on the covariant formalism of the quantum theory of fields, ii

65

Progress of Theoretical Physics, Vol. 6, No.1, January-February, 1951

On the Covariant Formalism of the Quantum Theory of Fields, II

Ryoyu UTIYAMA

Osaka University

(Received November 4, 1950)

In the preceding paper of the same author,l) he has investigated only those cases where the field equations have no functional dependence among themselves, or more exactly, the action integral admits no groups of transformations depending on arbitrary functions. If we consider, however, any field interacting' with the electromagnetic field, it

will not be possible to make use of the method stated in (1) without any modification, because the canonical conjugate quantity to the scalar potential vanishes identically, and the equations of the electromagnetic field are not mutually independent.

In order to remove this formal difficulty, Fermi has proposed to modify the Lagrange function of the electromagnetic field by adding an

adequate term- 1/2 (fJA(I./fJx(l.) 2 to it. An alternative method was suggested by Rosenfeld.21 He has formulated the electrodynamics by using the elegant theory of invariant variation. At first sight his formalism appears to be covariant under Lorentz transformation, but it is actually diffirult to consent to his assertion of the covariance of his theory. Accordingly the present author has tried to reformulate the same problem, following Rosenfeld's line of reasoning and at the same time preserving the required 'invariance.

In the quantum theory of fields, every field quantity is described with reference to anyone Lorentz frame (let us call this x-system for simplicity), and the field equations give how the field quantities change with the lapo;e of time xO

(in Heisenberg picture). But the identification of the time as the observation paramefer with the time coordinate of the x-system, in general, destroys the apparent covariance of the field equations and

the commutation relations ([C.R)). Therefore it is necessary to distinguish between these two time' concepts. The observation parameter T, if set equal to a constant, will describe a space-like hyperplane 11 (or in general a space-like hypersurface) in x-system. In order to show the Lorentz covariance of the whole theory, we must prove it for any deformation of 11. Hence it is convenient to introduce, from the outset, an arbitrary system of curvilinear coordinate, as waS done in (I).

In § 1, we shall present the outline of the classical electrodynamics from the standpoint stated above. In § 2 the quantization will be performed by the Heisenberg-Pauli's method by introducing a new quantity @o which is canonically conjugate to the scalar potential. Further the character of the time derivative of the scalar potential will be investigated which was assumed to be an arbitrary

c-number function in Rosenfeld's theory. In § 3 we shall show the covariance of our theory under both coordinate and gauge transformations. The field equations thus obtained differ from the usual ones. In particular, the so-called Lorentz condition. in our case, is not a condition imposed on the state vector, but a q-number relation. In §§ 4 and 5, the theory ,wiH be transformed into the interaction represeptation, where the Tomonaga-Schwinger equation emerges quite naturally, by virtue of our employment of the curved coordinate system.

Finally in the last two paragraphs the equi-, valence of our theory with the ordinary one will be proved and some remarks are given about the application of our method to the vector meson field.

1) R. Utiyama, Prog. Theor. Phys. 5 (1950), 437. This paper is cited as (1) in the present paper. 2) L. Rosenfeld, Ann. d. Phys. 5 (1930), 113.

K. Husimi, Buturigaku Koyenshu IV, 81. (in Japanese).

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66 R. UTJYAMA

§ 1. Classical theory (Lagrange formalism)

Let us consider the electronic field interacting with the electromagnetic field. The Lagrangian density of the total system is, as usual, given by

L=L",+L,- V,

L =i'/'+..I< a¢ +ix,~tdJ 77. 'r I ax.!: 'I' T'

(1.1)1>

where A l , A 2, As and Ao the scalar potential -~. as in (I).

represent res.pectively the vector potential A and minus In this paper we shall make use of the same notation

Now all the expressions in (1.1) are expressed with reference to an arbitrarily chosen Lorentz frame (x-system). If we represent these expressions in regard to an arbitralY system of curvilinear coordinates (~-system), they will be written as follows;

'i3=D.L='i3",+'i3,..,-~,

'i3 ... =iD. {¢tr!£(~) :f!£ +x¢t¢},

q - -!.-.D.P f!£~ .c,- 4 :/!£v ,

(1.1) ,

where

and PI' represents the covariant derivative with regard to ~!£, whereas A~,!£ merely denotes aAJa~!£. Here it should be noticed that from the definition mentioned above, AI' is a covariant vector, whereas ¢ and ¢t are world scalars under general transformations of ~-system (we shall call these ~-transformations).

and

The field equations of the system considered are

r!£ (a~!£ -ieA!£ )¢+x¢=O

(~l£ + ieA!£ )¢trl£_x¢t =0.

