abrupt changes of parameterization of the mechanical formalism—to reflect an absence of abrupt discontinuities in the physics. Eq. (3c) could never have been dropped from a rigorous formal Correspondence. Eq. (3) constitutes an operator analog of (1) and thus embodies both the size relativity principle and a rigorized formal Correspondence. Consequently, in the transition from “c-numbers” (the commuting real numbers of ordinary arithmetic) to “qnumbers” (operators), there is no abrupt change in form or parameterization. The classical canonical H-J theory is recovered from Eq. (3) by treating the


46 Thomas E. Phipps, Jr.
formal operand Ψf as a constant and cancelling it from all three of (3a,b,c). Hence the “Correspondence” becomes a two-way one... it works just as well going from the quantum to the classical side as going the other way. This is not true of Eq. (2), which allows a transition from classical to quantum, but not the
other way [for no analogs of the new canonical variables ( )
j, j
Q P are present in (2), so quantum straw is lacking to make classical bricks—sweeping statements to the contrary by some of the modern era’s most famous physicists to the contrary notwithstanding]. H-J theory is not a formal limit of accepted quantum theory. If we postulate Eq. (3) as holding for all mechanics—classical, quantum, and beyond—and thus avoid any formal difference among these quite different physical descriptive realms, how is our theorizing to reflect the vast differences we know to exist in nature? The answer exploits the fact that Eq. (3), a more complicated mathematical form than any set of mechanical equations previously considered by physicists, offers more solution options. Eq. (2), the Schrödinger equation, is really one equation in one unknown function Ψ . But in its most general form Eq. (3) is two equations in two unknowns, Ψf and S. Only in the special case of atomic solutions is it permissible to specialize to S = = / i = constant. In describing nuclear states it may prove advantageous to treat the “commutator” value S [see Eq. (4)] as a function of space coordinate values—in particular as a function of distance from a nuclear “force center.” We see thus that there are three distinct classes of solution of (3): Class I. Ψf = constant. The solutions for Hamilton’s principal function S describe the Newtonian states of motion (continuous trajectories). Class II. S = = / i , Ψf = Ψ . These are the ordinary quantum states descriptive of atoms. Class III. Both S and Ψf non-constant. These are states possibly descriptive of point particles within nuclei or “elementary particles.” (This is speculative.)
There is an abrupt discontinuity among these three solution-class options. But it is not a physical discontinuity—it is a descriptive choice discontinuity. Only in that altered sense can we agree with Dirac that quantum mechanical discontinuity sets a size scale to the world. The formalism itself, the equations of motion, set no such size scale. They are size invariant. Only our decision, our choice to pick one class of solutions or another, reflects a passage from one descriptive realm to another. And this is a good thing, since every new bit of empirical knowledge we acquire further blurs the line between quantum and classical worlds. There are observable particle-wave phenomena in the centimetre range, and non-localities of quantum action on the inter-stellar scale. We simply cannot rely on “size” to distinguish these worlds. One must know enough physics to use Eq. (3) wisely, to make the right solution choice to match the particular physical problem at hand—no formalism being foolproof. Such a necessity to make intelligent choices is nothing new. In treating Max


Prospects for the point electron 47
well’s equations, for instance, we need to know enough physics to choose between advanced and retarded solutions. And such descriptive choices are always discontinuous, although the physics is not.
Class-II solutions: atomic-level description
We passed a bit too quickly over the atomic solutions. Let us examine in more detail how ordinary quantum mechanics is extracted from Eq. (3), the form postulated for all mechanics. On setting S = = / i in Eq. (3), we see that it can be written as
ff
H it
∂
Ψ =− Ψ
∂
= , jf f j
p iq
∂
Ψ= Ψ
∂
= , jf f j
P iQ
∂
−Ψ= Ψ
∂
= (6a,b,c)
The first two of these equations are of the familiar form (2). The third is new.
We recall that in H-J mechanics the new canonical variables ( )
j, j
Q P are con
stants of the motion—unlike the old canonical variables ( )
j, j
q p , which morph here into “dynamical variable” operators. Naturally, by the arguments we have
already given, there can be no discontinuity in the interpretation of ( )
j, j
Q P as constants; so constants they remain in the operator calculus—constants being good operators. By inspection Eq. (6c) has the solution
( ,)
jj
j
fj
e− i Q P q t
Ψ= Ψ
=∑ . (7)
[Partially differentiate (7) with respect to Qj , to verify that it satisfies (6c)]. Thus the wave function Ψf satisfying Eq. (3) differs from the standard Schrödinger wave function Ψ only by the constant phase factor
jj
jQ P
i
ee
α −∑
=
=
(8) attached to the Schrödinger function Ψ . After cancelling this phase factor from both sides of Eqs. (6a,b), we get exactly Eq. (2), the Schrödinger equation. So it would appear that the Class-II solutions of Eq. (3) precisely duplicate ordinary quantum mechanics (OQM). (These solutions, then, constitute a “covering theory” of OQM.) But that is true only in a formal or mathematical sense. On the interpretational side the (uncancelled) phase factor eα makes a great difference, through its ability to affect quantum phases by undergoing abrupt changes. Let us examine this more closely.
Class-II solutions: quantum measurement theory
Einstein objected that quantum theory is “incomplete.” He probably meant that it lacked trajectories. Here we have instead asserted the accepted quantum formalism to be parametrically incomplete. This is an altogether different affair. Since we have recovered OQM as a viable class of solutions, it is apparent that we are just as far as ever from trajectories. But we may be in a position to correct another more serious loss occasioned by OQM—the loss of objectivity. Objectivity does not necessarily require trajectories, but it does need event points. That is, real, localized observable happenings must be described by any


48 Thomas E. Phipps, Jr.
physically valid theory... and OQM wholly lacks parameters to describe such objective point events. If we think of a quantum phase space in which a point is
specified by ( )
j, j
q p , we have to recognize that a point so specified is not an
observable event. Rather, ( )
j, j
q p is a running variable describing—if you like—a virtual event or presence or one of a sequence of virtual events forming a Feynman-like pseudo-trajectory. But it is not an observable localized happen
ing in real space or “real” phase space... for the simple reason that the ( )
j, j
qp
are operators. To be observable in real phase space, whose axes are labelled with ordinary real c-numbers, it is necessary that c-number parameters be included in the descriptive formalism. Otherwise the formalism is powerless to describe observable occurrences, or even to recognize that anything happens in the world. That is in fact the situation of OQM, based on Eq. (2). Logically, according to (2), nothing can happen in the quantum world, because the accepted formalism lacks c-number parameters to describe point events representing observable occurrences. What follows from such a lack? History has witnessed a great proliferation of ever more ingenious “interpretational” makeshifts, Band-Aids, and substitutes for a valid formalism. We have had three-quarters of a century of it now, and counting. This remarkable social phenomenon, only nominally related to physics, is known as “quantum measurement theory.” It has become a way of life, a source of steady income, an endless intellectual challenge, for a whole sub-culture among physicists. Its adherents dedicate their lives to avoiding recognition of the obvious: that OQM is under-parameterized. Their basic dogma is that mathematically OQM is an immaculate conception that must not be altered in any way. By contrast, the world (being defenceless) is their plaything... the rules of their game allow the world to be distorted into any shape that will fit their rigidly unyielding mathematics. I wish I could say that the Many Worlds Interpretation of OQM represents the apogee of their flights of fancy, but in fact there is no limit... they literally stop at nothing. By now the amount of professional interest vested in measurement-theory nonsense rivals that vested in string-theory nonsense(3). Indeed, the two groups of theorists could exchange professional concerns today without any externally detectable change—in either quality of product or effect on the rest of physics. In the beginning these interpretational makeshifts were simpler and less sophisticated than they are today; therefore they were more perspicuous. There used to be things called “quantum jumps,” and something called “severance of the von Neumann chain” of phase connections between observer and observed. Both of these approaches recognized that something had to be done about quantum phase connections—but both wandered in the wilderness because the theory had no parameters to do it with. Early on, there was a “Projection Postulate,” contrived to do postulationally what needed to be done parametrically. This has had a phoenix-like rebirth with the latest jargon of obscuration, “quantum (phase) entanglement.” The reason for this rebirth is that in order to make any connection at all with observation the OQM equation of motion has to be


Prospects for the point electron 49
discarded in order to let things happen locally in the world. Having postulated an equation of motion, Eq. (2), the “bold interpreteers” judged it natural to say, “Oops, that was wrong, we now add a Projection postulate that contradicts our equation-of-motion postulate by replacing a pure state with a mixture.” At one time there was even a vogue for carrying this one step farther with a “Selection Postulate”—via another, “Oops, that was wrong, we further postulate a singlestate Selection from the projected state mixture, thus contradicting our Projection that contradicted our postulated equations of motion.” But for some reason this busy postulational first-aid work, perhaps because it produced a structure resembling The House That Jack Built, fell into desuetude until observational necessity eventually forced something (anything) to be done about cutting phase connections. Aided by new jargon, these present-day champions of logic, mathematicians manqué, are still vying among themselves to build inconsistent axiomatic systems. The more postulates the merrier... for postulates whose only logical obligation is to contradict one another are always available in any number at no charge. It should be obvious to every child that the only way out of this thicket of obscuration is to get the equations of motion right in the first place. That is where and only where the postulational “corrections” should be applied. This can happen, as a social phenomenon, only if enough children find enough things laughable about the Emperor’s Parade. The necessary open-eyed innocents are going to have to come from the gene pool of uncommitted physicists. But with ever-more specialization and ever-more “professionalism” of mutual back-scratching among specialties, we have seen a steady trend toward less, rather than more, probability of an outcome favourable to physics as science. There is just not the requisite laughter in the air. As physics grows less respectable, the need of physicists for respect grows more urgent. In these conditions physics becomes a very serious business, indeed, or rather profession. Look at other professions. In my youth, when medicine was a calling, doctors made house calls. They went where needed. Then medicine became a profession. The word profession says it all. Well... has Eq. (3) finally got the equations of motion right? If so, it is a new ballgame, interpretionwise. For there are now extra parameters, constants of the motion, explicitly present in the theory. Moreover, those c-number constants appear in a phase factor (8) on the wave function (7). That is just where we should want them to be, if phase-connection severance or the description of “loss of phase knowledge” is our objective. And that, indeed, is precisely our objective. It is what all the postulational fuss was about—“Projection,” “Selection,” and the rest. But that must now be forgotten, if the new paradigm is to receive a fair trial. The questions will be the same, but the solutions will be new and the methods of arriving at them will be somewhat unfamiliar. Consider quantum particles “basking” in an atomic pure state. They obey Eq. (6). At some time in the past, we may suppose, this pure state was “pre
pared.” At that time or earlier the parameters ( )
j, j
Q P received some fixed


50 Thomas E. Phipps, Jr.
values, which they maintain throughout the duration of the pure state. They must stay constant in order that the phase factor eα , given by Eq. (8), may stay constant. During this basking period all processes may be considered virtual and all “time flow” in abeyance (without responsibility to causality), all phase connections remaining intact. Nothing observable can happen anywhere in the system. Then something happens. This occurs purely at nature’s initiative. Human thought or “mind” has nothing to do with it. The job of physics is to describe the happening, after the fact. The happening is localized—of the nature of a point event—to fit with relativity. But we may push beyond that in supposing the localization to take place in phase space, so that both Qj and Pj acquire numerical values. The result is an abrupt, unknowable change Δα in the phase angle of the wave function. (Heisenberg’s “Uncertainty” is not violated, since it concerns the old canonical variables, not the new ones.) The unknowability of Δα implies severance of phase connections and loss of phase “knowledge,” although this is a poor way of speaking, inasmuch as one cannot lose what one never had. So, here we have the “quantum jump,” a severance of the von Neumann chain, effected by a jump in numerical values of the new canonical variables. Something has happened locally in nature, described by a sudden change in c-number values of event-descriptive parameters. To accomplish this, the wave function phase discontinuously changes in an unpredictable way—thus severing the past from the future and ratcheting time flow at the most basic quantum descriptive level. As long as the pure-state phase stays constant, we cannot say that time flows at all. But when the phase jumps in a way we cannot know, it is allowable to say that phase “knowledge” is lost in an irreversible way, and that time flows irreversibly “forward,” in conformity with a postulated observance of causality. As an extra dividend for rigorizing formal Correspondence and thereby “completing” the equations of motion of quantum mechanics, we gain an accounting for the “arrow of time” at the quantum level. Like the Scarlet Pimpernel, that arrow has been sought high and low, even in the farthest reaches of the cosmos... and all the time it has been hiding in our sub-basement right at home. We also dispose of all those versions of quantum measurement theory that rest on deep Wignerian speculations about “mind” intervening as a causal agent in nature. Calling on mind to sever quantum phase connections is just as silly as calling on mutually contradictory postulates to do the job. Mind-fans will certainly prefer their “insights” to the more prosaic notion that the job of the mind, as applied to physics, is to describe nature, not to actuate it. To accomplish the description of nature requires descriptive parameters. Where parameters are lacking, mind and postulates are equally poor surrogates. What we have said so far about phase jumps reflecting locally completed processes applies strictly only to the simplest one-body and one-component (one-channel) problems. In many experimental situations, or generally in many-body problems, a quantum system may be described by numerous component wave functions, among which may occur only partial reductions of the


Prospects for the point electron 51
total system wave function. I have suggested(4) that these partial reductions be termed “virtual events,” and that they might be used to describe the type of absorber action whereby some observable effect(5) is “frustrated” by the mere presence of a potential absorber, without any actual localized absorption occurring. (The accepted jargon calls these “interaction-free” or “non-demolition” measurements. What it amounts to is the introduction of phase incoherence(4) into some but not all channels of a multi-channel pure-state process.) By the view I have suggested, the absorber always absorbs... but not always by a locally-completed “real” event; possibly by a virtual event that imposes phase incoherence on a single component or channel of a many-channel process. That virtual occurrence is not observable directly because the phase jump does not affect the quantum system as a whole (quanta by definition act only as a whole!); but it is indirectly inferred(5) through the observable aspects of “frustration.” This large and somewhat subtle subject is at present speculative and in need of development. I have been able to give only the most crude and fallible introduction to it(4). In summary concerning measurement theory: OQM endows factual history with ensemble attributes that have no basis in experience. To correct this, the theory needs to acquire c-number parameters descriptive of unique, factual point events. The best way to do this is to rigorize the formal Correspondence with H-J theory, thereby restoring analogs of the new canonical variables, or constants of the motion. The same is mandated by a Principle of Relativity of Physical Size, applied in the context of a point particle model.
Class-II solutions: the Dirac electron
By linearizing the classical one-body relativistic energy expression,
2 24 2 2 2 1 23
E = m c + p + p + p , (9)
through use of 4 × 4 anti-commuting unit matrices ( )
123
, , ,m
α α α α , Dirac(1) obtained an operator description of the free electron at the atomic level of de
scription. On introducing electromagnetism via the potentials,
(/)
jj j
p → p + e c A , j = 1, 2,3 (where e is the unsigned charge of the electron), he obtained a Hamiltonian (energy function) of the form
2 m
e
H c mc eV
cα
⎛⎞
=− ⋅ + − −
⎜⎟
⎝⎠
α p A , (10)
where 0
V = A is the scalar potential, etc. This Hamiltonian, which is a perfectly good classical one for any point particle, proved fabulously successful when applied on the quantum side to description of the electron-positron, by
means of the operator identifications ( )( )
→ = ∂∂
jj
p i q , in accordance with Eq. (2b). We need say no more about this, since it forms a cornerstone of modern physical theory, and is doubtless taught everywhere. Just one point need detain us. This is that the same formalism (of H-J pedigree) that on the classical side describes any point particle suddenly turns out on the quantum side to describe only one species of particle—the electron.


