zotero/storage/KRU4DG9M/.zotero-ft-cache

Progress of Theoretical Physics Vol. I, No.2, Aug.-Sept., 1946.
On a Relativistically Invariant Formulation of the Quantum Theory of Wave Fields.*
:By S. TOMO~AGA
(Received "May 11, 19(6)

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

§1.. The formalism of the ordinary quantum theory of wave fields.

Recently Yukawa(l) has made a comprehensive consideration about the basis of the quantum ,theory of wave fields. In his article he has pointed out the fact that the existing formalism of the quantum field, theory is not yet perfectly relativistic.
Let v(.:r)'s) be the quantity specifying the field. and l(.t'yz) denote 'its canonical conjugate. Then the quantum theory requires the commutation relations of the form:

J[v(xyzt), v(.ry'z't)]=[l(xyzt), l(.:rJlz't)]=O
hv (xyzt) , l (x'y'z't) ]=iM(x-x')8(y-y') 8(z-zl),

(1)**

but these have quite non-relativistic forms. The equations (I) give namely the commutation lelations between the
quantities at different- points (xyJ') and (.-r'y'z') at the same instant of time ,to The concept .. same instant of time at different points" has, however, a definite meaning only one specifies some definite Lorentz frame of reference. Thus this is not a relativistically invariant concept.
Further, the Schrodinger equation for the tfJ-vector representing the state of the system has the .form;

(2)

• Tnnsla.red ftom the paper, BulL I. P. C. R. (Ribn.·iho),U (1S43), MS,appeared ctriginally in Japanese. •• [A, Bl=.JB-B.{. We assulllC that the field oheys the Bose statilltics. Our consi. derations apply 'allio to the QliC of Fermi statistic5.

28

S. TOMONAGA

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

where jj is the operator representing the total energy of the field which is given bytIie space integral of a function of v and A. As we adopt here the Schrodinger picture, v and A are operators independent of time. The vector representing the state is in this picture a function of the time,and its dependence on t is determined by (2).
Also the differential equation (2) is no less non-relativistit. In this equation the time variable t plays' a role quite distinguished from the space coordinates x, y and z. This situation is closely connected with the fact that the notion of probability amplitude does not fit with the relativity theory.
As is well known, the vector l' has, as the probability amplitude, the
following physical meaning: Suppos.e the representation which makes the field quantity v(xyz) diagonal. Let 1'[v' (xyz)] denote the representative of
l' in this representation.* Then the representative ¢'[v' (x)',.;) ] is called
probability amplitude, and its absolute square

W[v'(xyz)]= l ¢[v'(xyz)] j2

(3)

gives the relative probability of v(x)'.$') having the specified functional form v' (x)'z) at the instant of time t. In other words: Suppose a plane** which is parallel to the x)'z-plane and intersepts the time axis at t. Then the probability that the field has the specified functional f9rm v' (xyz) on this plane is given by (3).
As one sees, a plane parallel to the xyz-plane plays here a significant role. But such a plane is only defined by referring ·to a certain frame of reference. Thus the probability amplitude is not a relativistically invariant concept in the space-time world.

§ 2. Four-dimensinal form of the
commutation relations.
As stated above, the laws of the quantum theory of wa\l'O! fielos are
'" We use the square blackets to indicate a functional. Thus tJI[v'(xyz)] means that tJI
is a functional.of the variable function '/l(,..),.8). When we use ordinary blackets ( ), as tP(v'(xys», we consider .p as an ordinary function of the function v'(xys). For example: the energy density is written 'as H(v{xya), A(xy.z» and this is also a function of x, y
and s, whereas the total energy H = SH('Z{xyz), JJ..iys»)dv is a functional of v(xyz) and
*'"A(xyz) and is written as H['Z{xyz), A(xyz)]. We call a three·dimensional minifold in the four·dimenstonal space-time world simply " surface ".

