zotero/storage/N6G5E6L2/.zotero-ft-cache

The visible emission spectra of iodine and bromine monofluorides 395

R eferences

Badger, R. M. & Yost, D. M. 1931 Phys. Rev. 37, 1548.

Brodersen, P. H. & Schumacher, H. J. 1947 Naturforsch. 2 a, 358.

Brown, W. G. 1932

Phys.Rev. 42, 355.

Brown, W. G. & Gibson, G. E. 1932 Phys. Rev. 40, 529.

Durie, R. A. 1950 Proc. Phys. Soc. 63, 1292.

Gibson, G. E. & Ramsperger, H. C. 1927 Phys. Rev. 30, 598.

Ruff, O. & Ascher, E. 1928 Z. anorg. Chem. 176, 258.

Ruff, O. & Braida, A. 1933 Z. anorg. Chem. 214, 81.

Ruff, O. & Braida, A. 1934 Z. anorg. Chem. 220, 43.

Wahrhaftig, A. L. 1942

J .Chem. Phys. 10, 248.

Yost, D. M. 1931 Z. Phys. Chem. 153, 143.

Description of plate 3
Figure 1. IF emission spectrum (Raman glass spectrograph; Ilford long-range spectrum plate—exposure 10 min. top, min. lower—iron arc comparison).
Figure 2. Rotational structure in IF bands. Note the overlap of the P and R branches in the 0, 4 (6031-2A) band (Glass Littrow spectrograph, H.P.3 plate, exposure 6 hr.—iron arc comparison).
Figure 3. BrF emission spectrum (Raman glass spectrograph, Rapid Process Panchromatic plate. Exposure (a) 10 min. (6) 1 min.—iron arc comparison).

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The renormalization method in quantum electrodynamics
B y F. J. D y so n , Warren Research Fellow Department of Mathematical Physics, University of Birmingham
( Communicatedby R. E. Peierls, F.R.S.—Received 23 January 1951)
A new technique has been developed for carrying out the renormalization of mass and charge in quantum electrodynamics, which is completely general in that it results not merely in divergence-free solutions for particular problems but in divergence-free equations of motion which are applicable to any problem. Instead of using a power-series expansion in the whole radiation interaction, the new method uses expansions in powers of the high-frequency part of the interaction. The convergence of the perturbation theory is thereby much improved. The method promises to be especially useful in applications to meson theory.
The present paper contains a preliminary and non-technical account ofa new method of handling problems in quantum electrodynamics. A full account of the method will be published in a series of papers of which the first only (Dyson 1951) is yet written. * That paper was occupied with a formal mathematical analysis of some expressions which arise in the matrix elements of Heisenberg operators. The analysis yielded a general rule by which any Heisenberg operator can be split into a sum of terms each of which has a simple structure. The results of th at paper, although mathematical and not physical in character, are an indispensable tool in the successful development of the physical ideas which will now be introduced.
* That paper, with the title ‘Heisenberg operators in quantum electrodynamics, I ’, is referred to hereafter as HOI.

396

F. J. Dyson

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These new ideas go considerably beyond the programme, outlined in the intro­ duction to HOI, of proving the finiteness of Heisenberg operators after renormaliza­ tion. The programme is now widened, so th a t the objective is a proof of the total disappearance of all divergences from quantum electrodynamics after the dynamical variables have been transformed by a suitable contact transformation. A unitary operator will be explicitly constructed which, when applied to the state-vector in the Tomonaga-Schwinger theory, gives a state-vector satisfying divergence-free equa­ tions of motion. The new equations of motion will describe exactly the behaviour of all electrodynamical systems, bound states being no longer excluded as they were from the ^-matrix formalism (Dyson 1949). Furthermore, it is intended th a t the new divergence-free formulation of the theory shall be practically useful and adaptable to approximate numerical calculations. Within this widened programme, the proof of the finiteness of Heisenberg operators will appear as a special and important limiting case.
I t is not surprising th at quantum electrodynamics can be transformed by a single contact transformation into an explicitly divergence-free theory. For if this were not possible, the successes ofthe theory in giving finite answers to numerous problems would be hard to understand. I t is also easy to foresee, from simple physical considerations, the general form of the required transformation. In the present introductory paper, the physical and heuristic ideas underlying the transformation method are briefly explained. Later papers will be concerned with the detailed mathematical verification th at a transformation operator of the kind indicated by heuristic principles in fact does all th a t is expected of it.
One may imagine a physical picture, which includes an intuitive explanation for the success of the renormalization technique in quantum electrodynamics, roughly as follows. The two interacting fields, the electromagnetic and the m atter field, are two characteristic properties of a single ‘fluid’ which fills the whole of space-time. The two fields are defined a t every point, like the velocity and stress in a fluid in classical hydrodynamics. The fluid is in a state of violent quantum-mechanical fluctuation, the fluctuations becoming more and more noticeable as the region over which they are observed is made smaller. The fluctuations have the property th a t a t sufficiently high frequencies and in sufficiently small regions they are essentially isotropic and uniform over the whole of space-time, like the fluctuations of a classical fluid in a state of isotropic turbulence. Statements concerning the behaviour of the fluid at a particular point are observationally meaningless; the description of the fluid in terms of operators defined a t points is possible only in a formal sense and involves mathematical divergences. However, because of the isotropy and uni­ formity of the fluctuations, the macroscopic properties of the fluid are observable and well defined. A divergence-free description of the fluid will be obtained as soon as its behaviour is expressed entirely in terms of new dynamical variables which are averages of the original instantaneous variables over finite intervals of time.
In accordance with the foregoing crude picture, it is found mathematically th a t the successful removal of divergences in quantum electrodynamics by renormalization is always associated with an averaging-out of high-frequency fluctuations of the fields. The averaging-out is achieved by integrating the equations of motion of the fields