52 Thomas E. Phipps, Jr.
What could this mean? It seems a very startling constriction of physical descriptive purview. Does it mean that our size relativity principle fails? That a sudden discontinuity at the quantum boundary is real and reflects an absoluteness of physical scale, as Dirac(1) thought? So it would seem... but I have ventured to suggest an alternative(6) appraisal that is both simple and drastic. This is embodied in the beta-structure hypothesis; namely, that all physical particles and the vacuum are composed of electrons. The steps of reasoning behind this are simple: (1) If a principle of relativity of physical size is valid, then a single mechanical equation of motion [viz., Eq. (3)] must govern particle mechanics on all size scales. (2) If Eq. (3) governs, then its atomic-level specialization S = = / i manifestly describes the Dirac electron. (3) If no physical discontinuity marks the transition between classical and atomic levels, as to either equations of motion or form (10) of the Hamiltonian, then there can be no abrupt change in physical nature of the particles described. (4) If there is no distinct boundary between the realms inhabited by “all particles” and by “electrons,” then all particles must in fact be electrons or their composites. (Here I do not distinguish between electrons and positrons.) This idea meets at once certain troubles, such as that protons seem quite different from positrons. But it is implicit in the beta-structure hypothesis that positrons (vacant electron states) are somehow captured within protons and held permanent prisoners there; and that the proton is in fact a very many-body relativistic system composed ultimately of electrons. Similarly the vacuum, as Dirac originally thought (“hole theory”) before he was brain-washed, is composed of electrons in negative-energy states. The relativistic very many-body problem is so difficult, so little explored, and so cleverly dodged (e.g., through field-theoretical devices such as second quantization), that I do not see how any prudent physicist, not adept at dodging, can dogmatically reject the betastructure hypothesis. Not included among prudent physicists is that vast majority who unhesitatingly bet the farm on field theory. It is my belief, to mix metaphors, that pure point particle mechanics has still plenty of mileage left in it. Admittedly, the leap from size relativity and form preservation to the above wild guess about world structure is a bold one... but the fact that field theory totally rejects it rather prejudices me in its favour. Is it not high time that a few physical theorists began to think outside the field-theory box?
Class-III solutions: formalism
From Eqs. (3a) and (5) we get
ff
Hs
it
∂
Ψ =− Ψ
∂
= . (11)
The Class-III solutions are those that treat S or s and Ψf as non-constant. This shows at once that there is a problem: Since s is to be treated as real (Hermitian), and −(= / i) ∂ /∂t is known to be Hermitian, we have H represented in (11) as a product of two Hermitian operators. It is a well-known theorem that


Prospects for the point electron 53
the product of two Hermitian operators is non-Hermitian. The reader will have to take my word for it that it is simply not physics to represent a physical energy as a non-Hermitian operator. That would say energy is unreal, which is physically incorrect, as far as is known. Fortunately, there is a ready fix for this that works like magic. We simply introduce a new Hermitian Hamiltonian H in place of the classical-analog Hamiltonian H, a Hermitian momentum pj in place of pj , and a transformed wave function Ψ , by means of the definitions
1
Hs−
H= , 1
pjs−
pj = , f
Ψ = sΨ . (12a,b,c)
Here, for the one-body problem, 1, 2,3.
j = This transforms Eq. (3) into
it
∂
Ψ=− Ψ
∂
=
H,
j
iq
∂
Ψ= Ψ
∂
=
pj , 1
j j
Ps i Q
−∂
− Ψ= Ψ
∂
= . (13a,b,c)
From this we see that the basic formalism of OQM is recovered, even for the Class-III solutions, but with a transformed Hamiltonian and an extra relation, Eq. (13c). In terms of the Hermitian operator H the equation of motion of a Heisenberg variable X,
()
dX X 1 X X
dt t i
=∂ +
∂ = H-H , (14)
is also recovered. Thus all the standard OQM techniques employing Hermitian operators are applicable [the operators on both sides of (13c) being Hermitian, as well as those of (13a,b)]. The formal operand Ψf is not useful for calculating observable probability distributions; but its transformed analog Ψ is [and is understood as the operand in (14)]. Similarly, any classical-analog Hamiltonian H, such as Eq. (10), is not the observable physical energy, but H is, since it is the generator of infinitesimal time displacements of the system [Eq. (13a)]. In short, the non-Hermitian classical-analog quantities entering Eq. (3) have served their purpose of form preservation over the whole physical range, and in the particular case of Class-III solutions are to be discarded in favour of their transformed Hermitian counterparts. This has proven a disappointment to mathematicians, who feel cheated of novelty by this return to familiar forms [Eqs. (13a,b), (14)]. But physicists will recognize that any alteration of the Hamiltonian [Eq. (12a)] entails “new physics”—which should console them for the lack of “new math.”
Class-III solutions: the electron on the nuclear scale
Given Eq. (3) as descriptive of particle mechanics on all size scales, we have seen that its Class-I solutions describe the motions of any classical point particles (possessed of trajectories). The Class-II solutions, descriptive of the atomic realm (without trajectories but with objective point events), given Eq. (10) as the (relativistic one-body) Hamiltonian, describe only electronpositrons. We may suppose that the same classical-analog Hamiltonian, with the help of Class-III solutions, might describe the same particle (electron) on a still smaller size scale. This is a speculation, but it proves fruitful. We recall


54 Thomas E. Phipps, Jr.
that the Class-III solutions obey a commutation rule [Eq. (4), left of arrow] that generalizes the Heisenberg postulate. Thus we are contemplating a new physics of the nuclear realm, whereby the Heisenberg postulate may be locally disobeyed. That is, the commutator of the position and momentum dynamical variables may become non-constant in the vicinity of a nuclear force center. This gives an entirely new meaning to the concept of “nuclear force,” and implies that it is not like other “forces” known from larger-scale (including atomic-scale) experience. To probe the general nature of point-electron dynamics in this nuclear or sub-nuclear domain, let us consider a relativistic central-force one-body problem. In this case the classical-analog Hamiltonian is given by Eq. (10) with V = Ze / r , where we allow for Z positive charges on the attractive force center, assumed to be a fixed point. The mathematics has been given in some detail in earlier references(6-8), and will only be summarized here. The Hermitian Hamiltonian is [from Eqs. (10), (12), (13)]
2
1 1 21 1 m
e Ze
Hs c s mc s s
ic r
α
− − −−
⎛⎞
= =− ⋅ ∇+ − −
⎜⎟
⎝⎠
αA
=
H . (15)
We consider the conservative case in which the Hamiltonian and s are time
independent, so that the substitution (i / )E't
eψ
−
Ψ = = reduces (13a) to the eigenvalue equation Hψ = E 'ψ . (16)
The assumption of spherical symmetry, s = s(r) , together with an identity given by Dirac(1), allows this (with A = 0 ) to be reduced to
2
21 1 33
1 i cj Ze
i c mc s s
rr r r
ε ερ − ρ −
⎛∂ ⎞
= +− − −
⎜⎟
⎝∂ ⎠
=
=
H , (17)
where j is an operator that commutes with any function of r (hence with s) and Dirac gives the representations
0
0
i
i
ε−
⎛⎞
=⎜ ⎟
⎝⎠
and 3
10
01
ρ⎛ ⎞
=⎜ ⎟
−
⎝⎠
(18)
The eigenvalue equation (16), with ψ a two-component wave function, then yields the two simultaneous equations
12 2
22 1
1'
'0
j s Ze E
mc
r c rc
ψψ ψ
−
⎡⎤
⎛⎞
+
+ + −− − =
⎢⎥
⎜⎟
⎝⎠
⎣⎦
= = , (19a)
12 2
11 2
1'
'0
j s Ze E
mc
r c rc
ψψ ψ
−
⎡⎤
⎛⎞
−
−+ + − − =
⎢⎥
⎜⎟
⎝⎠
⎣⎦
= = , (19b)
for the two ψ-components. We need another equation to determine s, and this is furnished by (13c) with j = 1,2,3. Introducing formal spherical polar coordinates by means of 2 2 2
1 23
R = Q + Q + Q , Q1 = R sin(θ) cos(φ) , Q2 = R sin(θ)sin(φ) , Q3 = R cos(θ) , we find


Prospects for the point electron 55
(,)
j jj
R
Q QR R
ξ θφ
⎛⎞
∂ ∂∂ ∂
==
⎜⎟
⎜⎟
∂ ∂∂ ∂
⎝⎠
, j = 1,2,3, (20)
the ξj being direction cosines. Let θ,φ specify an arbitrary fixed direction from the coordinate origin. Then the ξj are constants, and in view of spherical symmetry we can consider the Pj in Eq. (13c) to obey j j
P = Pξ , where P is some constant. By this means the three relations (13c) are reduced to the single (two-component) equation
1
Ps i R
ψψ
−∂
=− ∂
= . (21)
Recall that R is related to the Qj and thus is a constant of the motion. Both s and ψ may be considered to depend at least implicitly on this constant. However, we choose to eliminate the explicit appearance of R by seeking a solution that depends on r and ( )
123
R = Q ,Q ,Q only through the combination r − R . If such a solution exists, we can replace ∂ / ∂R by −∂ / ∂r , so that (21) becomes
1
Ps i r
ψψ
−∂
=∂
= . (22)
For consistency with our assumption of spherical symmetry, it is evidently necessary to impose a particular initial condition; namely,
( )( )
123
R = Q ,Q ,Q = 0, 0, 0 . (23)
This initial condition will unfortunately limit the usefulness of our solution, since it implies the presence at the coordinate origin of an infinitely massive force center. (Strictly speaking, we assume the electron to be “found” at an event point R coincident with the origin.) We accept this limitation and proceed, because we are here more interested in proving the existence of some solution than in finding the most general one. Since now ψ =ψ(r) , we can replace all partial derivatives by total ones, so that (22) becomes
1d
s iP dr
ψψ
− = = . (24)
The assumptions that s is some scalar (spin-independent) real function s = s(r) , possessing an inverse, and that P is a constant suffice, with (24), to establish the equality of logarithmic derivatives of the two ψ-components. Thus
11
1 21 1 2 21 2
ln .
d d const C
dr dr
ψ ψψ
ψ ψ ψψ
ψ
− − ⎛⎞
⎛⎞ ⎛ ⎞
= → = →=
⎜⎟
⎜⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ⎝⎠
, (25)
where C is some constant. Using (24), (25) to eliminate s and ψ2 from (19), we obtain from the two parts of the latter equation
()
2
11
1
'
'0
jC
imc iZe E
C P cPr c r
ψ ψ+
⎡⎤
⎡⎤
+ + + −+ =
⎢⎥
⎢⎥
⎣ ⎦⎣ ⎦
= , (26a)


56 Thomas E. Phipps, Jr.
2
11
'1
'1 0
imcC iZe C E C j
P cPr c r
ψψ
⎡ ⎤−
⎡⎤
−− + + − + =
⎢ ⎥⎢ ⎥
⎣⎦
⎣ ⎦ = . (26b)
As can be seen by multiplying (26a) by C, these two equations for ψ1 are compatible if and only if
21
1
j
Cj
−
= + and 1
imcC j
Pj
−
= + . (27a,b)
In order to recover the Heisenberg postulate (s = 1) at long distances from the force center, we shall require unity as the asymptotic value of s, lim ( ) 1
r sr
→∞ = . (28)
From (24) we see that in this limit both components of ψ behave like exp( iPr / = ). Since the wave function components must be bounded at infinity, P must have a positive imaginary part. In Dirac’s electron theory the operator j takes any positive or negative integral eigenvalues. Since j commutes with any function ( )
s r , we may expect j to have somewhat similar commutation properties and eigenvalues in the present formalism. However, it is easily seen from (27) that the eigenvalues ±1 must be excluded here. It follows from this exclusion that C is real; thus that P is pure imaginary and positive. Hence P can be written as 2
P = iK , where K is real and non-zero. Then Eq. (27) yields
2 21
j
mc
Kj
= −
1
1
jj
C jj
−
= − + (29a,b)
Eq. (29a) makes it obvious that 1
j = ± must be forbidden. All other non-zero integral eigenvalues of j are allowed. Substituting 2
P = iK and (26a) into (24), written for the first component ψ1 , we obtain
() 2
11 2 22
2 1
1
' 'j C
E mc Ze
sC
K c r K cK r
K
ψ ψ
−+
⎡⎤
⎛⎞ ⎡ ⎤
=− = − + + +
⎜ ⎟ ⎢ ⎥⎢ ⎥
⎣⎦
⎝⎠ ⎣ ⎦
==
= . (30)
Applying to this the asymptotic condition (28), we find
22
'
lim ( ) 1
r
mc E
sr C K Kc
→∞
⎡ ⎤⎡ ⎤
== + −
⎢ ⎥⎢ ⎥
⎣ ⎦ ⎣ ⎦ . (31)
Solved for the energy eigenvalues E ' , this yields with the help of (29)
2 22 J
mc
E E cK C mc j
′′
= = − − = − , 2, 3,
j = ± ± " , (32)
as the eigenvalue spectrum. The corresponding eigenfunctions can be found by integrating Eq. (30). However, it is easier to guess a solution of the form
()
11
r
j Aje r γ
α
ψψ β
−
= = + . (33)
From this we obtain
1
1
'
1/
r rr
γ αβ
α
ψγ
α
ψ ββ
−
−+ =− + =
+ + . (34)