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

On a Relativistically Invariant Formulation

29

usuany expressed as mathelllatical relations between quat ties having their me~nlDgs only in some specified Lorentz frame of referenc¢. But. since it is pro'(~ed that,the whole contents of the theory are of cource relativistically invariant, it must be certainly possible to build up the theory on the basis of concepts having relativistic space-time meanings. Thus, in his consideration, Yukawa has required with Dirac(2) to generalize the notion of probability amplitude so that it fits with the relativity theory. We shall now show below that the generalization of the theory on these lines is in fact possible to the relativistically necessary and"sufficient extent. Our results are, however, not so general as expected by Dirac and by Yukawa, but are already sufficiently general in so far as it is required by the relativity.theory.
Let us suppose for simplicity that there are only two fields interacting with each other. The case of more number of fields can also be treated in the same way. Let· 'l'1 and V2 denote the quantities specifying the fields. The canonically conjugate quantities be Al and ).2 respectively. Then between these quantities the commutation relations

(4)

must hold. The 4Jl-vector satisfies the Schrodinger equation
(5)
In this equation HI and ~ mean respectively the energy of the first and
the second field. HI is given by the space integral of a function of VI and
AI. il,. by the space integral of a function of V2 and A2• Further, ~~2 is the
interaction energy of the fields and is given by the space integral of a function of both Vlt At and Z'2. A2• We assume (i) that the integrand of H 12, i. e. the interaction-energy density, is a scalar quantity, and (ii) that the energy densities at two different points (but at the same instant of time) commute with eath other. In general, these two facts follow from the single assumption: the interaction term in the Lagrangean does not contain the time derivatives of VI and' V2.
If this energy density is denoted by H12• then we have

30

S. TOMONAGA

Hl!l= f Hl'J!ix d)' dc.

(6)

As we adopt here the Schrodinger picture, the quantifies 'v and A in Hf.,
Hz and HI2 are all operators independent of time. Thus far we have merely summarized the well known facts. Now, as
the first stage of making the theory relativistic, we suppose the unitary operator

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

(7)

and introduce the following unitary transformations ot v and ~. and the cor-
responding transfo}:mation of t/J :

fv,. = ll'llrU-t, Ar= U).rU-1

1 lJI'=Ut/J.

(8)

As stated above, v and ). in (5) are quantlties independent of time. But V and A obtained from them by means of (8) contain I through U. Thus they depend on I by

1~1t~= v,.~-~Vr ,.=1, 2

(9)

i1tAr =ArH,.- HrA...

These equations must necessarily have covariant forms against Lorentz transformations, because they are just the field equations for the fields when they are left alone without interacting with eath other.
Now, the solutions of these "vacuum· eguations ", the equations which the fields must satisfy when they are left alo~e, together with the commutaion relations (4), give -rise to the relations of the following forms:

[v,,{x)'ct) , V.(x';ls't')~=4r.(x-%, y-;/, c-s' I-I') {[Ar(%)'zt), A,(%I,sI/)]=Br~(x--%, :y-y', s--r,t-.t') (10)
[v,,(%,Pst), A,(%)ls't')]=c",(x-x', )'-)1, s-s'J t-i}
c... where Am Er• and are functions which. are combinations of the so-cal-
led four-dimensional 4~functions and their derivatives. CJlOne denotes usually these four dimensional cf-functions by D,..(x)'zt), ,.=1, 2.. They are defined by

0" (l Rllativ4tically Inaaria111 For1ltulation

81

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

with (12)
x,. being the constant characteristic to the field f'. It can be easily proved
that these functions are relativistically invariant.*
Since (10) gives, in contrast with (4), the commutation relations between the fields at two different world points (x)'zt) and (x'y:ll), it contains no more the notion of same instant of time. Therefore, (10) is -sufficiently relativistic presupposing no special frame of reference. We call (10) fourdimensional form of the commutation relations.
One property of D(xyst) will be mentioned here: When the world paint (X)'B'I) lies outside the light cone whose vertex is at the origin, .then D(x)'zt) vanishes identically:
(13)
It follows directly from (13) that, if the world point (x';I:lI) lies outside the light c 'me whose vertex is at the world point (x)'st), the right-hand sides of (10) always vanish. In words: Suppose two world points P and pl. When these points lie outside each other's light cones, the field quantities at P and field quantities at P commute with eath other.