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The renormalization method in quantum electrodynamics

397

explicitly with respect to the time. Thus in the Schwinger theory (Schwinger 1949 a,b) the state-vector of the system is transformed so th at the new state-vector refers to the behaviour of the system at a time which recedes in the limit into the infinite past. In the Feynman theory (Feynman 1949a, b, 1950) the description of events is directly in terms of an over-all space-time picture in which localization of processes in space and time is abandoned. In both theories, the time-averaging is performed not over a finite time-interval but over an infinite time. Averaging over an infinite time is also implicit in the definition of Heisenberg operators in HOI, because the multiple integrals in the series expansions of the operators extend into the infinite past.
Now it is precisely the averaging over an infinite time-interval which has hitherto introduced into every discussion of the renormalization method the two limitations mentioned in the introduction to HOI. Those limitations are: (i) because of the way in which the initial and final states are described, the customary $-matrix formalism ,is not applicable to problems involving bound states, (ii) the renormalization method has always been confined to quantities which are expanded as power series in the radiation interaction, while in many situations such expansions are demonstrably not convergent. The two limitations, of which the second is the more fundamental, are closely related. I t is reasonable to hope th a t the theory can be freed from both limitations, if the averaging over infinite time-intervals is dropped and the removal of high-frequency fluctuations is accomplished by integrating the equations of motion over finite intervals.
That is to say, the transformation which leads to divergence-free equations of motion may be expected to be of the following type. The original state-vector T of the interaction representation is replaced by a new one O according to

¥(<) = # ) 0 (0 ,

(1)

where t is the time and 8(t) is a unitary operator. The choice of S is guided by two

principles, (i) O is to be a smoothed-out average of T over a finite time-interval, or

in other words 8 is to follow accurately the high-frequency fluctuations of VF but not

the slow long-term variations, (ii)

8sito be a power-series exp

radiation interaction but only in the high-frequency fluctuating part of the inter­

action. By ‘high-frequency fluctuations’ are here meant Fourier components with

frequencies higher than a certain standard frequency which may be chosen arbi­

trarily. The standard will in general be chosen differently for different problems. I t

will probably be convenient to make it a little higher than the highest frequency th at

is physically im portant in a particular problem. Then the high-frequency p art of the

interaction is ineffective except in producing renormalization effects, and the

expansion in powers of the high-frequency interaction may be expected to converge

after the renormalizations have been carried out. In this series of papers no attem pt

will be made to prove the convergence. I t is plausible, and in accordance with the

original philosophy of the Schwinger theory, th at the high-frequency interaction

produces only small physical effects, and th a t an expansion in powers of the high-

frequency interaction should be rapidly convergent and convenient for practical

calculations.

A trial definition of 8, satisfying the above requirements, can now be .formulated.

^

H ^ H ^ t ) (2)

Vol. 207. A.