Prospects for the point electron 57
On putting (29) and (32) into (30) we evaluate s finally as
2
2
2
1
() 1 1
j
jj
jZe
s s r mc r mc r
⎡⎤
−
⎡ ⎤⎢ ⎥
= =+ −
⎢ ⎥⎢ ⎥
⎣ ⎦⎣ ⎦
= . (35)
Comparison of the ratio 1 1
ψ' /ψ from (30) with (34) then yields the following evaluation of the constants in (33):
j2 1
mc
j
α−
== ,
2
2
jZe
mc
β= ,
2
22
11
Ze j
jj
cj
γ= − + −
= . (36)
Eq. (33) then evaluates the first eigenfunction component ψ1 . The second, ψ2 , follows from (25) and (29b). The multiplier Aj in Eq. (33) is arbitrary and can be used for wave function normalization. This completes our formal demonstration of the existence of localized bound-state Class-III solutions on the sub-atomic scale. These are electron states, since m is the electron mass. It is seen that α is of the order of the electron Compton wavelength, and by (33) that this controls the “size” of the electron wave function. But the fact that the canonical “momentum” parameter P is pure imaginary [ 2
P = iK ] seems to imply that the point electron cannot be “found” or “detected” on any size scale larger than a mathematical pointwhich according to Eq. (23) is collocated with the force center at the origin. In other words, any event of “finding” is described by the new canonical variables Q1 = Q2 = Q3 = 0 . We know that nucleons have non-zero sizes. Therefore this solution is of no direct use for describing them. It treats merely an idealized limiting case of the infinite-mass force center localized at a point. Some of the simplifications we have pointed out along the way would have to be corrected in order to describe a finite-mass nucleon. That would ultimately involve solving a relativistic very many-body problem, and is beyond this writer’s capabilities. Still, the results so far seem encouraging.
Summation
We have seen that point particle mechanics is not dead, and that a nuclear dynamics founded on the Class-III solutions, which locally violate the Heisenberg postulate, lies easily within the realm of formal descriptive possibility. Such an enhancement of dynamics seems limited on the sub-atomic scale to a description of the electron-positron—a fact that suggests a “beta structure hypothesis,” viz., that only electrons exist on the finer scales in nature. Our derivation of eigenvalues, Eq. (32), and of eigenfunctions, Eq. (33), establishes that stable bound states exist, beyond any known on the basis of classical (Class-I) or atomic (Class-II) solutions of our postulated equations of motion for all mechanics, Eq. (3). The eigenvalues in question lie within what was termed by Pauli the “Zwischengebiet”—the region of real mass-energy, but imaginary momentum, lying between particle total energies 2
±mc . (That fits also with the imaginary value 2
P = iK of the canonical momentum parameter.) This furnishes a ready mechanical explanation for nuclear beta processes and encour


58 Thomas E. Phipps, Jr.
ages further speculations, for instance that all heavy particles may be aggregates of many imaginary-momentum electrons in real-mass states, i.e., electronic states within the Zwischengebiet... and that neutrinos (if found to be of zero rest mass) may be energy quanta associated with electronic transitions between states of real and imaginary momentum, as distinguished from photons, which are zero-mass quanta associated with electronic transitions between states of real momentum. The Class-I (Hamilton-Jacobi) solutions are exact solutions of Eq. (3), hence are as valid approximate descriptors of nature as the atomic (Class-II) solutions. Each solution class is available to describe its own appropriate aspect (and scale) of experience. None is subordinate to any other. Therefore we do not have to use the de Broglie wavelength of a planet to get a more “accurate” description of its motion; nor do we have to picture a “wave function of the universe.” Our basic theme has been a rigorization of formal Correspondence, motivated by a Principle of Relativity of Physical Size. An immediate consequence has been the parametric restoration of formal analogs of the new canonical variables (constants of the motion). The parameter count must not change under formal Correspondence, there being no corresponding discontinuity in nature. The restoration of c-number parameters in the Class-II equations of motion clears up all the OQM mysteries that have provided full employment for quantum measurement theorists (by providing a parametric mechanism for phase-connection severance that replaces “Projection”). Prospects for a resurgence of the dynamics of the point electron have never been brighter. Still, I have found during forty years that such ideas are of little interest to professional physicists... who remain supremely assured that quantum field theory, not particle dynamics, is the mathematical language by which nature communicates her inmost secrets. Thus they conform to the definition of an expert, as one who makes no small mistakes.
References
1. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford, Clarendon, 1947)
2. H. Goldstein, Classical Mechanics (Addison-Wesley, Cambridge, 1950)
3. P. Woit, Am. Scientist 90, 110-112 (2002)
4. T. E. Phipps, Jr., Phys. Essays 11, 155-163 (1998)
5. P. Kwiat, H. Weinfurter, T. Herzog, and A. Zeilinger, Phys. Rev. Lett. 74, 4763 (1995)
6. T. E. Phipps, Jr., Heretical Verities: Mathematical Themes in Physical Description (Classic Nonfiction Library, Urbana, IL, 1986)
7. T. E. Phipps, Jr., Phys. Rev. 118, 1653 (1960)
8. T. E. Phipps, Jr., Found. Phys. 3, 435 (1973); 5, 45 (1975); 6, 71 (1976); 6, 263 (1976).


What is the Electron? 59 edited by Volodimir Simulik (Montreal: Apeiron 2005)
The Spinning Electron
Martin Rivas Theoretical Physics Department University of the Basque Country Apdo. 644-48080 Bilbao, Spain e-mail:wtpripem@lg.ehu.es
A classical model for a spinning electron is described. It has been obtained within a kinematical formalism proposed by the author to describe spinning particles. The model satisfies Dirac’s equation when quantized. It shows that the charge of the electron is concentrated at a single point but is never at rest. The charge moves in circles at the speed of light around the centre of mass. The centre of mass does not coincide with the position of the charge for any classical elementary spinning particle. It is this separation and the motion of the charge that gives rise to the dipole structure of the electron. The spin of the electron contains two contributions. One comes from the motion of the charge, which produces a magnetic moment. It is quantized with integer values. The other is related to the angular velocity and is quantized with half integer values. It is exactly half the first one and points in the opposite direction. When the magnetic moment is written in terms of the total observable spin. one obtains the g = 2 gyromagnetic ratio. A short range interaction between two classical spinning electrons is analysed. It predicts the formation of spin 1 bound states provided some conditions on their relative velocity and spin orientation are fulfilled, thus suggesting a plausible mechanism for the formation of a Bose-Einstein condensate.
1. Introduction
The spin of the electron has for many years been considered a relativistic and quantum mechanical property, mainly due to the success of Dirac’s equation describing a spinning relativistic particle in a quantum context. Nevertheless, in textbooks and research works one often reads that the spin is neither a relativistic nor a quantum mechanical property of the electron, and that a classical interpretation is also possible. The work by Levy-Leblond [1] and subsequent papers by Fushchich et al. [2], which show that it is possible to describe spin 1⁄2 particles in a pure Galilean framework, with the same g = 2 gyromagnetic ratio, spin-orbit coupling and Darwin terms as in Dirac’s equation, lead to the idea that spin is not strictly a relativistic property of the electron. The spin is the angular momentum of the electron, and the classical and quantum mechanical description of spin is the main subject of the kinematical formalism of elementary spinning particles published by the author [3]. This work presents the main results of this formalism and, in particular, an analysis of a model of a classical spinning particle whose states are described by Dirac’s spinors when quantized. Other contributions are also discussed.


60 Martin Rivas
2. Classical elementary particles
To understand what a classical elementary particle is from the mathematical point of view, we consider first the example of a point particle. It is the simplest geometrical object with which we can build any other geometrical body of any size and shape. The point particle is the classical elementary particle of Newtonian mechanics and has no spin. Yet we know today that spin is one of the intrinsic properties of all known elementary particles. The description of spin is related to the representation of the generators of the rotation group, and we know it is an intrinsic property since it is related to one of the Casimir operators of the Galilei and Poincaré groups. From the Lagrangian point of view, the initial (and final) state of the point particle is a point on the continuous space-time manifold. In fact what we fix as boundary conditions for the variational problem are the position r1 at time t1 and the position r2 at the final time t2 . We call kinematical variables of any mechanical system the variables which define the initial (and final) configuration of the system in this Lagrangian description, and kinematical space the manifold covered by these variables. The point particle is a system of three degrees of freedom with a four-dimensional kinematical space. In group theory, a homogeneous space of any Lie group is the quotient structure between the group and any of its continuous subgroups. The important property of the kinematical space of a point particle, from the mathematical viewpoint, is that it is a homogeneous space of the Galilei and Poincaré groups. In the example of the point particle, the kinematical space manifold is the quotient structure between the Poincaré group and the Lorentz group in the relativistic case, and also the quotient between the Galilei group and the homogeneous Galilei group in the non-relativistic one. We use this idea to arrive at the following definition.
Definition: A classical elementary particle is a mechanical system whose kinematical space is a homogeneous space of the kinematical group.
The spinless point particle fulfils this definition, but it is not the most general elementary particle that can be described, because we have larger homogeneous spaces with a more complex structure. The largest structured particle is the one for which the kinematical space is either the Galilei or Poincaré group or any of its maximal homogeneous spaces. With this definition we have a new formalism, based upon group theory, to describe elementary particles from a classical point of view. It will be quantized by means of Feynman’s path integral method, where the kinematical variables are precisely the common end points of all integration paths. The wave function of any mechanical system will be a complex function defined on the kinematical space. In this way, the structure of an elementary particle is basically related to the kinematical group of space-time transformations that implements the Special Relativity Principle. It is within the kinematical group of


The spinning electron 61
symmetries that we must look for the independent and essential classical variables to describe an elementary object. When we consider a larger homogeneous space than the space-time manifold, for both Galilei and Poincaré groups, we have variables additional to time and space to describe the states of a classical elementary particle. These additional variables will produce a classical description of spin.
3. Main features of the formalism
When we write the Lagrangian of any mechanical system in terms of the introduced kinematical variables, and the dynamics is expressed in terms of some arbitrary evolution parameter τ(not necessarily the time parameter), we get the following properties: • The Lagrangian is independent of the evolution parameter τ. The time evolution of the system is obtained by choosing ( )
t τ =τ.
• The Lagrangian is only a function of the kinematical variables xi and their
first τderivatives xi . • The Lagrangian is a homogeneous function of first degree in terms of the derivatives of the kinematical variables xi and therefore Euler’s theorem implies that it can be written as ( , ) ( , ) ,
ii
L x x = F x x x where i i
F = ∂L ∂x .
• If some kinematical variables are time derivatives of any other kinematical variables, then the Lagrangian is necessarily a generalised Lagrangian depending on higher order derivatives when expressed in terms of the essential or independent degrees of freedom. Therefore, the dynamical equations corresponding to these variables are no longer of second order, but, in general, of fourth or higher order. This will be the case for the charge position of a spinning particle. • The transformation of the Lagrangian under a Lie group that leaves the dynamical equations invariant is L(gx, gx) = L(x, x) + dα(g; x) dτ,
  where (;)
α g x is a gauge function for the group G and the kinematical space X. It only depends on the parameters of the group element and on the kinematical variables. It is related to the exponents of the group [4]. • When the kinematical space X is a homogeneous space of G, then ( ; ) ( , x)
α g x = ξ g g , where 1 2
ξ(g , g ) is an exponent of G. • When quantizing the system, Feynman’s kernel is the probability amplitude for the mechanical process between the initial and final state. It will be a function, or more precisely a distribution, over the X × X manifold. Feynman’s quantization establishes the link between the description of the classical states in terms of the kinematical variables and its corresponding quantum mechanical description in terms of the wave function. • The wave function of an elementary particle is thus a complex square integrable function defined on the kinematical space. • The Hilbert space structure of this set of functions is achieved by a suitable choice of a group invariant measure defined over the kinematical space.


62 Martin Rivas
• The Hilbert space of a classical system, whose kinematical space is a homogeneous space of the kinematical group, carries a projective, unitary, irreducible representation of the group. In this way, the classical definition of an elementary particle has a correspondence with Wigner’s definition of an elementary particle in the quantum case.
4. The classical electron model
The latest LEP experiments at CERN suggest that the electron charge is confined within a region of radius 19
10 m
Re −
< . Nevertheless, the quantum mechanical effects of the electron appear at distances of the order of its Compton’s wavelength 13
/ 10 m
λC mc −
= = ≅ , which are six orders of magnitude larger. One possibility to reconcile these features is the assumption, from the classical viewpoint, that the charge of the electron is a point, but at the same time this point is never at rest and it is affected by an oscillating motion in a confined region of size λC . This motion is known in the literature as Zitterbewegung. This is the basic structure of spinning particle models that will be obtained within the proposed kinematical formalism, and also suggested by Dirac’s analysis of the internal motion of the electron [5]. It is shown that the charge of the particle is at a single point r, but this point is not the centre of mass of the particle. Furthermore, the charge of the particle is moving at the speed of light, as shown by Dirac’s analysis of the electron velocity operator. Here, the velocity corresponds to the velocity of the point r, which represents the position of the charge. In general, the point charge satisfies a fourth-order differential equation, which is the most general differential equation satisfied by any three-dimensional curve. We shall see that the charge moves around the centre of mass in a kind of harmonic or central motion. It is this motion of the charge that gives rise to the spin and dipole structure of the particle. In particular, the classical relativistic model that when quantized satisfies Dirac’s equation shows, for the centre of mass observer, a charge moving at the speed of light in circles of radius R0 = = / 2mc and contained in a plane orthogonal to the spin direction [6,7]. This classical model of electron is what we will obtain when analysing the relativistic spinning particles. To describe the dynamics of a classical charged spinning particle, we must therefore follow just the charge trajectory or, alternatively, the centre of mass motion and the motion of the charge around the centre of mass. In general the centre of mass satisfies second-order, Newton-like dynamical equations, in terms of the total external force. But this force has to be evaluated not at the centre of mass position, but rather at the position of the charge. We will demonstrate all these features by considering different examples.