§3. Generalization of the Sehrodinger equation.
Next we observe the vector 7Jf obtained from t/J by means of the unitary transformation U. We see from (5), (7) and (8) that this 7J1', considered as

• Suppose that a &urface in the R" ky ..t. k-space is defined by means of the equation

AIl=AIl", +AIl, +AIl. +K!_ Then this surface has the invariant'meaning in this space, since

Y+( ::)'i+(- ::. r + All,. All, + Il'. - All is invariant against Lorentz transformations. The al'ea of the ~ur-

face element of this surface is given by dS=J(-:!

-1 dk", dk/ldk.

=.K dk,. d;v dk. _ Now, since dS has the invariant meaning, we can thus conclude that

d..t" d;" d..t. is an invariant, and tUis results that the function defined by (11) is invariant..

82

s. TOMONAOA

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

a function o( I.. satisfies
{JH,I( P; (~y.~t). A, (x)'!!!). JI';(x)'3'!). AI(~J'St) )tlx d)' t/g
+ ~ :/}" =0. (14)
One sees that t plays also here a role distinguished from x, )' and II: also here a plane parallel to the x)'z-plane has a special significance. So we must in some way rem<we this unsatisfactory feature of the theory.
This improvement can be attained in the way similar to that in which Dirac(4) has built up the so-called many-time fonnalism of the quantum mechanics. We will now recall this theoty.
The SchrOdinger equation (or the system containing N charged particles interacting with the electromagnetic field is given by
3} { H- ol+ ~N H..(q... I .., a (q..» +Tt ·W 1'=0. (15)

Hert'" HOI means the energy of the electromagnetic field, "" the energy of the tl-th particle. H.. contains, besides the kinetic energy of the n-th particle, the interaction energy between this particle and the field through 0(;0,,), q.. being the coordinates of the particle and a the potential of the
field. I.. in (15) means as usual the momentum of the n-th particle. We consider now the unitaty operator

(16)

and introduce the unitary transformation o( a :

~=uau-l

(17)

and the corresponding transformation of "':

(18)

++ :, Then we see that ~ satisfies the equation {~H..(q..,/Il, ~(q.., t»

}dJ=O.

(19)

In contrast with Or which was independent of times (Schriklinger picture), ~

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

On a Relativistically IttfJa"ianl FfWmulation

ss

contains I through tl. To emphsize this, we have written I expliciteiy as

argument of~. We can prove that 2'( satistles the maxwell equations in

vacuum {accurately speaking, we need special considerattons for the equa-

tion div (f:::;0).

The equation (19) is the stalting pomt of the many-time theory. In

this theory one introduces then the function (J)(qlfh qs"s, ''', qN, IN) contaln-

ine- so many time variables 110 t2• •••••• IN as the number of the particles in

place of the function fP(qlo qt•... , q,N' I) containing only one time variable,*

and. suppose that this tP(qltlo qS'2..... qR"N) satisfies slmultaneously the fol-

lowingN equations;

+! ;'jfP(qJt {n. (q,., Pi.. ~(q.., I..)

1• q2t2' .... q~N) =0

n=l, 2...., N.

(20)

This '(tI ,4, ... , 'N)~ whioh is a fundamental quantity in the many-time theory. is related to the ordinary probability amplitude lP(t) by

lP(t)=iP(t. I, ..., t).

(21)

Now, the simultaneous equations (20) can be solved when and only when the !vI conditions

are satisfied for all pairs of n and n~. ' If the world point (qntu) lies outside the light cone whose vertex is at the point (q..'.t.:). we can prove H ..H..' -0':H,.=O. As the result, the function satisfying (20) Can eXIst in the region where
(23)
is satisfied simultaneously for all values of nand ,,', According to Bloch(&) we can give (j)(qltIQ2t2' ...• qNlN) a physical
meaning when its arguments lie in the region given by (23). Namely

gives the relative probability that one finds the value ql in the measurement of the position of the first particle at the instant of time 111 the value q2 in
* Here we suppose the representation which makes the coordinates 911 IJ" .... fiN dia-
gonal. Thus the vector ~ is represented by a function of these coordinates.