3) The natural units are used, i.e. !=c=l.

(1.2)

(1.3)

(1.3)t

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On the Covariant Formalism of the Quantum Theor), of Fields, II 67

It is easily seen that these equations are covariant under e:-transformations.

Now the action integral I=JuS(d04 is invariant under the gauge transforma

tion generated by the fonowing infinitesimal one,

¢'-H/" =¢'+ aG¢" aG¢, = ie).¢" (1.4)

¢'t....,¢'t'=¢'t + aG¢,t, aG¢'t = -ie).¢'t.

Hence following the line of reasoning in § 2 (1), we obtain the following identifies;

(U5)

and

a .. ~ { (ie as ¢' + [S]A)). + ~)., .. } ==0. E* a¢',~ aA .. ,~

(1.6)

Taking notice of the fact that). and its derivatives can be taken quite arbitrarily, the following identities are derived from (1.6)

(1.7)a

[S] -D'''+~~==O A.. ~ ae:" aA .. ,~ , (1.7)b

as as aA .. ,~=- aA~, .. ·

(1.7)c

The identity (1.7)b can be written as follows;

~ = - ie~¢'-D .j". aA.. a¢" ..

(1.7)'b

From this expression we know that the Lagrangian density must contain A .. only in the combination a¢' ta~ .. -ieA .. ¢,. Furthermore, from (1.7)c, it is seen that aA .. /a~'II must be contained in S in the combination 1 .. '11' Making use of the equations of the electromagnetic field, we obtain from (1.7)a the equation of continuity;

a~ .. (D.}") =D· fJ .. j"=O· (1.7)'a

Finally because of (1.7) c the canonical conjugate to the scalar potential vanishes identically. This fact interferes with the construction of the canonical formalism as already mentioned in the introduction.

Now if the field equations are used, the identity (1.6) gives us a constant of motion expressed as follows,

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68

where

R. UTIYAMA

~1"=~=D..tI"O, aAl",o

(1.8)

(1.9)

and the integration should be performed over a space-like hypersurface iT which is defined by e:o=constant.

In § 3 we shall know that G is nothing but the infinitesimal generating operator for the gauge transformation (1.4). But in the present case we know from (1.9) and the field equations that G is actually equal to zero.

Apart from these characters of G, it is easily seen that G is invariant under both e:- and Lorentz-transformations.

§ 2. Quantum theory

Let us introduce the canonical conjugate quantities as -usual;

(2.1)

The last expression of (2.1) prohibits us from imposing the usual commutation relation ([C.R]) for ~o. Thus, following the reasoning of Rosenfeld let us introduce a new quantity ~o which is canonically conjugate to Au, and does not vanish identically. Then the [C.RJ's are written, as usual;

(2.2)

All the other combinations are set to be commutative (or anticommutative). In order to obtain the Hamiltonian density, the time derivative of any field

quantity must be expressed in terms of canonical variables and their derivatives with regard to space-coordinates. But in the present case aAo/a~o is indeterminate as we see from (2.1), i.e.,

Au,a =A(e:) = indeterminate.

Hence the Hamiltonian of £he total system contains an indeterminate term and is given by

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On the Covariant Formalism 0/ the Quantum Theory cf Fields, II 69

where

.i)=~o+m+(f°A,

.i)o = - rv:;- .. (fi(f; + (f<t A Ii - + fv:~. (ro;(fii - roi(f~) 2D. rOo .11- 2roO

Now Rosenfeld has assumed A to be a c-number function. But we shall propose, for the present, that A is an undeterminate linear combination of canonical variables (fll- and A~. The restriction of linearity seems to be necessary in order to retain the linearity of the equations of the electromagnetic field. The precise determination of the form of A, will be given later.