The spinning electron 63
5. Non-relativistic elementary particles
Let us first consider the non-relativistic formalism because the mathematics involved is simpler. In the relativistic case the method is exactly the same, [3,6,7] and we limit ourselves here to giving only the main results. We start with the description of the Galilei group to show how we obtain the variables that determine a useful group parameterization. These variables associated with the group will later be transformed into the kinematical variables of the elementary particles. We end this section with an analysis of some different kinds of classical elementary particles.
5.1 Galilei group
The Galilei group is a group of space-time transformations characterised by ten
parameters g ≡ (b, a,v,αG)
G G . The action of a group element g on a space-time
point x ≡ (t, rG) , represented by x gx
′ = , is considered in the following form
exp( ) exp( ) exp( ) exp( )
x bH a P v K α J x
′= ⋅ ⋅ ⋅
GG G
G
GG
It is a rotation of the point, followed by a pure Galilei transformation, and finally a space and time translation. Explicitly, the above transformation becomes
t t b,
′ = + (1)
() .
r R α r vt a
′= + +
G G G G (2)
The group action (1)-(2) represents the relationship between the coordinates (, )
t rG of a space-time event, as measured by the inertial observer O, and the
corresponding coordinates (t′, rG′) of the same space-time event as measured by
another inertial observer O′. Parameter b is a time parameter, aG has dimen
sions of space, vG of velocity and αG is dimensionless, and these dimensions will be shared by the corresponding variables of the different homogeneous spaces of the group.
The variables b and aG are the time and position of the origin of frame O
at time t = 0 as measured by observer O′. The variables vG and αG are respectively the velocity and orientation of frame O as measured by O′. The composition law of the group g′′= g′g is: b′′= b′+b, (3) () ,
a Rα a vb a
′′ ′ ′ ′
= ++
G
G G G G (4)
() ,
v Rα v v
′′ ′ ′
=+
G
G G G (5)
( ) ( ) ( ).
Rα Rα Rα
′′ ′
=
G G G (6)
The generators of the group in the realization (1, 2) are the differential operators
,, ,
i i li i i k kli
H = ∂ ∂t P = ∂ ∂x K = t ∂ ∂x J = ε x ∂ ∂x (7)
and the commutation relations of the Galilei Lie algebra are
[ , ] , [ , ] , [ , ] , [ , ] 0,
J J = −J J P = −P J K = −K J H =
G G G G G G G G G G (8)
[ , ] 0, [ , ] , [ , ] 0, [ , ] 0.
H P = H K =P P P = K P =
G G G G G G G (9)


64 Martin Rivas
The Galilei group has the non-trivial exponents [4]
2
1
( , ) ( ) '.
2
gg m vb v R a
ξα
⎛⎞
′′
= +⋅
⎜⎟
⎝⎠
G
G G G (10)
They are characterised by the non-vanishing parameter m. The gauge functions for the Lagrangians defined on the different homogeneous spaces of the Galilei group are of the form
2
1
(;) ()
2
gx m vt v R r
αα
⎛⎞
= +⋅
⎜⎟
⎝⎠
G
GG G
They all vanish if the boost parameter vG vanishes. This implies that a Galilei Lagrangian for an elementary particle is invariant under rotations and translations, but not under Galilei boosts. In the quantum case this means that the Hilbert space for this system carries a unitary representation of a central extension of the Galilei group. In the classical case, the generating functions of the canonical Galilei transformations, with the Poisson bracket as the Lie operation, satisfy the commutation relations of the Lie algebra of the central extension of the Galilei group [4]. The central extension of the Galilei group [8] is an 11-parameter group with an additional generator I which commutes with the other ten,
[I , H ] = [I , P] = [I , K ] = [I , J ] = 0,
G G G (11)
while the remaining commutation relations are the same as above (8, 9), the only exception being the last, which now appears as [,] .
i j ij
K P = −mδ I (12)
If the following polynomial operators are defined on the group algebra
2
11
,,
2
W IJ K P U IH P
mm
=− × = −
G G G G G (13)
we see that U commutes with all generators of the extended Galilei group and
that WG satisfies the commutation relations
[ , ] , [ , ] , [ , ] [ , ] [ , ] 0.
W W = −IW J W = −W W P = W K = W H =
G G G GG G GG GG G
We find that 2
WG also commutes with all generators. It turns out that the extended Galilei group has three functionally independent Casimir operators. In those representations in which the operator I becomes the unit operator, for instance, in the irreducible representations they are, respectively, interpreted as the mass, M = mI, the internal energy 2
H0 = H − P / 2m , and the absolute value of the spin
2 2
11
,.
S J KP S J KP
mm
⎛⎞
=− × ⇒ = − ×
⎜⎟
⎝⎠
G G G G G G G (14)
In what follows we take the above definition (14) as the definition of the spin of a nonrelativistic particle. In those representations in which I is the unit operator, the spin operator S
G
satisfies the commutation relations:
[S, S ] = −S, [J , S] = −S, [S, P] = [S, K] = [S, H ] = 0,
GG G G G G G G
G GG


The spinning electron 65
i.e., it is an angular momentum operator, transforms like a vector under rotations and is invariant under space and time translations and under Galilei boosts, respectively.
Furthermore, it reduces to the total angular momentum operator JG in those frames in which P = K = 0
GG .
5.2 The spinless point particle
The kinematical variables of the point particle are {t, rG} , time and position, respectively. The nonrelativistic Lagrangian written in terms of the τderivatives of the kinematical variables is the first order homogeneous function
2
,
2
NR
mr
L Tt R r
t
= = +⋅
G G G


where we define T = ∂L / ∂t and / i
Ri = ∂L ∂r . The constants of motion obtained through the application of Noether’s theorem to the different subgroups of the Galilei group are
2
energy ,
2
m dr
H T dt
⎛⎞
=− = ⎜ ⎟
⎝⎠
G
linear momentum ,
dr
P R m dt
==
G
GG
kinematical momentum K = mr − Pt,
GG
G
angular momentum .
J =r×P
GG
G
The spin for this particle is / 0
S =J −K×P m=
GGGG .
5.3 A spinning elementary particle
According to the definition, the most general nonrelativistic elementary particle [9] is the mechanical system whose kinematical space X is the whole Galilei
group G . The kinematical variables are, therefore, the ten real variables ( ) { ( ), ( ), ( ), ( )}
x τ ≡ t τ r τ u τ ρG τ
G G , with domains 3 3
t∈ , r∈ , u∈
GG
R R R and
(3)
ρG ∈ SO . The latter, with ρ= tanα/ 2 , is a particular parameterization of the rotation group. In this parameterization the composition law of rotations is algebraically simple, as shown below. All these kinematical variables have the same geometrical dimensions as the corresponding group parameters. The relationship between the values x′(τ) and x(τ) take, at any instant τ, for two arbitrary inertial observers t (τ) t(τ) b,
′ = + (15)
() ( )() () ,
r τ R μ r τ vt τ a
′= + +
G G G G G (16)
() ( )() ,
u τ R μu τ v
′= +
G G G G (17)
() ()
() .
1 ()
μ ρτ μ ρτ
ρ τ μ ρτ
+ +×
′ = −⋅
GG
GG
GG
G (18)


66 Martin Rivas
The way the kinematical variables transform allows us to interpret them, respectively, as the time (15), position (16), velocity (17) and orientation (18) of the particle. There exist three differential constraints among the kinematical variables: u(τ) = r (τ) / t(τ)
G G  . These constraints, and the homogeneity condition on the Lagrangian L in terms of the derivatives of the kinematical variables, reduce from ten to six the essential degrees of freedom of the system. These degrees of freedom are the position ( )
rG t and the orientation ρG(t) . Since the Lagrangian
depends on the derivative of uG it thus depends on the second derivative of
rG(t) . For the orientation variables the Lagrangian only depends on the first de
rivative of ρG(t) . It can be written as
L = Tt + R ⋅ r + U ⋅ u + V ⋅ ρ,
G G GG
GG

 (19)
where the functions written in capital letters are defined as before as /, / , / , /
i ii i ii
T = ∂L ∂t R = ∂L ∂r U = ∂L ∂u V = ∂L ∂ρ

  . In general they will be
functions of the ten kinematical variables (t, r , u, ρG)
G G and homogeneous func
tions of zero degree of the derivatives (t, r , u, ρG)
GG 

.
If we introduce the angular velocity ωG as a linear function of ρG , then the
last term of the expansion of the Lagrangian (19), VG ⋅ ρG , can also be written as
WG ⋅ωG , where / i
Wi = ∂L ∂ω .
The different Noether constants of motion are related to the invariance of the dynamical equations under the Galilei group, and are obtained by the usual Lagrangian methods. They are the following observables:
energy ,
dU
H T u dt
=− − ⋅
G
G (20)
linear momentum ,
dU
P R dt
=−
G
G G (21)
kinematical momentum K = mr − Pt − U ,
G GG
G (22)
angular momentum J = r × P + u ×U + W .
G G GG
G G (23)
From 0
KG = , comparing with (21), we find RG = muG , and the linear momentum has the form P = mu − dU / dt
GG
G . We see that the total linear momentum does
not coincide with the direction of the velocity uG . The functions UG and WG are what distinguishes this system from the point particle case. The spin structure is thus directly related to the dependence of the Lagrangian on the acceleration and angular velocity.
We see that KG in (22) differs from the point particle case K = mr − Pt
GG
G,
in the term −UG . If we define the vector kG = UG / m , with dimensions of length,
then KG = 0 leads to the equation:
( ).
dr k
P m dt
−
=
G
G
G
The vector q = r − kG
G G , defines the position of the centre of mass of the particle.
It is a different point from rG , whenever k
G
(and thus UG ) is different from zero.


The spinning electron 67
In terms of qG the kinematical momentum takes the form
K = mq − Pt,
GG
G
which looks like the result in the case of the point particle, where the centre of mass and centre of charge are the same point.
The total angular momentum (23) has three terms. The first term r × PG
G
resembles an orbital angular momentum, and the other two Z = u ×U + W
G GG
G can be taken to represent the spin of the system. In fact, the latter observable is an angular momentum. It is related to the new kinematical variables and satisfies the dynamical equation /
dZ dt = P × u
G G G . Because PG and uG are not collinear vec
tors, ZG is not a conserved angular momentum. This is the dynamical equation
satisfied by Dirac’s spin operator in the quantum case. The observable ZG is the classical spin observable equivalent to Dirac’s spin operator.
One important feature of the total angular momentum is that the point rG
is not the centre of mass of the system, and therefore the r × PG
G part can no longer be interpreted as the orbital angular momentum of the particle. The an
gular momentum ZG is the angular momentum of the particle with respect to
the point rG , but not with respect to the centre of mass. The spin of the system is defined as the difference between the total angu
lar momentum JG and the orbital angular momentum of the centre of mass motion L = q × P
GG
G . It can assume the following different expressions:
1.
dk
S J q P J K P Z k P mk W
m dt
= − × = − × = + × =− × +
G
GG
GG GG GG G G G
G (24)
The second form of the spin S
G
in (24) is exactly expression (14) which leads to one of the Casimir operators of the extended Galilei group. It is expressed in terms of the constants of the motion J , K
G G and PG , and it is therefore another constant of motion. Because the particle is free and there are no external torques acting on it, it is clear that the spin of the system is represented by this constant angular momentum and not by the other angular momentum observ
able ZG , which is related to Dirac’s spin operator.
The third expression in (24) is the sum of two terms, one ZG , coming from
the new kinematical variables, and another kG × PG , which is the angular mo
mentum, of the linear momentum located at point rG , with respect to the centre of mass. Alternatively we can describe the spin according to the last expression
in (24) in which the term −kG × mdkG / dt
suggests a contribution of (anti) orbital type coming from the motion around the centre of mass. It is related to the Zit
terbewegung, or more precisely to the function U = mkG
G , which comes from the
dependence of the Lagrangian on acceleration. The term WG comes from the dependence on the other three degrees of freedom ρi , and thus on the angular velocity. This Zitterbewegung is the motion of the centre of charge around the
centre of mass, as we shall see in an example in section 5.6. That the point rG represents the position of the centre of charge has also been suggested in previous works for the relativistic electron [10].


68 Martin Rivas
To analyse the different contributions to the spin of the most general elementary particle we shall consider now two simpler examples. In the first one, the spin is related to the existence of orientation variables, and in the second, to the dependence of the Lagrangian on the acceleration.
5.4 Spinning particle with orientation
The kinematical space is G / GK , where GK is the three-dimensional subgroup which consists of the commutative Galilei boosts, or pure Galilei transformations at a constant velocity. The kinematical variables are now { , , }
t r αG
G , time, position and orientation, respectively. The possible Lagrangians are not unique
in this case. They must be functions only of the velocity uG = drG / dt and of the
angular velocity ωG . They have the general form
L = Tt + R ⋅ r + W ⋅ω,
G GG
G

where T = ∂L / ∂t, R = ∂L / ∂r , W = ∂L / ∂ω
G GG
G
. The basic conserved observables are: energy ,
H = −T
linear momentum PG = muG,
kinematical momentum K = mr − Pt,
GG
G
angular momentum .
J = r ×P+W
G GG
G
For such a particle rG = qG , the centre of mass and centre of charge coincide and the spin 0
S =W ≠
G G . A particular Lagrangian which describes this system is the Lagrangian of a spherically symmetric body:
2 2
1,
22
dr I
L m dt ω
⎛⎞
=+
⎜⎟
⎝⎠ where the spin is S = W = Iω
G G G.
5.5 Spinning particle with Zitterbewegung
The kinematical space is the manifold G / SO(3) , where SO(3) , is the threedimensional subgroup of rotations. The kinematical variables are ( ) {, , }
x τ = t rG uG , time, position and velocity, respectively. The possible Lagrangians are not unique as in the previous case, and must be functions of the
velocity uG = drG / dt and the acceleration aG = duG / dt . The Lagrangians have the general form when expressed in terms of the kinematical variables and their τ-derivatives
L = Tt + R ⋅ r + U ⋅ u,
GG
GG


where T = ∂L / ∂t, R = ∂L / ∂r , U = ∂L / ∂u
GG
GG

 . A particular Lagrangian could be, for example
22
2,
22
mr m u
L t ωt
=−
GG

  (25)
If we consider that the evolution parameter is dimensionless, all terms in the Lagrangian have dimensions of action. The parameter m represents the mass of