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

34

S. TOMONAGA

the measurement of the position of the second patticll! at the insta'l't ~f

time 4, ... and the value tjN in the .measurement of the POSltiott 01 the N-

th particale at the instant of time tAo

This is the outline of the many-time formalism of the quantum

mechaUlcs.

We will now return to ol1T main subject. If we compare oUt' e<RJation

(14) WIth the equation (19) of themany-ttme theory, we notice a marked

similarity between these two equations. In (19) stands the snffix n, which

designates the particle, while in (14) stand' the-variables x, o"P and S.I whiell

designate the position in space~ Further. tP is a function of the N inde~ndem

variables tj1.tj:. • ... tjN. tj" giving the position of the 1t-th particle. while'"

is a functional of the infinitely many II independent variables" Vl(X)'S) and

V~ (x)'s), Vi (x)'s) andv2 (.t')'S:) giving the fields at the position (x)'z). Cor-
. responding to the sum ')JH.. in (19) the i.ntegral IflJ.prdJltlsstands in (14).
In this way. to the suffix 11 1n(19) which takes the values 1. 2, 3, ... , N

correspond the variables x. )' and s whlce take continuously all values from

+ -00 to

00-

Such a similarity suggest,; us to introduce infinitely many time variables
t"'lIn whi"ch we may call local time * each for one position (x)'3') in the space

as we have introduced N time variables. particle times, t10 ft•...• tN. each

for one particle. '{he only difference 'Consist in that we use in our case

infinitely many time variables whereas we have used N time variables in

the ordinary many-time theory

Corresponding to the tranSition from the use of the func"tion with one

time variable to the use of the function of N time variables, we must now

consider the transition from the use of 'I" (t) to the use of a functionaIfF[t,.,.}

of infinitely many time variabfes t"IJ1'

We regard now !"'IP as a function of (xy3') and consider its variation
,"US which differs from zero- only in a small domain Vo in. the neigllbouf'o

hood of the point (Xo.)'o8'o). We will define the partical differential coeffi-

cient of the functional W[t"us] with respect to the variabre 'zouoro tn the

following manner:

(25)

• The notion of local time of tbis kind bas been occasionary introduced by Stuecker· bergJ6)

85

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

We ten generalize (14), and regard

m.,. 1j~t(x-, ".~, t) +-,!. .!>~ 1f 1'=0

(26)

the infinitely many simultaneous eqaations eorresponding to the N equatiolls (20), as the fundamental equationS of our theory. In (26) we have written, for simplicity. JhJ~' y, fI, t) in place of li1'J(v,,(X:JlfI, I). V';(xy..c. I), ......)k In general, when we have a 'function P(P: A) of V and A.. we will write
simply F(z,)I, fI, t) for F(V(xyz'~J')' A(X)'3", t.I/.))' or still simpler F(P)
P denoting the world point with the coordinates (xy::;-, tig.)' Thus F(P) means F(x'~y', z'J I} or, more precisely, F(V(x';lz!. 1."pI.o')' (#;1:1, "'y'~'))'
We will now adopt the equation (26) as the basis of our theory. For v,,(P). V.(~t, .A1 (P) and At(P) ilt Hj2 the commutation relations (10) hold. where D(xygt) has the property (13). As the consequence, we have

(27)

when the point P lies a finite distance apart 'from pI and outside the light «>ne whose vertex is at P. Further, from our assumption (ii) the relation (27) holds also when P and P' are two adjacent points approaching in a space-like direction. Thus our system of equations (26) is integrable
when the surface defined by the equations t=t.."., considering t",v' as a func-
tion of x, y and fI, is space,.like. In this way, a functional of the variable surfilCe in the spate-time world
is determined by the .functional partial differential equations (26). Corresponding to the relation (21) in case of many-time theory. W[t..,1.] reduces to the ordinary- IT{I} when the surface reduces to a plane parallel to the "ys-plane.
The dependent variable surface I=t"'/I' can be of any (space-like) form in the space~time world, and we need not presuppose any Lorenz frame of
reference to define Stich a surface. Therefore, this !lTlty.J is a relativistically
invariant eoncept. The restricti.on that the surface must be sp:rce-like makes no harm since the property that a surface is space-like or time-like does not depend on a special choise of the reference system. It is not necessary, from the stand-point of the relativity theory, to admit also time-like surfaces for the variable surface, what was-'reQuired by Dirac and by Yukawa. Thus we consider that ,"[I.:,.] introduced above is already the sufficient generalization of the ordinary- tJ-vector, and assume that the quantum-

S. TOMONAOA

theoretical state* of the fields is represented by this functional vector.