The field equations are given by

d(/). =i[~, (/)] d~o

where (/) is any field quantity. Especially, the equations of the electromagnetic field run as follows:

8~l£o: = 1 .. rii;;(f.+AIi.P:+ r::/v:;+iJ (f°(e') [A(e:'), AiiJdt: 0.. D.roo rOo a

(2.4) a

8~o =A +ir (fO(~')[ A(~'), AO]d!,' 8~0 Ja (2.4)b

8~~ _ ~(D ·IV:~) + iJ'fO (e:') [;1 (~') , (f~ ]it + D-j~ 8;0 8~. a .

(2.4)c

Q~o = 8! +iJ(fO(~')[A(~')' (fO(~)]d~+D.i. 8,° 8~1£ a .

(2.4) d

From (2.4)c (2.4)d, we obtain

82~O =iJ (f0(,') [A(e), 8(f~]dl+ir 8(fO(~') [A (,'), ~O]ai' 8,02 a 8;1£ J a 8~0

(2.5)

4) As to the orders of non-commutative q-numbers, we assume, in this paper, that any suitable symmetrization procedure has been performed.

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70 R. UTIYAMA

by virtue of the equation of continuity (1.7)'a which holds also in quantum theory. Further from (2.4)a, (2.4)c and (2.4)d, it results in

D!P'J"~-j"l =(terms including ~o and ~~o). (2.6)

Therefore, in order to make our canonical equations (2.4) agree with the Maxwell's ones (1.2), it is necessary to impose the following condifions on the state vector 1f!o (in Heisenberg representation) j

011' the surface (1. (2.7)

Now, it is easily seen that the conditions (2.7) are not only necessary but also sufficient, from the fact that the time derivative of any order of ~o can be expressed in terms of the linear combination of ~o and d~o / d~o by virtue of eq. (2.5) and the linear character of A stated above. Therefore (2.7) holds at eVery instant and consequently it holds over the entire space-time j

(2.8)

This equation implies that ou~ canonical equations (2.4)a, c, d are practically covariant.

Any transformation of the system oLe-coordinates defined by

~"-?:'''=e'' + a~" (~), A,,-?A~(~') =A"(~) +aA", aA"=- ~~:~ A~,

(2.9)

can be expressed by a unitary transformation;

(2.10)

where (/J is any field quantity and T is a unitary operator which is given as follows in case of the infinitesimal transformation (2.9) ;

T=l+iK,

K= L(~"aA,,-:t~a~")d~,

From (2.10) and (2.11) it holds:

a*(/J=i[K, (/J],

where

(2.11)

(2.12)

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On the Covariant Formalism of the Quantum Theory of Fields, II 71

Now, according to the definition of a*r/J, we know the following equality:

~a* r/J = a*dr/J . d~'!- d~'!-

Substituting from the field equations and the eq. (2.12), into the above equation, we obtain

(2.13)&)

or at least,

(2.13),

provided that the state vector 1Jfo satisfies the condition (2.7). If (2.13) holds for every field quantity r/J then we can conclude that the expression ( ...... ) has the following form j

dK o-M -. -a*.p=c-number,

d;o -(2.14)

and we can easily show the covariance of all the equations under the transfOlmation (2.9). On the contrary, if all the equations are covariant, then M must be a c-number.

In fact, (2.13)' holds for every field quantity r/J except for Au as we can see from (2.8). But if we assume A to be a C-111 mber function, then it holds j

i[M, Au]= {~a* Au - a*A}. d~o

Therefore M must have the following form

M ==f ~u(~a* Ao - a* A)d~ + f M'dt o d~o 0

where 111' is commutative with Ail and hence does not contain ~u. On the other hand We know that the following equation holds for any field quantity r/J except for AU. ,

[M, r/J]1Jfo=O. Hence M' must be a c-number. Therefore M gets the following form j

M =f ~o(~a* AU - a* A)d~ + (c-number). o dEo

This expression contradicts with the requirement (2.14). Thus the assumption that A is a c-number interferes with the cavariance of the field equations.

5) The notation $* represents the variation due to the explicit dependence on the quantities h~. h~ etc.

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72 R. UTIYAMA

Our next task is to determine the form of A so as to retain the covariance of the field equations.