The spinning electron 69
the particle while the parameter ω, with dimension 1
time− , represents an internal frequency: it is the frequency of the internal Zitterbewegung. In terms of the essential degrees of freedom, which reduce to the three position variables
rG , and using the time as the evolution parameter, the Lagrangian can also be written as
2
22
2 2.
22
m dr m d r
L dt ω dt
⎛⎞
⎛⎞
= − ⎜⎟
⎜⎟
⎝⎠ ⎝ ⎠
GG
(26)
The dynamical equations obtained from the Lagrangian (26) are:
42
24 2
1 0,
dr dr
ω dt + dt =
GG
(27)
whose general solution is
( ) cos sin ,
r t = A + Bt + C ωt + D ωt
GG
GG
G (28)
in terms of the 12 integration constants A, B, C
GG
G and DG .
We see that the kinematical momentum KG in (22) differs from the point
particle case in the term −UG . The definition of the vector kG = UG / m , implies
that KG = 0 leads to the equation P = md (r − kG) / dt
G G , as before, and q = r − kG
GG
represents the position of the centre of mass of the particle. It is defined in this example as
2
22
1 1.
dr
qr Ur
m ω dt
=− =+
G
G
G G G (29)
In terms of the center of mass, the dynamical equations (27) can be separated into the form
2
2 0,
dq
dt =
G
(30)
2 2
2 ( ) 0,
dr r q
dt +ω − =
G G G (31)
where (30) is just equation (27) after twice differentiation of (29), and equation (31) is (29) after all terms on the left hand side have been collected.
From (30) we see that the point qG moves in a straight trajectory at con
stant velocity while the motion of point rG , given in (31), is an isotropic har
monic motion of angular frequency ωaround the point qG . The spin of the system S
G
is defined as
S J q P J 1 K P,
m
= −× = − ×
GG GG GG
G (32)
and since it is written in terms of constants of motion it is clearly another constant of motion. Its magnitude S 2 is also a Galilei invariant quantity which characterizes the system. From its definition we get
()() ,
d dk
S u U k P mr q r q k m
dt dt
= × + × =− − × − =− ×
G
GG
GGG
G G G G G (33)
which appears as the (anti)orbital angular momentum of the relative motion of
the point rG around the centre of mass position qG at rest, so that the total angu


70 Martin Rivas
lar momentum can be written as
J = q × P + S = L + S.
GG
GG G
G (34)
The total angular momentum is the sum of the orbital angular momentum LG , associated with the motion of the centre of mass, and the spin part S
G
. For a
free particle both LG and S
G
are separate constants of motion. We use the term (anti)orbital to suggest that if the vector k
G
represents the position of a point of mass m, the angular momentum of its motion is in the opposite direction from what we obtain here for the spin observable. But, as we shall see in a moment, the vector k
G
represents not the position of the mass m, but the position of the charge of the particle.
5.6 Interaction with an external electromagnetic field
If the point qG represents the position of the centre of mass of the particle, then
what position does point rG represent? The point rG represents the position of the charge of the particle. This can be seen by considering interaction with an external field. The homogeneity condition of the Lagrangian in terms of the derivatives of the kinematical variables suggests an interaction term of the form
(, ) (, ) ,
LI = −eφ t r t + eA t r ⋅ r
G
G G G
 (35)
which is linear in the derivatives of the kinematical variables t and rG , and
where the external potentials are only functions of t and rG . The dynamical equations obtained from the Lagrangian I
L + L are
()
42
24 2
1 (, ) (, ) ,
d r d r e Etr u Btr
ω dt + dt = m + ×
GG G G
G G G (36)
where the electric field EG and magnetic field BG are expressed in terms of the potentials in the usual form / ,
E = −∇φ− ∂A ∂t B = ∇ × A
GG
G G . Because the inter
action term does not depend on uG , the function U = mkG
G has the same expression as in the free particle case. Therefore the spin and the centre of mass definitions, (33) and (29) respectively, remain the same as in the previous free case. Dynamical equations (36) can again be separated into the form
()
2
2 (, ) (, ) ,
d q e Etr u Btr
dt = m + ×
GG G
G G G (37)
2 2
2 ( ) 0.
dr r q
dt +ω − =
G G G (38)
The centre of mass qG satisfies Newton’s equations under the action of the total
external Lorentz force, while the point rG still satisfies the isotropic harmonic
motion of angular frequency ωaround the point qG . But the external force and
the fields are defined at the point rG and not at point qG . It is the velocity uG of
the point rG which appears in the magnetic term of the Lorentz force. The point
rG clearly represents the position of the charge. In fact, this minimal coupling we have considered is the coupling of the electromagnetic potentials with the particle current, which, in the relativistic case, can be written as j Aμ
μ . The
current jμ is associated with the motion of a charge e at the point rG .


The spinning electron 71
The charge has an oscillatory motion of very high frequency ω, which in the case of the relativistic electron will be 2 21 1
ω 2mc / 1,55 10 s−
= = ≈ × , as shown later. The average position of the charge is the centre of mass, but it is this internal orbital motion which gives rise to the spin structure and also to the magnetic properties of the particle.
When analysed in the centre of mass frame (see Fig. 1), q = 0, r = kG
G G , and the system reduces to a point charge whose motion is in general an ellipse. If we choose C = D, and 0
C⋅D=
G G , it reduces to a circle of radius r = C = D, orthogonal to the spin. Because the particle has a charge e, it produces a magnetic moment, which according to the usual classical definition is [11]
3
1,
2 22
e dk e
r jd r k S
dt m
μ= × = × = −
∫
G
GG
G
G G G (39)
where j = eδ3 (r − kG)dkG / dt
G G is the vector current associated with the motion of a charge e located at the point k
G
. The magnetic moment is orthogonal to the Zitterbewegung plane and opposite to the spin if e > 0. The particle also has a non-vanishing electric dipole moment with respect to the centre of mass
dG = ekG
. It oscillates and is orthogonal to μG , and therefore to S
G , in the centre of mass frame. Its time average value vanishes for times larger than the natural period of this internal motion. Although this is a nonrelativistic example, it is interesting to compare this analysis with Dirac’s relativistic analysis of the
electron, [5] in which both momenta Gμ and d
G
appear, giving rise to two possible interacting terms in Dirac’s Hamiltonian.
6. Relativistic elementary particles
The Poincaré group can be parameterised in terms of exactly the same ten parameters { , , , }
b a v αG
G G as the Galilei group and with the same dimensions as before. We therefore maintain the interpretation of these variables respectively as the time, position, velocity and orientation of the particle. The homogeneous spaces of the Poincaré group can be classified in the same manner, but with some minor restrictions. For instance, the kinematical space of the example of the spinning particle with orientation as in section 5.4, / K
X = G G , can no longer be defined in the Poincaré case, because the three dimensional set GK of Lorentz boosts is not a subgroup of G; but the most general structure of a spinning particle still holds.
Figure 1: Charge motion in the C.M. frame.


72 Martin Rivas
The Poincaré group has three different maximal homogeneous spaces
spanned by the variables {b, a, v, αG}
G G , which are classified according to the
range of the velocity parameter vG . If v < c we have the Poincaré group itself. When v > c , this homogeneous space describes particles whose charge is moving faster than light. Finally, if v = c , we have a homogeneous space which de
scribes particles whose position rG is always moving at the speed of light. This is the manifold which defines the kinematical space of photons and electrons [6,7]. The first manifold gives, in the low velocity limit, the same models as in the nonrelativistic case. It is the Poincaré group manifold, which is transformed into the Galilei group by the limiting process c → ∞ . But this limit cannot be applied to the other two manifolds. Accordingly, the Poincaré group describes a larger set of spinning objects.
6.1 Spinning relativistic elementary particles
We shall review the main points of the relativistic spinning particles whose kinematical space is the manifold spanned by the variables { , , , }
t r u αG
G G , interpreted as the time, position, velocity and orientation of the particle, but with u = c . This is a homogeneous space homomorphic to the manifold G/V, where V is the one-dimensional subgroup of pure Lorentz transformations in a fixed arbitrary direction. For these systems the most general form of the Lagrangian is
L = Tt + R ⋅ r + U ⋅ u + W ⋅ω,
G G GG
GG


where / , / , /
ii ii
T = ∂L ∂t R = ∂L ∂r U = ∂L ∂u
   and / i
Wi = ∂L ∂ω will be, in general, functions of the ten kinematical variables { , , , }
t r u αG
G G and homogeneous functions of zero degree in terms of the derivatives { , , , }
t r u αG
GG 

.
The Noether constants of motion are now the following conserved observables:
energy ,
dU
H T u dt
=− − ⋅
G
G (40)
linear momentum ,
dU
P R dt
=−
G
G G (41)
22
kinematical momentum K = Hr / c − Pt − SG × u / c ,
GG
G G (42)
angular momentum J = r × P + SG
GG
G , (43)
where
S = u ×U +W.
G GG
G (44) The difference from the Galilei case comes from the different behaviour of the Lagrangian under the Lorentz boosts when compared with the Galilei boosts. In the nonrelativistic case the Lagrangian is not invariant. However, the relativistic Lagrangian is invariant and the kinematical variables transform in a
different way. This gives rise to the term SG × uG / c2 instead of the term UG which appears in the kinematical momentum (42). The angular momentum observable (44) is not properly speaking the spin of the system, if we define spin


The spinning electron 73
as the difference between the total angular momentum and the orbital angular momentum associated with the centre of mass. It is the angular momentum of
the particle with respect to the point rG , as in the nonrelativistic case. Nevertheless, the observable S
G
is the classical equivalent of Dirac’s spin observable because in the free particle case it satisfies the same dynamical equation,
,
dS P u
dt = ×
G GG
as Dirac’s spin operator does in the quantum case. It is only a constant of motion for the centre of mass observer. This can be seen by taking the time deriva
tive of the constant total angular momentum JG given in (43). We shall keep the notation S
G for this angular momentum observable, because when the system is quantized it gives rise to the usual quantum mechanical spin operator in terms of the Pauli spin matrices.
6.2 Dirac’s equation
Dirac’s equation is the quantum mechanical expression of the Poincaré invari
ant linear relationship [6,7] between the energy H and the linear momentum PG
0,
du
H Pu S u
dt
⎛⎞
− ⋅−⋅ × =
⎜⎟
⎝⎠
G
G
GG G
where uG is the velocity of the charge (u = c), du / dt
G the acceleration and
u
S S Sα
=+
GG G
Dirac’s spin observable (see Figure 2). This expression can be obtained from (42) by making the time derivative of that constant observable and
a final scalar product with the velocity uG . The Dirac spin has two parts: one Su = u ×U
GG
G , is related to the orbital motion of the charge, and S W
α=
G G is due to the rotation of the particle and is directly related to the angular velocity, as it corresponds to a spherically symmetric object. The centre of mass observer is defined as the observer for whom K = P = 0,
G G because this implies that 0
qG = and / 0
dq dt =
G . By analysing the observable (42) in the centre of mass frame where H = mc2, we get the dy
namical equation of the point rG ,
2
r = SG × u / mc
GG
where S
G
is a constant vector in this frame. The solution is the circular motion depicted in Figure 2.
Figure 2. Motion of the centre of charge of the electron around its centre of mass in the C.M. frame.


74 Martin Rivas
The radius and angular velocity of the internal classical motion of the charge are, respectively, R = S / mc , and ω= mc2 / S . The energy of this system is not definite positive. The particle of positive energy has the total spin S
G
oriented in the same direction as the Su
G
part while the orientation is the opposite for the negative energy particle. This system corresponds to the time reversed motion of the other. When the system is quantized, the orbital component Su
G
, which is directly related to the magnetic moment, quantizes with integer values, while the rotational part Sα
G
requires half integer values. For these particles of spin 1⁄2, the total spin is half the value of the Su
G
part. When expressing the magnetic moment in terms of the total spin, we thus obtain a pure kinematical interpretation of the g = 2 gyromagnetic ratio [12]. For the centre of mass observer this system appears as a system of three degrees of freedom. Two represent the x and y coordinates of the point charge, and the third is the phase of its rotational motion. However this phase is exactly the same as the phase of the orbital motion of the charge. Because the motion is at constant radius at constant speed c, only one independent degree of freedom is left—say the x variable. Therefore the system is reduced to a onedimensional harmonic oscillator of angular frequency ω. When the system is quantized, the stationary states of a one-dimensional harmonic oscillator have the energy
1 , 0,1, 2,...
2
En n ω n
⎛⎞
=+ =
⎜⎟
⎝ ⎠=
But if the system is elementary, then it has no excited states, and in the C.M. frame it is reduced to the ground state of energy
2 0
1.
2
E mc
= =ω=
If we compare this with the classical result ω= mc2 / S we see that the constant classical parameter S takes the value S = = / 2 when quantized. The radius of the internal motion is / 2
C
R = λ , half Compton’s wavelength. We see that all Lagrangian systems with the same kinematical space as the one considered in this model have exactly the same dynamics for the point r, describe spin 1⁄2 particles and satisfy Dirac’s equation when quantized. The
formalism describes an object whose charge is located at a single point rG , but it is nevertheless moving in a confined region of radius of order λC . It has a magnetic moment produced by the motion of the charge, and also an oscillating electric dipole moment, with respect to the centre of mass, of average value zero. To conclude this section, and with the above model of the electron in mind, it is convenient to remember some of the features that Dirac obtained for
the motion of a free electron [5]. Let the point rG be the position vector in terms of which Dirac’s spinor ( , )
ψ t rG is defined. When computing the velocity of the
point rG , Dirac arrives at:


The spinning electron 75
1. The velocity u = i / [H , r ] = cαG
GG
= , is expressed in terms of the αG matrices and he writes, ... “a measurement of a component of the velocity of a free electron is certain to lead to the result ±c .”
2. The linear momentum does not have the direction of this velocity uG , but must be related to some average value of it: ... “the x1 component of the velocity, 1
cα , consists of two parts, a constant part 2 1
1
c p H− , connected with the momentum by the classical relativistic formula, and an oscillatory part, whose frequency is at least 2
2mc / h ,.....”
3. About the position rG : “The oscillatory part of x1 is small... which is
of order of magnitude = / mc ....”
And when analyzing the interaction of the electron with an external electromagnetic field in his original 1928 paper [13], after taking the square of Dirac’s operator, he obtains two new interaction terms:
,
22
e ie
BE
mc Σ ⋅ + mcα⋅
GG
GG
==
Here Dirac’s spin operator is written as / 2
S= Σ
G =G where
0,
0
σ
σ
⎛⎞
Σ=⎜ ⎟
⎝⎠
G
GG
in terms of σ-Pauli matrices. EG and BG are the external electric and magnetic fields, respectively. He says, “The electron will therefore behave as though it
has a magnetic moment (e= / 2mc)Σ and an electric moment (ie / 2mc)αG
= . The magnetic moment is just that assumed in the spinning electron model” (Pauli model). “The electric moment, being a pure imaginary, we should not expect to appear in the model.”
In the last sentence it is difficult to understand why Dirac, who did not reject the negative energy solutions, disliked the existence of this electric dipole, which was obtained from the formalism on an equal footing with the magnetic dipole term. Properly speaking this electric dipole does not represent the existence of a particular positive and negative charge distribution for the electron. The negative charge of the electron is at a single point but because this point is not the centre of mass, there exists a non-vanishing electric dipole moment with respect to the centre of mass even in the centre of mass frame. This is the observable Dirac disliked. It is oscillating at very high frequency and basically plays no role in low energy electron interactions because its average value vanishes, but it is important in high energy processes or in very close electronelectron interactions. All real experiments to determine very accurately the gyromagnetic ratio are based on the determination of precession frequencies. But these precession frequencies are independent of the spin orientation. However, the difficulty separating electrons in a Stern-Gerlach type experiment suggests polarization experiments have to be done to determine in a direct way whether the spin and magnetic moment for elementary particles are either parallel or antiparallel to each other. One of the predictions of this formalism is that for both particle and