Let C denote the surface defined by the equation 1=1",.. Then. IF is

c.. a functional of the surface

We write this as IF[C]. On C we take a

point P, whose coordinates are (x)'$, t,.,.},and suppose .asurface C' which

overlap C except in a small domain about P. We denote the volume of

the small world. lying between Cand C' with dClJp. Then we may write

(25) also in the form:

lim F[C'J-lF[C].

C/+O

dwp

(28)

Then (26) can be written in the form:

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

This equation (29) has now a perfect space-time form. In the first place, Hu is a scalar accgrding to our assumption (i); in the second place; the commutation relations between V(p) and A(P) contained in /Ii! has
the four-dimensional f01"lJlsas (10), and finany the differentiation 8~p is
defined by (28) quite independently of any frame of reference. A direct conclusion obrained from (29) is that .(1"[C'] is obtained from.
IF{C] by the following infinitesimal transformation:
(30)
When there exist ih the space-time world two surfaces c,. and ~. a
finite distance apart, we need only to repeat the infinitesimal transformations
e in order to obtain IF[Cil from W[ l ]. Thus
(31)
The meaning of this equation is as follows: We devide the world region
lying between c;. and C; in small elemdnts dwp (it is necessary that each
world element is surrounded by two spaceplike surfaces). We consider for
• The word state is here ~d in the relativistic space-time meaning. Cf. Dira'li. book
(sec:olld eddition) f 6.

On II R,lativislieaP)' Invariant Form,datlon

87

each world element the infinitesimal trartsformation 1- ~ ~!(P)dcl)p. Then
we take the product of these transformations, the; ord~ of the factor being
taken fro'm c; to C';. This product transforms then if'[C';] into ~[C2]' The surfaces c; and C2 must. be here both space-like, but otherwise
they may have any form and any connguration. Thus c; does not necessarily lie afterward against 4 i Ci and£:; may even cross with each other.
The relation of the form (31) has been already introduced -by Heisen-
berg.('/) It can be regarded as the integral (orm of our generalizedSchrodin-
ger equation (29)

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

§ 4. Generalized probability ampUtude.

We must now find the physiCal meaning of the functional f"[C). As regards this we can make a similar cDnsideration as BIDch· has done for the case of ordinary many-time theDry. Besides the fact that in our case there
appear mfinitely many time variables, .one pDint differs from Bloch's case that in (16) the unitary .operator u is cDmmutable with the ·coordinates q1t qt, ..... qNt our U is not cDmmutable with the field quantities Vl(XYZ) and 'lJ2(Z.rs). Noting this difference and treating the cDntinuum infinity -as the
limit .of an ennumerable infinity by some ilrtifice, fDt instance, by the. prDcedure .of Heisenberg and Pauli,(8) Bloch's cDnsideration can be applied also here
almD!)t without anyalteratiDn. We shall give here only the results.
Let us suppose that the fields are in the state represented .bya vector
q CJ. We suppose that we make measunnents .of a function I(Vh V2, AlJ ~)
at every pDint on a surface 4 in the space.·timewotld. Lct: PI denote the variable point on 4, then, if f(PI ) at any tWD .. values" .of Px commute with each other, the measurement of f at each of these two points dD -not
interfere with each other. Our first conclusion says that in this case the expectatiDn value .of f(Px) is given by

I(PI ) =«t"{~],/(~)"[Ci]»

(32)

where 1(11.) means J(~(Px) • ...••. ) accDrding to our convention on p:;tge 35, and the symbol «A, .B}) with dDuble blackets is the scalar product of two yectDrs A and B. It is impDssible in case ofcDntinuously many degree .of freedom to represent this scalar product by an integral of the prDduct of two funCtiDns. For this -purpose we must replace the ·.continuum infinity by an at least ennumerable infinity.