The eq. (2.4)b can be generalized into the following tensor equations;

(2;15)

the (0, 0) component of which is just our eq. (2.4)b. Now let us, for the moment, deal with those cases where the e--system is a Lorentz frame and the transformation of it is restricted to the Lorentz transformations. Now (2.U5) is reducible under the Lorentz transformation in the sense of the group representation, i.e. (2.15) can be separated into three irreducible parts. The first one is a symmetric tens.or of the 2nd rank with the vanishing trace. The second is an anti symmetric tensor and the third is a scalar.

Our eq. (2.4)b is contained in the first and the third part. Therefore let us investigate these two cases separately.

i) The case of th first irreducible fa,.t

In this case we have to interpret eq. (2.4)b as it means

BAli-8~",-=A(I.'II-=O

i.e.

If we adopt again the curvilinear system, (2.16) must be replaced by

p~II-=O,

i.e.

(2.16)

(2.16)1

r0i> rop -' 1 BrI'll- rii~ 1 rll-A=--.-. Ao, P _-----,-;-/1'0_-.-. -,,-Ap--.-. A~. Ii --.-. (I.~A~. (2.16)"

rOo rOo rOo u~'" rOo rOo

Substituting from (2.1) into the second terni of the right hand side of (2.16)", and further from (2.16)" into the Hamiltonian, then we obtain, instead of (2'4), the following eq.;

BAy. 1 __ -;-, • _ rO~ _ r0-; • --. =---.-. r",.{;!;-+Ao, "'+----;-;-:f"'~---.-. {;!;O B~o DrOO roo Droo

or

- _. r°ji; ~(I.=D'f(l.o+-.-. {;!;o.

rOo (2.17)

This equation contradicts to the original definition (2.1) of {;!;ri". However, the additional term containing Q;o does not give rise to any change in eq. (2.17), by virtue of the commutativity of {;!;O and Aji;, on replacing the second term of (2.16)" by

- ;;;o{{;!;p- :~f {;!;o},

and further aAji;la~o in the Hamiltonian by the righthand side of (2.17) instead of (2.1). The new Hamiltonian density, thus obtained, has the following form;

(2.18)

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where .po and fll are given in (2.3), and A' is defined by

1 {. - ro pC - 7 pO • ) f)rPII- - } A'---.-. rOPAo,p+-D ~P_-.-.~o +oe-Ap+rll-~A~ji+lll-A~ • rOo 2roo u,,11- • II-~

(2.16)111

In this way we can obtain new field eqs. without self.contradiction, which run on rearrangement, thus :The first set are

If we introduce the new quantity E by

~O=_DrOOE,

(2.17) and (2.19) can be brought into a single compact form;

~II-=D· U",o-rll-oE). The remaining sets of the field equations instead of (2.4)b, c, d, are

p'II-AII-=O, and

f)E P' ~.fll-~-rll-~--= jll-.

f)~~

Substituting from (2.17) b into (2.17)c, we can rewrite (2.17)c into

f)E . DAII-+ f)~11- =-,JII-,

since our underlying space-time is flat. Further from this equation, we deduce

DE=O

provided that the equation of continuity and the eq. (2.17)b are taken into account. but the eq. (2.5). The condition (2.7) can, in the present case, be replaced by

~o . Wo=O, lJ· Wo=O

where

if we make use of the field equation.

ii) The case o.f the third irreducible pari

In this case we have to take (2.4)b as meaning

f)AII---=(a world scalar)=.9. f)~11-

The discussion of this case will be given in the later § 6.

(2.17)

(2.19)

(217)a

(2.17)b

(2.17)c

(2.17)'c

(2.17)"c

This is nothing

(2.20)

(2.21)

In concluding this somewhat lengthy paragraph, we may summarize our problems in view to be investigated in the next paragraph. These consist in proving the following facts:

i) E is a world scalar. ii) The condition (2.20) is covariant under ~-transformation.

iii) The two conditions (2.20) are m)ltually compatible. iv) The covariance of [C.R] 5.

v) The gauge-invariance of the whole system of equations and conditions.

§ 3. Coordinate- and gauge-transformation

We shall begin with the discussion of the ~-transformation (2.9). The infinitesimal generating operator of (2.9) has been already given by (2.11).

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74 R. UTIYAMA

Making use of the field equations (2.17)a, b, c, we can see the transformation character of (fP which is given by

a*(fp = i[' K. (fP] = a(fp _ a(fp a~'" , a~'"

(3.1)

whele

(fP"'=D· {/P"'-rP"E}, (fP=(fpo.