76 Martin Rivas
the corresponding antiparticle the spin and magnetic moment have to have the same relative orientation, either parallel or antiparallel.
6.3 Dynamical equation of the relativistic spinning electron
We recall from elementary differential geometry some basic properties of any arbitrary three-dimensional curve ( )
rG s . If it is expressed in parametric form in terms of the arc length s as the parameter, it has associated the three orthogonal unit vectors , 1, 2,3
vi i =
G called respectively tangent, normal and binormal. These unit vectors satisfy the so called Frenet-Serret differential equations:
12
21 3
32
() () ()
() () () () ()
() () ()
v s sv s
v s sv s sv s
v s sv s
κ
κτ
τ
=
=− +
=−
GG

GG G

GG

,
where κand τare respectively the curvature and torsion. Since the unit tangent vector is (1)
v1 = r ≡ r
G GG
 , when successive derivatives are taken it yields
(1)
1
(2)
2
(3) 2 22 12 3
(4) 3 2 1 23
,
,
,
3 ( ) (2 ) .
rv
rv
r v v vv v
rv v v
κ
κ κ κ κ κτ
κκ κ κ κτ κτ κτ
=
=
= + =− + +
=− + − − + +
GG
GG
G G G GG G

 GG G G
   
The elimination of the vGi vectors between these equations implies that the most general curve in three-dimensional space satisfies the fourth-order ordinary differential equation:
2
(4) (3) 2 2 (2) 2 (1) 2
r 2 r 2 r r 0.
κ τ κτ κ κκ κ τ
κτ κ
κ τ κτ κ κ τ
⎛⎞
−
⎛ ⎞ ⎛⎞
− + + ++ + + − =
⎜⎟
⎜ ⎟ ⎜⎟
⎝ ⎠ ⎝⎠
⎝⎠
    
 
GG G G
All the coefficients in brackets, in front of the s-derivatives rG(i) , can be expressed in terms of the scalar products ( ) ( ) , , 1, 2,3
ij
r ⋅r i j =
G G . For helical motions there is a constant relationship κ/τ = constant, and therefore the coefficient of (1)
rG vanishes. Our example of the nonrelativistic spinning particle also satisfies the
fourth order differential equation (27). Similarly, the point rG of the relativistic spinning electron also satisfies a fourth order ordinary differential equation which has been calculated from invariance principles [14]. It takes the following form for any arbitrary inertial observer:
(2) (3) (4) (3) (2) (2)
(3) (3) (2) (3) 2 (2) (2) 1/ 2 (2) (2) (2) (2) (2) 2
3( )
()
2( ) 3( ) ( ) 0.
( ) 4( )
rr
rr
rr
rr rr rr r
rr rr
⋅
−+
⋅
⎛⎞
⋅⋅
− −⋅ =
⎜⎟
⋅⋅
⎝⎠
GG
GG
GG
GG GG GG G
GG GG
(46)
It corresponds to a helical motion since the term in the first derivative (1)
rG is lacking, and it reduces to circular central motion at constant velocity c in the centre of mass frame. Here we use space-time units such that the internal radius R = 1 and the Zitterbewegung frequencyω= 1 . The centre of mass position is defined by


The spinning electron 77
(2) (2) (2)
(2) (3) 2 (2) (2) 3/ 2 (3) (3) (2) (2)
2( ) .
3( )
( ) ( ) 4( )
r rr
qr r r
rr rr rr
⋅
=+ ⋅
⋅ +⋅− ⋅
G GG
GG G G
GG GG GG
(47)
We can check that both qG and (1)
qG vanish for the centre of mass observer. The fourth order dynamical equation for the position of the charge (46) can also be rewritten as a system of two second order ordinary differential equations for
the positions of the points qG and rG
(1) (1) (2) (2)
2
1
0, ( ),
()
qr
q r qr
qr
−⋅
== −
−
GG
G G GG
G G (48)
i.e., a free motion for the centre of mass qG and a kind of central motion for the
charge position rG around the centre of mass. Equation (46) emerges from (47) after differentiation twice with respect to time. The last equation of (48) is just
(47) written in terms of qG and (1)
qG .
For the relativistic electron, when the centre of mass velocity is small, q(1) → 0
G , and because q − r = 1
G G in these units, we obtain the equations of the Galilei case
(2) (2)
q = 0, r = q − r
G G G G (49)
i.e., a free motion for the centre of mass and a harmonic motion around qG of angular frequency ω= 1 , for the position of the charge, as happened in the nonrelativistic example analysed in (30) and (31).
6.4 Interaction with an external field
The free equation for the centre of mass motion q(2) = 0
G represents the conservation of linear momentum / 0
dP dt =
G . But the linear momentum is written in terms of centre of mass velocity as (1) (1)
PG = mγ(q )qG , so that the free dynamical equation (48) in the presence of an external field should be replaced by
(1) (1) (1) (2)
2
1
, ( ),
()
qr
P F r qr
qr
−⋅
== −
−
GG
G G G GG
G G (50)
where FG is the external force and the second equation is left unchanged. We consider the same definition of the centre of mass position (47) as in the free particle case, because it corresponds to the fact that the internal structure of an elementary particle is not modified by any external interaction, and the charge moves in the same way around the centre of mass as in the free case. Since
(1) (2) (1) 3 (1) (2) (1)
( ) ( )( )
dP m q q m q q q q
dt = γ + γ ⋅
G G GGG
it yields
(1) 3 (1) (2) (1)
mγ(q ) (q ⋅ q ) = FG ⋅ q
GG G
and by leaving the highest derivative (2)
qG on the left hand side we finally obtain the differential equations that describe the evolution of a relativistic spinning electron in the presence of an external electromagnetic field:


78 Martin Rivas
()
(2) (1) (1) (1) (1)
(1) [ ] ,
()
e
mq E r B q E r B q
q
γ⎡ ⎤
= + ×− + × ⋅
⎣⎦
GGGG
G G G G G (51)
(1) (1) (2)
2
1 ( ).
()
qr
r qr
qr
−⋅
=−
−
GG
G GG
G G (52)
7. Gyromagnetic ratio
The Hilbert space which describes the wave functions of the spinning electron is a complex vector space of squared integrable functions ( , , , )
ψ t r u αG
G G of the kinematical variables. The general structure of the quantum mechanical angular momentum operator acting on this Hilbert space, in either the relativistic or nonrelativistic approach, is
J r S r P S,
i
= × ∇+ = × +
GG
GG
=
G G (53)
where the spin operator takes the form S = Z+W
.
u
Su W
i
= × ∇+
GG
=
G (54)
The operator ∇u is the gradient operator with respect to the velocity variables
and WG is a linear differential operator which depends only on the orientation
variables αG ; it therefore commutes with ∇u . For example, in the
ρG = nG tan(α/ 2) parameterization WG is written as
[ ( )],
2
W iρ ρ ρ
= ∇ + ρ× ∇ + ρ ρ⋅ ∇
G G GG
= (55)
where ∇ρ is the gradient operator with respect to the ρG variables. The first part Z in (54) is related to the Zitterbewegung spin and has only integer eigenvalues. This is because it has the form of an orbital angular mo
mentum operator in terms of the uG variables. Half-integer eigenvalues come only from the operator (55). This operator takes into account the change of orientation, i.e., the rotation of the particle. We have seen, in both relativistic and non-relativistic examples, that if the only spin content of the particle S
G
is related to the Zitterbewegung part Z = u ×U
GG
G , then the relationship between the magnetic moment and Zitterbewegung spin is given by
,
22
e dk e
kZ
dt m
μ= × =
G
GG
G (56)
i.e., with a normal gyromagnetic ratio g = 1. If the electron has a gyromagnetic ratio g = 2, this necessarily implies that another part of the spin arises from the angular velocity of the body, but makes no contribution to the magnetic moment.
For the electron, therefore, both parts WG and ZG contribute to the total
spin. But the WG part, which is related to the angular variables that describe its orientation in space, does not contribute to the separation k
G
between the centre


The spinning electron 79
of charge and the centre of mass. It turns out that the magnetic moment of a general particle is still related to the motion of the charge by the expression
(56), i.e., in terms of the ZG part, but not to the WG part. It is precisely when we express the magnetic moment in terms of the total spin S
G
that the concept of gyromagnetic ratio arises.
We now assume that both ZG and WG terms contribute to the total spin S
G
with their lowest admissible values. In the model of the spinning electron ZG
and WG have opposite orientation. For Dirac’s particles, the classical Zitterbewegung is a circular motion at the speed of light of radius R = = / 2mc and angular frequency 2
ω= 2mc / = , on a plane orthogonal to the total spin. The total spin S
G and the ZG part are both orthogonal to this plane, and parallel to each other. Let us define the gyromagnetic ratio by Z = gS. For the lowest admissible values of the quantized spins z = 1 and w = 1⁄2 in the opposite direction, this gives rise to a total s = 1⁄2 perpendicular to the Zitterbewegung plane, and therefore g = 2.
8. Bound motion of two electrons
If we have relativistic and nonrelativistic differential equations satisfied by the spinning electrons we can analyze the interaction between them by assuming, for example, a Coulomb interaction between their charges. This leads to a system of differential equations of the form (37-38) or (51-52) for each particle. For example, the external field acting on the charge e1 is replaced by the instantaneous Coulomb field created by the other charge e2 at the position of e1 , and similarly for the other particle. The integration is performed numerically by means of the numerical integration program Dynamics Solver [15]. Figure 3 represents the scattering of two spinning electrons analysed in their common centre of mass frame [14]. We send the particles with their spins parallel and with a non vanishing impact parameter. In addition to the helical motion of their charges, we can also depict the trajectories of their centre of mass. If we compare this motion with the Coulomb interaction of two spinless electrons coming from the same initial position and with the same velocity as the centre of mass of the spinning electrons, we obtain the solid trajectories marked with an arrow. Basically, this corresponds to the trajectory of the centre of mass of each spinning particle, provided the two particles do not approach each other below the Compton wavelength. This can be understood because the average position of the centre of charge of each particle approximately coincides with its centre of mass, and if they do not approach each other too closely
Figure 3. Scattering of two spinning electrons with parallel spins, in their centre of mass frame. It is also depicted the scattering of two spinless electrons with the same energy and linear momentum.


80 Martin Rivas
the average Coulomb force is the same. The difference comes out when we consider a very deep interaction or very close initial positions. Figure 4 represents the initial positions of a pair of particles with parallel spins. Recall that the radius of the internal motion is half the Compton wavelength. The initial separation of their centres of mass a is a distance smaller than the Compton wavelength. The centre of mass of each particle is consid
ered to be moving with a velocity vG , as depicted. That the spins of the two particles are parallel is reflected by the fact that the internal motions of the charges, represented by the oriented circles that surround the corresponding centre of mass, have the same orientation. It must be remarked that the internal motion of the charge around its centre of mass can always be characterised by a phase. The phases of the particles are chosen opposite to one another. We also depict the repulsive Coulomb force F computed in terms of the separation of charges. This interaction force F has also been drawn attached to the corresponding centre of mass, so that the net force acting on the point m2 is directed toward the point m1 , and conversely. This external force determines the motion of each centre of mass. We thus see that a repulsive force between the charges represents an attractive force between their centres of mass when located at such a short distance. In Figure 5 we depict the evolution of the charges and masses of this twoelectron system for 0, 4 C
a = λ and v = 0,004c during a short time interval. Figure 6 represents only the motions of the centres of mass of both particles for a longer time. It shows that the centre of mass of each particle remains in a bound region. The evolution of the charges is not shown in this last figure because it
Figure 5: Bound motion of two electrons with parallel spins during a short period of time
Figure 4: Initial position and velocity of the centre of mass and charges for a bound motion of a two-electron system with parallel spins. The circles would correspond to the trajectories of the charges if considered free. The interacting Coulomb force F is computed in terms of the separation distance between the charges.


The spinning electron 81
blurs the picture, but it can be inferred from the previous figure. We have found bound motions at least for the range 0 0,8 C
≤ a ≤ λ and velocity 0 ≤ v ≤ 0, 01c . We can also obtain similar bound motions if the initial velocity v has a component along the OX axis. Bound motion can also be obtained for initial charge positions different from the ones depicted in Figure 4. This range for the relative phase depends on a and v, but in general bound motion is more likely if the initial phases of the charges are opposite to each other. If, instead of the instantaneous Coulomb interaction between the charges, we consider the retarded electromagnetic field of each charge, we obtain a similar behaviour for the bound motion of this electron-electron interaction. We thus see that if the separation between the centre of mass and centre of charge of a particle (Zitterbewegung) is responsible of part of the spin structure, then this attractive effect can be easily interpreted. A bound motion for classical spinless electrons is not possible. We can conclude that one of the salient features of the present formalism is the existence, from the classical viewpoint, of possible bound states for spinning electron-electron interaction. If the centres of mass of two electrons are separated by a distance greater than the Compton wavelength, they always repel each other as in the spinless case. But if the centres of mass of two electrons are separated by a distance less than the Compton wavelength, then from the classical viewpoint they can form bound states, provided certain initial conditions regarding their relative initial spin orientation, position of charges and centre of mass velocity are fulfilled. The difficulty may be to prepare a pair of electrons in the initial configuration depicted in Figure 4. A high-energy deep scattering can bring electrons to a very close approach. At low energy, if we consider the electrons in the conduction band of a solid, their interaction with the lattice could do this job. If we have a very thin layer under a huge external magnetic field perpendicular to the surface, as in the quantum Hall effect measurements, most of the electrons in this layer will have the spins parallel. If this happens to be true, we have a mechanism associated with the spin structure of the elementary particles for the plausible formation of a spin 1 Bose-Einstein condensate. This is just a classical prediction, not a quantum prediction, associated with a model which satisfies the Dirac equation when quantized. The possible quantum mechanical bound states must be obtained from the corresponding analysis
Figure 6. Evolution of the centres of mass of both particles for a longer time


82 Martin Rivas
of two interacting quantum Dirac particles, a problem which has not been solved yet. From the classical viewpoint, bound states for a hydrogen atom can exist for any negative energy and any arbitrary angular momentum. The quantum analysis of the atom gives the correct answer for the allowed stationary bound states.
References
[1] J. M. Levy-Leblond, Non-relativistic particles and wave equations, Comm. Math. Phys. 6, 286, (1967). [2] V.I. Fushchich, A. Nikitin and V.A. Sagolub, On the non-relativistic motion equations in the Hamiltonian form, Rep. Math. Phys. 13, 175 (1978).
[3] M. Rivas, Kinematical theory of spinning particles, (Kluwer, Dordrecht) 2001.
[4] V. Bargmann, On unitary ray representations of continuous groups, Ann. Math. 5, 1 (1954). [5] P.A.M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Oxford U. P.), §69, 1958. [6] M. Rivas, Quantization of generalized spinning particles. New derivation of Dirac’s equation, J. Math. Phys. 35, 3380 (1994). [7] M. Rivas, Classical Relativistic Spinning Particles, J. Math. Phys. 30, 318 (1989). [8] J.M. Levy-Leblond, Galilei Group and Galilean Invariance, in E.M. Loebl, Group Theory and its applications, Acad. Press, NY 1971, vol. 2, p. 221.
[9] M. Rivas, Classical particle systems: I. Galilei free particle, J. Phys. A 18, 1971 (1985). [10] A.O. Barut and A.J. Bracken, Zitterbewegung and the internal geometry of the electron, Phys. Rev. D 23, 2454 (1981). [11] J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, NY, 3rd. ed. p.186, 1998. [12] M. Rivas, J.M. Aguirregabiria and A. Hernandez, A pure kinematical explanation of the gyromagnetic ratio g = 2 of leptons and charged bosons, Phys. Lett. A 257, 21 (1999). [13] P.A.M. Dirac, The quantum theory of the electron, Proc. Roy. Soc. Lon. A117, 610 (1928). [14] M. Rivas, The dynamical equation of the spinning electron, J. Phys. A, 36, 4703 (2003) . [15] J.M. Aguirregabiria Dynamics Solver, 2002. Computer program for solving different kinds of dynamical systems. It is available from his author through the web site <http://tp.lc.ehu.es/jma.html>.