26

398

F. J. Dyson

be the radiation interaction appearing in the Tomonaga-Schwinger equation, inte­

grated over all space a t a given time t. According to the results of the earlier ^-matrix

analysis (Dyson 1949),

Hxdepends upon two divergent constants e

formal power-series (with divergent coefficients) in the finite exwhich is the physically

observed electronic charge. Thus Hxis supposed to be expressed explicitly as a power-

series in el5the higher terms representing the effects of mass and charge renormaliza­

tion. Let a function g(a) of the positive real variable a be chosen with the properties:

g r ( 0 ) = 1, g(a) -^Oasa->oo, and g(a) varies smoothly as a varies from 0 to 00. Consider

a fictitious world in which the radiation interaction a t time t', instead of being given

by (2), is for t' < t given by

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) j ^ )•

(3)

The fictitious world is one in which the charge ex of the electron is smoothly, but

not adiabatically, varied, rising from zero in the remote past to its correct value a t

time

t.In the fictitious world, there is a unitary transformation operator trans

forming the state-vector representing a system without interaction a t

00 into

the state-vector representing the same system with interaction at

this trans­

formation operator* is S(t). In other words, 0(£) is the state-vector a t

00

which, developing with time t’in the fictitious world, coincides with the actual state

T(<) of the system in the real world at time t.

The definition which will finally be adopted for S(t) is essentially th a t given above.

Some modifications have to be made in the definition of Hg, in order to compensate

a transient photon self-energy effect which appears while the charge ex is being varied, f Also, some more restrictive conditions will be imposed on the function g.
In the specification of g there will necessarily appear some constant T with the

dimension of a time. Then T~x is the standard frequency defining the division of frequencies into high and low. Consider first the meaning of the two limiting cases

T -> 0 and T->co. In the limit T ->0, all frequencies are considered as low, g(a) = 0

and S is the identity operator. In this case O is the state-vector of the interaction

representation, there is no smoothing-out of the fluctuations of T , and there is no

removal of divergences. In the limit 00, all frequencies are considered as high,

g(a)= 1, and $ satisfies the same equation of motion as T*. In this case O is the state-

vector of the Heisenberg representation and is constant in time, all the fluctuations

of T* have been smoothed out, the formal removal of divergences is complete, but

S is an expansion in powers of the whole interaction which is generally not con­ vergent. When T is given a finite value, the situation is intermediate between the

two limiting cases. The representation in which O is the state-vector will be called

the ‘intermediate representation’, meaning th at it is intermediate between the

interaction and Heisenberg representations. An expression of the form

Q M = S - Ht) Q(t ) S (t)4( )

* The operator S(t) has been previously studied in a series of papers by Ferretti (1950 a, 6, c,d).
t The author is deeply indebted to Mr Abdus Salam for suggesting to him the correct definition to use for H g. This suggestion was based upon an unpublished treatm ent of the renormalization technique by S. N. Gupta.

Therenormalization method in quantum electrodynamics

399

where Q(t) is an interaction representation operator referring to the time t, will be called an intermediate representation operator. In the intermediate representation, high-frequency fluctuations of a system are described by the field operators as in the Heisenberg representation, while low-frequency processes are described by the state-vector as in the interaction representation.
The programme ofthis series ofpapers is to prove th at the intermediate representa­ tion provides a complete divergence-free formulation of quantum electrodynamics. The programme is divided into three parts, the first of which is an immediate general­ ization of the programme of the introduction to HOI. Let Q(r, t) be a field-operator in the interaction representation defined a t the point (r, t), for example, an electro­ magnetic field component. Let Qg(r,t) be the corresponding intermediate repre­ sentation operator defined by (4). Let B(r, t) be a scalar function of the point (r, t), vanishing outside a finite space-time region, and satisfying certain requirements of continuity. Then the operator

^Qg{r,t)R{r,t)dzrdt(5)

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is called an intermediate representation field-average; such operators represent a general class of locally defined physical quantities which are in principle precisely measurable. The first part of the programme is to prove th at all matrix elements of intermediate representation field-averages are finite after the renormalizations of mass and charge are carried out.
The second part of the programme is to prove th at the state-vector in the inter­ mediate representation satisfies a divergence-free Sch^odinger equation. The Schrodinger equation in the intermediate representation is

ih(d®ldt) = H'(t) O,

(6)

where by (1) the Hamiltonian is

H'it) =

-« !]« (< )■

(7)

The objective is to prove th at the operator H' is divergence-free. The physical importance of the operator H' is made clearer by transforming back
to the Schrodinger representation, in which all operators are independent of time. If H0is the Hamiltonian of the non-interacting fields, then

S(t) = exp (iH0tlh) S exp ( —iH0t/h),

(8)

where $ is a constant Schrodinger representation operator. The state-vectors and O0, defined by

T*o(0 =. exp (

- iH 0tlh)'¥(), O0(£) = exp

0 (<), (9)

are related by the time-independent unitary transformation

Y 0(«) =

(10)

Now T*0is the state-vector in the standard Schrodinger representation of quantum electrodynamics with Hamiltonian (Hq+ H^. In view of (10), <t>0 may equally well

400

F. J. Dyson

be regarded, as the state-vector of the system in the .Schrodinger representation, satisfying a Schrodinger equation

ih(d%/dt)= (H0+ H') <D0,

where H' is the time-independent operator corresponding to the time-dependent

The identity

H<s+H ' =

+

H

, ) (12)

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from which (11) is derived, is equivalent to (7).