This shows that (fP( = (fP") is the (p,O) component of a tensor density (fP"'. Hence E must be a world scalar. Thus the first problem in §2 has been proved. From this statement we can easily understand the covariance of the equations of field (2.17)a, b, c. Therefore we obtain the following result from the reasoning of § 2;

dK o_

M -. -J*~'=c-number. d~o

In fact we can show that M is equal to zero by somewhat laborious calculation. Putting p equal to 0 in (3.1), we get

a*(fo =(2 aa~o rO'" + 2.r"'~ar )(fO _ a(fo a~'" a~'" rOo 2 "'~ a~""

hence it holds a*(f°.1Jf 0=0,

if we make use of the condition (2.20). In the same way we can also show

a*~ ·lJfo=O'

(3.2)

(3.2)'

Thus the 2nd problem has been solved. The proof of the covariance of [C.R]'s is trivial, i.e.

~ ~

=[A(~), B(~)]olla(=c-number),

where (7' is a hypersurface defined by the same numerical value of ~O, as that of ~ defining the original surface (7. Thus we have given the proof of c0variance of the whole system of formulae under any transformation of ~-system.

In the second place let us consider the gauge transfOlmatioll (104), where A was an arbitrary c-number and scalar function. But in the present case, A. must satisfy the following wave equation

DA.=y",v"'A.=o, because of the gauge invariance of the eq. (2.17)b.

(3.3)

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Now the gauge transformation (1.4) is expressed by a unitary transformation

r/)-r/)'=TGr/)T"(/=r/)+aGr/), (3.4)

where TG is given by the following expression in case of the infinitesimal transformation;

(3.5)

From (3.4) we see at once the gauge covariance of [C.RJ's. As the gauge transformation is commutative with the f-transformation, the

following equality holds for any field quantity r/);

aGa*r/)=a*aGr/). (3.6)

Substituting from (3.4) and (2.12) into (3.6), we obtain

If we restrict the f-transformation to the special one

a;o= E = constant parameter,

then (3.7) becomes

(3.7)

(3.8)

(3.7)'

Eq. (3.7) means the invariance of G under any f-transformation while (3.7)' means that G is a constant of motion. Further, if the f-transformation is restricted to another special one:

and

,.' {<O near one particular point ~p on a, a"o= =0 at all the other points

aafO --=0 near the surface a,

afo aG[a]

aa" we obtain from (3.7) o

where G[n] is considered to be a functional of a.

(3.9)

(3.7)"

From (3.7) and (3.7)', it can be proved that the equation (2.12) and the field equations are gauge-invariant,

Now (3.7)' may, on the ground of (3.3), be computed to the form

It follows 2{=0, and ~=O, (3.10)

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76 R. UTIYAMA

where

(3.11)

From (3.10) and (3.11) we can aiso show the compatibility of the condition (2.20) with the eqs. of field. In quite the same way it can also be proved from the eq. (3.7) that (2.20) is covariant under ~-transformation.

The third problem mentioned in § 2, i.e. the compatibility between the two -conditions (2.20) is easily proved because of the commutability of (Eo and ~. Further from this character of commutability, we know also the gauge invariance <Of the condition (2.20).

Thus we have proved the gauge invariance of all the formula. We have, so far, expressed all the formulae in reference to the e:-system.

Now let us rewrite all the expressions with regard to the original Lorentz frame in order to show the apparent covariance under the Lorentz transformation. The field equations are

And the [C.R], s are;

r"(~- ieA,,)¢,+x¢,=O, axk

a/Tel _aE = l", OE=O, ax-' aXk

_aA" -0 ,£k=lkO_...kO. E. ax" - , 5.