What is the Electron? 83 edited by Volodimir Simulik (Montreal: Apeiron 2005)
Relativistic Wave Equations, Clifford Algebras and Orthogonal Gauge Groups
Claude Daviau La Lande, 44522 Pouille-les-coteaux, France e-mail : daviau.claude@wanadoo.fr Fondation Louis de Broglie, 23 rue Marsoulan 75012 Paris, France
The Dirac equation is considered the wave equation for a particle with spin 1⁄2. We write this equation in the frame of real linear spaces. We present the change resulting from new frames: we can construct new relativistic wave equations, which may not be equivalent to the Dirac equation. One of these new wave equations is proposed for neutrinos. The diversity of relativistic waves is connected to with the diversity of particles with spin 1⁄2.
Classical electromagnetism itself was not sufficient to explain the electron, because it predicted neither the electron’s stability, nor quantification of energy levels, nor the electron’s spin. Louis de Broglie discovered [1] the electron’s wave. The non-relativistic equation for the electron was introduced by Erwin Schrödinger. From relativistic considerations the Klein-Gordon relativistic equation was proposed for the electron’s wave. However, this equation contains two kinds of defects. It gives a conserved current without the probability density, and it does not give expected quantum numbers and the expected number of states in the case of the hydrogen atom. In order to obtain a current with the probability density, Dirac introduced a relativistic wave equation [2] based on the Pauli equation, and sought a wave equation with first order derivatives only. This equation lead to interesting results. In the case of the H atom, the Dirac equation gives the expected quantum numbers, the expected number of states, and precise energy levels. Moreover, on the basis of the Pauli principle the Dirac equation (not the Schrödinger equation) leads to the periodic classification of chemical elements. This equation also gives a correct calculation of the Zeeman effect and explains the Lande factors. Consequently, the Dirac equation is the basis for quantum field theories. Further, when experimental physicists discovered new particles with spin 1⁄2 (muons, neutrinos and quarks), the Dirac equation was used as the wave equation for all these particles. Today it is possible to write the Dirac equation differently. Moreover, these real formalisms lead to new relativistic wave equations that are not equivalent to the Dirac equation. The diversity of these wave equations may be


84 Claude Daviau
linked to the diversity of particles. For instance, it is possible to obtain a chiral relativistic wave equation with a mass term but without the possibility of a charge term, which corresponds to the neutrino’s properties. It is possible to derive the different wave equations in triplicate, equivalent, but distinct equations, and this fact may be related to the existence of three and only three generations of particles. Thus, a study of relativistic wave equations provides simple explanations of well-established but not yet understood facts, such as the insensitivity of leptons and sensitivity of quarks to strong interactions, or the presence of charged and uncharged leptons in each generation.
1. Classical framework of the Dirac equation
We use the usual matrices here
1
20 0 3
4
0
0
;; ;
0
0
j
j j j
I
I
ψ σ
ψ
ψ γγ γ γ σ
ψ
ψ
⎛⎞
⎜⎟ −
⎛⎞
⎛⎞
⎜⎟
= = = =− =⎜ ⎟
⎜⎟
⎜⎟ −
⎝⎠ ⎝ ⎠
⎜⎟
⎝⎠
(1)
01 01
23 23
10 01
;;
01 10
0 10
;;
0 01
I
i
i
σσ σ σ
σσ σσ
⎛⎞ ⎛⎞
= = = =− =
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠ −
⎛⎞ ⎛ ⎞
=− = =− =
⎜⎟ ⎜ ⎟
−
⎝⎠ ⎝ ⎠
(2)
5 0123
;.
γij = γiγj γ = −iγ (3)
Thus, the Dirac equation reads
[ ( ) ] 0,
iqA im
μ
μμ
γ ∂ + + ψ = (4)
0
,,
mc
e
qm
c
==
= = (5)
where e is the negative electron’s charge and Aμ are the covariant components of the electromagnetic potential vector. The probability vector current is one of the tensorial quantities of this theory. These quantities have the form:
† 10
Ω =ψψ; ψ =ψ γ , (6)
Jμ μ
=ψγ ψ , (7)
Si
μν μ ν
= ψγ γψ , (8)
5
Kμ μ
= −ψγγ ψ, (9)
25
Ω = −iψγψ , (10)
where †
ψ is the adjoint, Ω1 is an invariant scalar; J is the probability current, whose time component
42
00
1
i i
J ψγψ ψ
=
= = ∑ (11)
is the probability density. Authors who present the Dirac theory are usually very happy to get 16 tensorial densities without derivatives, because the 4 × 4


Relativistic wave equations, Clifford algebras and orthogonal gauge groups 85
complex matrix algebra is 16-dimensional above the complex field. And we have 16 densities: the Ω1 , scalar, the J, vector, the S, bivector, the K, pseudovector, the Ω2 , pseudo-scalar, but these densities are real, not complex. More generally, it is very difficult to adjust the Dirac theory to the quantum theory, because the Dirac matrices are not Hermitian. We have †
00
γ = γ , but, on the contrary, we have †
11
γ = −γ . Many difficulties arise from this fact in terms of adjusting the Dirac equation to quantum principles derived from nonrelativistic quantum mechanics. We shall see that these sixteen tensorial densities are not the only existing densities of the theory. The γmatrices and the ψwave are not uniquely defined in the Dirac equation, and this fact has led some to see the wave only as a tool for calculations, without physical reality. It is always possible to replace the γmatrices and ψby
1 1†
S S; S; S S
μμ
γ γ ψψ
−−
′′
= = = , (12)
where S is any fixed unitary matrix. The gauge transformations with iaI4
S =e
are among the gauge transformations (12), where In is the unitary n × n matrix. This gauge transformation is very important, because it is both global and local . The gauge (12) can be made local, but the U(4) gauge group is too small to give an acceptable frame for local gauge invariance of the standard model.
2. Real mathematical frames for the Dirac equation
Due to the fact that each ψ has a real and an imaginary part, the Dirac spinor is made of eight real components. Clifford algebras present two kinds of eightdimensional algebras over R which can be used to obtain the Dirac equation, the Clifford algebra Cl3 of the three-dimensional physical space, and the even subalgebra of the space-time algebra Cl1,3.
A. Space algebra
This Clifford algebra, isomorphic to the Pauli algebra, is generated by the eight
elements 1 2 3 23 31 12 123
1, σ , σ , σ , σ , σ , σ , σ . To get the Dirac equation with this frame it is sufficient [3] to associate with each ψof the Dirac theory the φ= f (ψ) , defined by
1 2 32 3 31 4 12 5 123 6 1 7 2 8 3
f (ψ) =φ= a + a σ + a σ + a σ + a σ + a σ + a σ + a σ (13)
where a j are
11 4 2 3 2
38 5 46 7
;
;
a ia a ia
a ia a ia
ψψ
ψψ
= + =− −
= + = + (14)
The space algebra Cl3 is isomorphic to the Pauli algebra M 2 (C) , but this iso
morphism is not an isomorphism of linear space above C , it is only an isomor
phism of linear space above R , because we get
12
f (iψ) =φσ (15)
but not f (iψ) = if (ψ) . Therefore, only linear spaces and algebras above R are convenient here. We chose the notation ψ* for the complex conjugate of ψ. We also use


86 Claude Daviau
†
1 2 32 3 31 4 12 5 123 6 1 7 2 8 3
φ=φ = a − a σ − aσ − aσ − aσ + aσ + aσ + aσ
 , (16)
1 2 32 3 31 4 12 5 123 6 1 7 2 8 3
ˆa a a a a a a a
φ= + σ + σ + σ − σ − σ − σ − σ , (17)
1 2 32 3 31 4 12 5 123 6 1 7 2 8 3
φ = a − a σ − a σ − a σ + a σ − a σ − a σ − a σ , (18)
and for each A and B we get
^
† †† †
ˆ ˆˆ
() ; ; ;
AB = B A AB = B A A = A AB = AB . (19)
The f isomorphism yields for each ψ
*
2ˆ 2
() ;
f ψ =σφσ ˆ
()
fμ
μ
γ ψ =σ φ. (20)
And we get
()
( )0
μ
μμ
γψ
⎡⎤
∂+ + =
⎣⎦
f iqA im .
This is
12 12
ˆˆ 0
qA m
μμ μμ
σ ∂ φ+ σ φσ + φσ = (21)
Within space algebra we use
;A A
μμ μμ
∇ =σ ∂ =σ , (22)
0
0
ˆ;
ˆ;
AA A
∇=∂ +∂
=−
G
G
11 2 2 3 3
12 3
A A 1 A 2 A 3.
σσ σ
σσσ
∂= ∂ + ∂ + ∂
=+ +
G
G (23)
And the Dirac equation in the space algebra frame is
12 12
ˆˆ 0
∇φ+ qAφσ + mφσ = (24)
Tensorial densities without derivatives are
†
J μJ
μ
=φφ =σ , (25)
† 3
K μK
μ
=φσφ =σ , (26)
1 2 123
R =φφ = Ω + Ω σ , (27)
23 31 12 10 20 30
S =φσ3φ = S σ1 + S σ2 + S σ3 + S σ23 + S σ31 + S σ12. (28)
If J and R are single, we immediately see that K and S, with σ3 , may be chosen as a case (3) (3)
K = K , S = S of
†
() ()
;
jj jj
K =φσφ S =φσφ (29)
We therefore get the old 16 tensorial densities without derivatives, and 20 new tensorial densities without derivatives. More generally, from a spinor with 2n real components it is possible to construct 1
(2 1) 2
n n−
+ × tensorial densities without derivatives. The 16 densities of the classical theory are electric gauge invariant. Many attempts have been made to reduce the Dirac spinor to its tensorial densities [4] [5]. It is well known that we cannot know the entire wave from the only 16 gauge invariant tensorial densities, because they tell us nothing about the phase, which changes with the gauge. The gauge transformation
12
;σ
ψ ψ φφ
′′
= =a
eia e (30)
induces a rotation between (1)
K and (2)
K and between (1)
S and (2)
S as
†
(1) 1 (1) (2)
K φσφ cos(2a)K sin(2a)K ,
′ ′′
= = − (31)


Relativistic wave equations, Clifford algebras and orthogonal gauge groups 87
†
(2) 2 (1) (2)
K φσφ sin(2a)K cos(2a)K .
′ ′′
= = + (32)
Space algebra also enables us to describe the chirality of relativistic waves, and that is made, in the complex formalism, by the Weyl spinors. These spinors are defined by
1
05
13 13
24 24
1
; () 2
11
;. 22
U UU
ξ
ψ γγ
η
ψψ ψψ
ξη
ψψ ψψ
−
⎛⎞
= == +
⎜⎟
⎝⎠ +−
⎛⎞ ⎛⎞
==
⎜⎟ ⎜⎟
+−
⎝⎠ ⎝⎠
(33)
But we simply have
( *)
13
φ= 2 ξ σ η ; ( *)
13
ˆ2 .
φ= η σ ξ (34)
Thus, using the matrix representation of the space algebra, ξis simply the left column of φ and ηis the left column of φˆ . From (21) it is easy to write the wave equation for ξand η
∇η+ iqAη+ imξ= 0, (35)
ˆ
∇ˆ ξ+ iqAξ+ imη= 0 (36)
Chirality in the Dirac theory is linked to the existence of two different representations of the proper Lorentz group. If M is an element of the SL(2,C) group, and if V is a space-time vector
μ μ
V =σ V (37)
the transformation †
: ′=
r V 6 V MVM is a Lorentz rotation, a component of the restricted Lorentz group ↑
L+ . With
††
φ φ; ;
′′ ′
= ∇= ∇ =
M M M A MAM (38)
we have † ˆ 1
M M−
= , and 1
M M−
= , and we get
12
ˆˆ
φσ φ φ
′′ ′ ′′
∇ = m + qA (39)
We notice that the linkage between φ, ξ and η is invariant under the restricted Lorentz group ↑
L+ , because ξ and η are transformed as
† 1 * ** * 22
** 13 13
ˆˆ
; () ;
ξ ξ η η η η η σ ση
ση ση
−
′′ ′
= = = ==
′=
M MM M M
M
(40)
The Ω1 and the Ω2 are invariant:
1
1 2σ123 φφ MφφM MM φφ 1 2σ123 ,
−
′ ′ ′′
Ω + Ω = = = = Ω + Ω (41)
1
() () ,
jj j
S φσφ MS M −
′ ′′
= = (42)
While the J and ( j)
K vectors become
†† † † () ()
;.
jj j
J φφ MJM K φσφ MK M
′ ′′ ′ ′ ′ ′
= = = = (43)
These transformations under a Lorentz rotation are sufficient to prove the tensoriality of ( j)
K and S( j) . Therefore we must regard the complex formalism of the Dirac equation as very deficient.