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

38

S. TOMONAGA

More generally, we suppose a functional F[f(P')] of the independent variable function I(P'), regarding I(p]) as a function of~. Then the expectation value of this F is given by
(3l})

A physically interesting Fis the projective operator M['ll/(P,) , v/(P,) , VJ(P,), V2(Fl)] belonging to the .. eigen-value" vi'(P1), v/(Pl) of Vl(P,), P;(P,). Then its expectation value
M[7J/(P1), vi (P1)jV1(P'), ~(P')]
=«1[£:;], M[lJ/(P') , vi (p]) j V't(Pi),V2(P,).J.!l'I£:;])) (34)

gives the probability that the field 1 and the field 2: haye .respectively the
functional form v/(Pt) and 7Jl(P') on the surface Cl' As C; is assumed to be space-like, the measurement of the functional Mis possible (the measurements of.. VI(p]) and V2 (P,) at all pointsbn C 1 mean just the measurement. of M).
Thus far we have made no mention of the representation of P'[C]. We use now the special representation in which V1(P,} at -aU points on C1 are
simultaneously diagonal. It is always possible to make all r;. (P') and
~(P,) diagonal when the surface Cl is space-like. In this representation vr[C,] is represented by a functional W[v/(Pl ) , 'lIl(P1) j £:;] of the eigenvalues v/(P,) and 'l'/(l~) of VI(P,) and~(Pl)' The projection operator M has in this representation such diagonal form that (34) is simplified as
follows

W['l>/(P1 ), 11/(P,)]=M[11/(P') , vl(P,.); VJ(p,r"lI;(p,)]

=1 W['ll/(P]) , v/(~)j £:;] I~.

(35)

In this sence we can call W[11/(P'), v/(P')j £:;] .. generalized probability amplitude ".

§5. Generalized transformation functional.
We have stated adove that between W[£:;] and W[C~] the relation (31) holds, where C. and C2 are two spece-like surfaces in the space-time world. We see thus that the transformation operator

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

0,1 a Relativistically Invariant Formulation

39

(36)
plays an important 1·01e. It is evident that also this operator has a spacetime meaning.
Similarly as the special representative of the ifJ-vector, the probability
a amplitude, has a distinct physical meaning, there is special representation
in which the representative of the transformation operator rrC2 ; ~] has
a distinct physical meaning.
We introduce namely the mixed representative of T[e2 ; Cx] whose
rows refer to the representation in which Vt(PI ) and V;(~) at all points
on Cx become diagonal and wltose column refer to the representation in
which Vt(~) and ~(~) at all potnts on C; become diagonal. We denote this representation by
[v/' (~), vi' (P2) I I[ C;; Cx] I v/ (~), V2' (ED], (37) *
or simpler:
(38)*
If we note hoce the relation (35), we see· that we can give the matrix elements of this representation the following meaning: One measures the field quantities VI and V; at all points on C; when the fields are prepared in such a. way that they have certainly the values v/(~) and v/(I{) at
all points on Cx. Then

gives the probability that One obiains the result vt" (~) and vi' (~) in this
measurement. In this proposition we have assumed that C; lies afterward against Cl.
From this physical interpretation we may regard the matrix element
(37), OJ· (38), considered as a functional of v/, (P2 ), vi' (~) and v/ (I{) ,
vl (~), as the generalization of the ordinary transformation function (qta" I
qt.').
• As the matrix elem:lnts are CunctionaIs of v(P), we ust here the square blackets.

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

40

S. TOMONAGA

As a special case it may happea

that Cs lies apart from C; only in
a portion S. and a portion Sl of c;.

and C; respeca.ely, the other parts
of C; and q overlapping with- each-

other (see Fig. 1).
In this case the matrix elements
of 11Ct; C;] depend only on the

----~rO~--------~--------~
"~,'-

values of the fields on the portions

SI and S. of the surfaces C; and Ct. In this case we need for calculating

7[Ct; C;] to take the product in (36) only in the closed domain surroYllded

by SI and SI, thus

(40)