The transformed Schrodinger equation (11) gives a complete description of the

behaviour of all systems in quantum electrodynamics, but is especially appropriate

to the treatm ent of bound states and eigenvalue problems. The proof of finiteness

of the operator H' will enable all such problems to be formulated in finite terms.

The third part of the programme is to devise methods for the approximate solution

of the Schrodinger equation in practical situations, especially in circumstances where

expansions in powers of the whole radiation interaction are forbidden. An important

advantage of the intermediate representation is its flexibility, due to the fact th a t

the function

gmay be left unspecified until the third part ofthe programme

When a particular problem is attacked, it should be possible to minimize the labour

involved in calculating the solution by making an appropriate choice of g. The

change from one g to another is equivalent merely to a finite contact transformation

which leaves all physically observable quantities invariant.

I t is hardly necessary to repeat here the remarks made earlier (Dyson 1949)

concerning the manipulations of infinite expressions which are involved in the

renormalization method. The original equations of motion of quantum electro­

dynamics, and the ^-operator itself, contain numerous divergent quantities; the

equations of motion become divergence-free only after the infinities have been

cancelled out as a result of formal analytical manipulations. The transformation of

the theory by means of the ^-operator is not a mathematically rigorous operation;

the method must be justified a posteriori by the fact th at it yields well-defined and

physically reasonable results. I t is regrettable th at the S-transformation is not only

divergent but also entirely non-covariant in form. The intermediate representation

conceals both the Lorentz invariance and the gauge invariance of the theory. This

lack of apparent Lorentz and gauge invariance does not, however, interfere with the

consistent carrying out ofrenormalizations, and the practical inconvenience resulting

from it is much less than might have been expected.

Though the methods discussed in this paper are here applied to quantum electro­

dynamics, they can be transferred without serious modifications to meson theory,

whenever the meson theory is such as to give a divergence-free ^-matrix. The

removal of divergences from the ^-m atrix, using the renormalization technique, has

recently been carried through by Salam (1951) for the meson theory of greatest

practical importance, the theory of charged pseudo-scalar mesons interacting with

a nucleon field and with the electromagnetic field.* The formal possibility now

therefore certainly exists, of constructing a divergence-free formulation of pseudo­

* The removal of divergences in charged meson theory is, of course, very much more com­ plicated than in electrodynamics.

The renormalization method in quantum electrodynamics

401

scalar meson theory, the results of which might be compared unambiguously with experiment. Such a divergence-free formulation will be a practical possibility, if the expansion of operators as power-series in the high-frequency part of the interaction is found to be convergent even when the coupling constant is large. On the basis of rough estimations, it seems probable th a t the expansion will be sufficiently con­ vergent, if the function g is chosen suitably. The intermediate representation method thus already offers good prospects of overcoming the mathematical obstacles which have so long delayed a decisive verdict for or against the pseudo-scalar meson theory as a workable theory of nuclear phenomena.

The author is indebted to The Royal Society, the University of Michigan and the Institute for Advanced Study, Princeton, for financial support and hospitality. He wishes to thank Professor W. Pauli, Dr Res Jost, and above all Mr Abdus Salam, for much help received in discussions and correspondence.

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R eferences

Dyson, F. J . 1949 Phys. Rev. 75, 1736.

Dyson, F. J . 1951 Phys. Rev. 82, 428.

Ferretti, B. 1950a Nuovo 7, 79.

Ferretti, B. 19506 Nuovo 7, 375.

Ferretti, B. 1950c Nuovo Cim. 7, 783.

Ferretti, B. 1950^ Nuovo

7, 899.

Feynman, R. P. 1949a Phys. Rev. 76, 749.

Feynman, R. P. 19496 Phys. Rev. 76, 769.

Feynman, R. P. 1950 Phys. Rev. 80, 440,

Salam, A. 1951 Phys. Rev. 82, 217.

Schwinger, J . 1949 a Phys. Rev. 75, 651.

Schwinger, J . 19496 Phys. Rev. 76, 790.