88 Claude Daviau
B. Space-time algebra
The Clifford algebra Cl(1,3) constructed above the space-time with a signature + – – – was used by Hestenes [6], Boudet [7], Lasenby [8] to write a complete relativistic physics, and particularly the Dirac equation. It is impossible to summarize these works here. We will simply describe how we can use the matrix representations of space algebra and space-time algebra to go from one to the other. The space-time vector A reads ˆ
0
A 0
μ μ
⎛⎞
==
⎜⎟
⎜⎟
⎝⎠
AA
A γ (44)
0 5
= γ ; = 1, 2, 3; = γ
γ j j j γ (45)
The gradient μ
μ
∂ = γ ∂ reads
ˆ
0
0
⎛ ∇⎞
∂=⎜ ⎟
⎜⎟
⎝∇ ⎠
(46)
With each ( )
φ= f ψ we associate the ( )
Ψ = g φ defined by
1 2 23 3 13 4 21 5 0123 6 10 7 20 8 30
ˆ0
0
φ
φ
⎛⎞
Ψ=⎜ ⎟
⎜⎟
⎝⎠ =+ + + + + + +
a aγ aγ aγ aγ aγ aγ aγ
(47)
We notice that Ψ has value in the even subalgebra of the space-time algebra, and g is an isomorphism of the space algebra to the even subalgebra. The Dirac equation takes the Hestenes form
∂Ψγ12 = mΨγ0 + qAΨ (48)
Invariance under the restricted Lorentz group reads
; A A A;
′′
∂→∂ = ∂ → =

R R R R (49)
† 1
ˆ0
0
, ,0
0
−
⎛ ⎞⎛ ⎞
′
Ψ→Ψ = Ψ = = =
⎜ ⎟⎜ ⎟
⎜ ⎟⎝ ⎠
⎝⎠
M
M
RR R R
M
M (50)
where ~ “tilde” is reversion, defined by
;( ) .
μμ
= = 
γ γ AB BA (51)
As f and g are isomorphisms, each result in one of these mathematical frames may automatically be translated into another. But we may also use a third mathematical frame, with real matrices.
C. The algebra of real matrices
Quantum mechanics gives great importance to hermiticity and unitarity, because quantum theory always uses a Hermitian scalar product, which is, for Dirac spinors
4
1
ψψ ψ* ψ .
=
=
⎛⎞
′′
=⎜ ⎟
⎝⎠
∑
∫∫∫
i
ii i
dv (52)
This Hermitian scalar product is associated with the norm


Relativistic wave equations, Clifford algebras and orthogonal gauge groups 89
4
20
1
( *) .
i
ii i
ψ ψψ ψ ψ dv J dv
=
=
== =
∑
∫∫∫ ∫∫∫ (53)
Transposition of this norm to real algebras is very easy, because
8
02
1
.
i
i i
Ja
=
=
= ∑ (54)
And we get the norm
8
22 0 2
1
φ.
=
=
⎛⎞
=Ψ = = ⎜ ⎟
⎝⎠
∑
∫∫∫ ∫∫∫
i
i i
J dv a dv (55)
It is always possible to translate something from one formalism to another, so it is possible to transpose this Hermitian scalar product to the real Clifford algebra, by calculating the real and imaginary parts of the scalar product separately. But with a linear space above the real field a Hermitian scalar product is pure nonsense. The only scalar product naturally linked to the norm of φ is the Euclidean scalar product
8
1
φφ .
=
=
⎛⎞
′′
⋅= ⎜ ⎟
⎝⎠
∑
∫∫∫
i
ii i
a a dv (56)
But we get this scalar product very simply as
φ φ′ t ′dv
⋅ = ΦΦ
∫∫∫ , (57)
associating with each φof space algebra the real matrix
1
2
8
.
a
a
a
⎛⎞
⎜⎟
⎜⎟
Φ=⎜ ⎟
⎜⎟
⎝⎠
# (58)
With this real matrix the Dirac equation reads
( )3
3 0,
μ
μμ
⎡⎤
Γ ∂ + + Φ=
⎣⎦
qA P mP (59)
where the Γμ and Pj are
013
4
01 01 013
4
0
0, ,
0
0
I
I
γ
γ
⎛⎞
⎛⎞
Γ = Γ = Γ = −Γ = ⎜ ⎟
⎜⎟
−
⎝⎠ ⎝ ⎠
(60)
03 01
23 23 03 01
00
,,
00
γγ
γγ
−−
⎛⎞ ⎛⎞
Γ = −Γ = Γ = −Γ =
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
(61)
013 05 135 12 3 013 05 135
00 0
,, .
000
PP P
γ γγ
γ γγ
−−
⎛ ⎞⎛ ⎞⎛ ⎞
===
⎜ ⎟⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠⎝ ⎠
(62)
These matrices yield
8
2; .
jj
gI P P
μ ν ν μ μν μ μ
Γ Γ + Γ Γ = Γ = Γ (63)
The P3 matrix, with square −I8 , replaces the “i” of the classical formalism. For example, the gauge invariance of the wave equation now reads


90 Claude Daviau
1,
A AA a
q
μ μμ μ
′
→ = − ∂ (64)
3.
eaP
′
Φ → Φ = Φ (65) The real matrix formalism uses
0.
t
Φ = Φ Γ (66)
Transposing the Dirac equation, and multiplying by Γ0 by the right, with
t
jj
P = −P and 0 0
( μ)t μ
Γ Γ = Γ Γ we obtain the equation
( 3 ) 3 0.
μ μμ
⎡⎤
Φ ∂− Γ− =
⎣⎦
qA P mP (67)
Tensorial densities without derivatives are Ω1 = ΦΦ, (68)
Jμ μ ,
= ΦΓ Φ (69)
() () ,
jj
SP
μν μν
= ΦΓ Φ (70)
() () ,
jj
KP
μνρ μνρ
= ΦΓ Φ (71)
0123
Ω2 = ΦΓ Φ. (72)
This formalism immediately shows the similarity between (3) (3)
S ,K
μν μνρ
of the complex formalism and the new densities (1) (2) (1)
S ,S ,K
μν μν μνρ and
(2)
Kμνρ . But the real formalism yields much more. While with the complex formalism the algebra generated by the γμ and their products is the complete 4 × 4 matrix algebra, the matrix algebra generated by the Γμ and their products is also 16-dimensional, but it is an algebra above R and the real 8 × 8 ma
trix algebra is 64-dimensional above R , so these two algebras are not identical.
It is possible to establish that any matrix of M8 (R) may be written in only one way under the form
0 11 2 2 3 3
M = M + M P + M P + M P , (73)
where the j
M are linear combinations of the Γμ and their products. The Pj
commutes with the Γμ , while I8, P1, P2, and P3 generate an algebra isomorphic to the quaternion field. From the existence of three matrices with square −I8 which replace the indistinct “i” of quantum mechanics results that we may write, in addition to the Dirac equation, two more equations
( 1 ) 1 0,
μ
μμ
⎡⎤
Γ ∂ + + Φ=
⎣⎦
qA P mP (74)
( )2
2 0.
μ
μμ
⎡⎤
Γ ∂ + + Φ=
⎣⎦
qA P mP (75)
And so we get three similar wave equations. These equations are equivalent, since an arbitrary solution of one can be associated with a single solution of another. There is a small but very interesting difference between these three equations. In the case of the hydrogen atom, when the Dirac equation is solved, the kinetic momentum operators J 2 and J3 are diagonalized. The third axis is always used. As early as 1934 Louis de Broglie [9] showed this shocking failure of symmetry, and tried to save the Dirac theory, indicating that with a rota


Relativistic wave equations, Clifford algebras and orthogonal gauge groups 91
tion it is always possible to put the third axis into any direction. But when a rotation is made (with the Dirac equation) a multiplication is done on ψ or on φ with a left-acting matrix, which changes nothing on the third preferred axis. This number is the index of one P matrix, corresponding to a multiplication of φ by the right. After rotation, the third axis is always used again. And this third axis will always be used when the Zeeman effect is calculated: this calculation requires a magnetic field in the direction of the third axis. What is the result of a magnetic field in any other direction, not orthogonal to the plane of the trajectory? If this calculation is made, the Dirac equation does not give the experimental results. The first axis is the preferred axis for equation (74). If we solve the equation in the case of the hydrogen atom, it will be J 2 and J1 which become diagonalized. And to calculate the Zeeman effect, we must take the magnetic field in the direction of the first axis to obtain the experimental results. Evidently it is the same with (75), where the second axis is the preferred axis.
The 1 2 3
P , P , P matrices are matrices from right multiplication of φ by
23 31
σ , σ and 12
σ . They commute with the Γμ matrices, which are matrices from left multiplication of φ by σμ . Therefore they commute with Γμν , which are the generators of the Lorentz rotations. The index of the preferred axis is a relativistic invariant. But in experimental physics, a very similar situation exists. We know that in addition to electrons there are muons and tauons. A muon acts exactly as an electron, but it is not an electron, does not have the same mass, is not forced by the Pauli principle into an electron cloud. Absolutely nothing explains the existence of three kinds of electrons. Nevertheless these three kinds exist, and now the study of the Z0 boson indicates that only three kinds exist. Well, the simplest hypothesis that we can make is to associate each generation with one of the three possible indices, with one of these three possible objects with square –1, which give a Dirac equation. Separately, these objects will act in a very similar way, as these three equations are equivalent. But together the three objects will be different, because if we make a rotation, it is impossible to put a direction into the third and the first axis at the same time. Therefore, the spin of one cannot be added to the spin of the other. A muon decays into an electron by emitting two neutrinos: one has the muonic preferred direction, while the other carries the preferred direction of the electron family. From our hypothesis, we cannot know if the muon’s family is the first, the second or the third family. But since inversion between a first and a second axis is a spatial symmetry, we should not be astonished if chirality, the difference between left and right, plays a fundamental role. The 64 matrices ΓP —where Γ is a product of matrices Γμ and where P is I8 or one of the Pj —form a basis for M8 (R) . This basis splits into two sub
sets: 36 have square I8 , yield t 1
M M M−
= = and give the 36 tensorial densi


92 Claude Daviau
ties Φt M Φ ; 28 have square −I8 , yield t 1
M M M−
= − = , do not give tensorial densities, but generate the Lie algebra of the orthogonal group SO(8). This orthogonal group is a global gauge group of the Dirac equation, since (59) is invariant under the transformations
1
, t,
A AA
−
′
Φ → Φ = Φ = (76)
1,
μ μ μ−
′
Γ → Γ = AΓ A (77)
1
3 3 3 −.
′
→=
P P AP A (78)
In quantum mechanics, replacing unitarity by orthogonality is a heresy, which should immediately lead to catastrophic results. For example, solving the Dirac equation in the case of the H atom, we orthonormalize the different solutions corresponding to the different possible quantum states, and we use this orthonormalization when we calculate the Zeeman effect. But there is a particular coincidence, and the orthonormalization for the Hermitian scalar product is exactly identical to the orthonormalization for the Euclidean scalar product (56) in this case [10].
3. New relativistic wave equations
A. Chiral wave with mass for the neutrino
One of the most unexpected discoveries of particle physics was parity violation by weak interactions, and even maximal violation of this parity. This violation led some to think that a wave of neutrinos is purely chiral and that charge conjugation reverses chirality. The charge conjugate of the left neutrino is the right antineutrino. The chirality of relativistic waves may be described with Weyl spinors, and we have seen that they transform as ˆ
ξ ξ; η η.
′′
==
M M (79)
We obtain them in the complex formalism by considering the left part ψL and the right part ψR of a Dirac spinor ψ:
() ()
45 45
11
;.
22
ψ = −γ ψ ψ = +γ ψ
LR
I I (80)
Translation into real formalisms is
( ) ()
( ) ()
3
3
1ˆ
1 , 2 0,
2
11 2 0,
2
φφ σ φ η
φφ σ ξ
=− =
= +=
LL
R
(81)
() ()
30 30
11
1γ , 1γ ,
22
Ψ =Ψ − Ψ =Ψ +
L R (82)
() ()
0123 0123 83 83
11
,.
22
Φ = −Γ Φ Ψ = +Γ Φ
LR
I P I P (83)
The Dirac equation gives (35) and (36); the mass term links ξ with η. Consequently we have only two possibilities: either a mass term exists and the wave


Relativistic wave equations, Clifford algebras and orthogonal gauge groups 93
has a left and a right part, or the wave is purely chiral, with only a left or a right part, and the mass term must be zero. It is easy to give a relativistic chiral wave equation, with only a left wave and a mass term [11]:
2
ˆ.
LL
∇φ = mφσ (84)
This equation gives for the left column, which is the Weyl spinor η: ∇η= mσ2η*. (85)
Conjugating, we get
2
ˆ
ˆ.
LL
∇φ = −mφσ (86)
And for the second order we get
() 2
φ = ∇∇φ = − ∇φσ2 = − φσ2 σ2 = − φ ,

,
LL L L L
m m m m (87)
(,+ 2 )φL = 0.
m (88)
To obtain the plane wave solutions we let
() ()
12
cos sin ,
μμ μμ
φ= φ + φ
L LL
p x p x (89)
where φ1L and φ2L are fixed left terms. With p μp
μ
=σ , the wave equation is equivalent to
1 22
ˆL L
− pφ = mφ σ (90)
2 12
ˆ.
LL
pφ = mφ σ (91)
The first equation gives
2 12
ˆ
LL
p
m
φ = − φ σ (92)
and substituting in the second equation we obtain
( 2)
12
ˆ
ˆ 0.
− φσ =
L
pp m (93)
A not identically null wave results only if
2 02 2 2
ˆ , ( ) () ,
pp = m p − p = m
G (94)
which is the relativistic condition between mass and impulse. If this condition is found, φ1L is anything and φ2L is given by (92). With space-time algebra, the wave equation (84) reads
2
ˆ0
;.
0
φ
φ
⎛⎞
∂Ψ = Ψ Ψ = ⎜ ⎟
⎜⎟
⎝⎠
L LL L
L
m γ (95)
It is possible to build a Lagrangian formalism for this wave equation. But to get this Lagrangian, we must consider in the same time a left wave ΨL and a right wave ΨR . We shall use here the method explained by Lasenby in [12]. <A> is the scalar part of a multivector A. The Lagrangian density is
= ∂Ψ 1Ψ + Ψ 12Ψ .

LR L R
L γ m γ (96)
Lagrangian equations are
()
Ψ ∂Ψ
∂ =∂ ∂

LL
L L , (97)