The matrix elements of the mixed representation of this T is a funcbonal
of v/ (PI), vz' (PI) and v/' (P1), v/' V.) where PI denots the moving point
on the portion S" and P. the moving point on the portion S.. This matrix
is independent on the field quantities on the other portions of the surfaces
Cl and Ct.
The matrix element of 11SI; SI] regarded as a functional of vl (PI) ,
vl (PI) and vi' (Ps), vi' (P.) has the properties of g. t. f. (generalized trans-
formation functional) of Dirac. But in defining our g.t.f. we had to restrict
the surfaces Sl and SI to be space-like, while Dirac has required his g. t. f.
to be defined also referring to the time-like surfaces. As mentioned above,
however, such a generalization as required by Dirac is superflous so far as
the relativity theory concerns. It is to be noted that (or the physical mterpretation of [vl'(~), Vl'(PI)1
v/ (~), vi (~) ] it is not necessary to assume Cs to lie afterward against Ct. Also when the inverse is the case, we can as well give the physical
meaning for W of (39): One measures. the field quantities P; and V at all
points on Ct when the fields are prepCJred in such a way that they would have certainly the values v/(P,) and vl(P,.) at all points on C; if the fieldswere left alone until Cl without being measured before on Ct. Then W
gives the probability that one finds the results vs" (Pt) and -Va" (~) in this
measurement on Ci.

On a Relativistically /Jtaanant Formulation

41

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

§ 6. Concluding remark..
We have thus shown that the quantum tneory of wave fields can be really brought into a form whie}. reveals directly the invariance of the theory against Lorentz transformaoons. fhe reason why the ordinary formalism of the quantum field theory is so unsatisfactory lies in the fact that one has built up this theory in the way which is too much analogous to the ordinary non-relativtstic mechanics. In this ordinary formalism of the quantum theory of fields the theory is devided into two distinct sections: the section giving the kinematical relations between various' quantities at the same mstant of time, and the section determining the causal relations between quantities at different instants of time. Thus the commutation relations (1) belong to the first section and the Schrodinger equ~tion (2) to the second.
As stated defore, this way of separating the theory into two sections is very unrelativistic, since here the concept" same instant of time" plays a distinct role.
Also in Qur formalism the theory is devided into two sections. But now the separafion is introduced in another place: In our formalism the theory consists of two sections, one of which gives the laws of behavior of the fields when they are left alone, and the other of which gives the laws determining the deviation from this behavior due to interactions. This way of separating the theory can be carried out relativistically.
Although in this way the theory can be brought into more satisfactory form, no new contents are added thereby. So, the well known divergence difficulties of the theory are inherited also by our theory. Indeed, our fundamental equations (29) admit only catastrophal solutions as can be seen directly in the fact that the unavoidable infinity due to non-vanishing zeropoint amplitudes of the fields inheres in the operator Hu(P). Thus, a more profound modification of the theOry is required in order to remove this fundamental difficulty.
It is expected that such a modification of the theory would possiblly be introduced by some revision of the concept of interaction, because we meet no such· difficulty when we deal with the non-interacting fields. This revision would then result that in the separability of the theory into two sections, one for free fields and one for interactions, some uncertainty would be introduced. This seems to be implied by the very fact that, when we formulate the quantum field theory in a relativistically satisfactory manner,

42

S. TOMONAGA

this way of se'il<tration has revealed itself as the fundamental elp:nent of the theory.

Physics Department, Tokyo Bunrika University.

Downloaded from http://ptp.oxfordjournals.org/ at Bibliotheque de l'Universite Laval on June 28, 2014

References
(I.) H. Yukawa: Kagaku, 12 (1924), ~51, 282 and 322. (2) P. A. M. Dirac: Phys. Z. USSR., II (1933), 64-
(3) W. Pauli: Solvey Ber,chte (1939).
(4) P. A. M. Dirac: Proc. Roy. Soc. London, IS6 (1932), 453.
(5) l~. Bloch: Phys. Z. USSR., 5 (1943), 301.
(6) E, Stueckelberg: lIelv. Pbys. Acta, 11 (1928), 22.5, i 5.
(7) W. I-IeisenLerg: Z. Pbys., 110 (1938), 251. (8) W. Heisenberg and W. Pauli: Z. Phys., 56 (1929), 1.