LUlkl(x') _gkIE(x')}, A",(x)]do"/= -i8:"

L {(¢,t(x')r") .. , ¢'~(x) }da,.'=3 .. a•

The condition (2.20) is expressed by

E·lJl'o=O, nk(x)aE ·lJl'o=O, aX'''

(3.12)

(3.13)

(3.14)

where n" is the k-th component of a unit normal of the surface a at the point (x) and is defined by

Finally, the generating operator G of the gauge transformation can be expressed by

(3.15)

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If we make use of the eqs (3.12), the G can be transformed into the the following form;

G=f {'iJE. A-E~}d(J". "'iJz" 'iJz"

From this expression, we can easily verify the eq. (3.7)";

§ 4. Interaction representation

Let us consider the following unitary transformation;

A~A=UAU-l

(3.15)'

(4.1)

where A is any field quantity in Heisenberg representation and A is the same quantity in the interaction representation (operators in Heisenberg representation will be denoted by bold lace letters). U is a unitary operator whose transformation character under the ';-transformation is defined by

ia*u=- J" USa;O)dt.U,

Y!3=DUVU-1•

Then from (4.1) we obtain the transformation rule of any field quantity

a*A=i[Ko, A],

(4.2)

(4.3)

if we make use of the definition (4.2) and the eq. (2.12). Here K o is the infinitesimal generating operator of the free fields, that is, Ko does not contain the interaction term. If we restrict the ';-transformation to the special one (3.8), we obtain from (4.3)

dA .--. =t[C"" A], d;o 'Ii.

(4.4)

where ~J=~o+Q;OA',

i.e. the eq. (4.4) is nothing but the equation of the free field. In this case, (4.2) is replaced by

.dU(';O) t------'------'--

d;o (4.5)

On the other hand, if the ';-transformation is restricted to the second special case (3.9), then (4.2) becomes

(4.5),

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78 R. UTIYAMA

where U is considered to be a functional of n rather than a function of ~o. The state vector 1J!(~O) in the interaction representation corresponding to (4.1).

is defined as follows;

or

(4.6)

From (4.6) and the eqs. (4.5), (4.5)', we obtain the Tomonaga-Schwinger equation;

or in general

. d-fJ!' (~O) t .

d~o

i a1J![n] an"

(4.7)

V(P).1J![n], (4.7)'

(4.7)"

The [C.R]'s and the condition (2.20) retain their original expressions also in this representation. Especially the latter is given by;

~O(P) . 1J![n] =0, g:(P) .1J![nJ=O, (4.8)

where the point P lies on the surface n.

Next let us consider the gauge transformation, the generating operator of which keeps the same form as in case of the Heisenberg representation. Now let us separate G into two parts, one of which does not contain the interaction constant e while the other depends on ~, i.e.

G=GO+G1,

Go=J {(fo~_a~~. ltdf, a a~o a~p )

Gl=-JfiM~. (4.9)

The eq. (3.7) is, now, replaced by

a*G-{J(mafO)df, GJ=o. (4.10)

Making use of (4.3), this is written as follows;

a*G+i[Ko, G]-{J (ma;O)df: G J-O. (4.10)'

(4.10)' is an identity because it contains only canonical field quantities and their derivatives with regard to space coordinates. Hence there result the following identities from (4.10)' corresponding to each power of e;

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a*Go=a*Go+ i[Ko, GoJ===O, (4.11) a

a*G1-{L(ma~O)d{, Go]===O, (4.11)b

[L(ma~O)dJ, G1]===O. (4.11)c

Now from (4.10). the following relation is obtained;

TG(ia* + L (f8a~O)ii) T-;/,1Jf

={ia*+a*G+ L(ma~O)d;+i[G, L(ma;O)dl]}1JI'

= (ia* + L (ma~O)dt)IJf=O, if we make use of the eq. (4.7)". Let us separate TG into two factors in correspondence to (4.9), and put

TG= To' T1=T1, To,

To=l+iGo' T1=1+iG1,

then, from (4.11)a it follows;

Hence from (4.12) we obtain

To{ia*+ L(ma;O)dl}T;1. (TIt, 1Jf)

={ia*+ L(m,.a~O)dJ}IJI"=O

where

(4.12)

(4.13)

(4.14)

(4.1,,)

Therefore, if we define the transformation character of the state vector IJf and the field quantity (j) under the gauge transformation by the following equations;

IJf ~1Jf' = Tj1.1Jf = IJf -iGl lJf,

(j)~(j)' = To (j)T;1 = (j) + i[ Go, (j)], (4.16)

then the eq. (4.14) guarantees the gauge invariance of the eq. (4.7)". Thus we have learned the two transformation character of IJf ;

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80 R. UTIYAMA

and (4.17)6)

As to the compatibility of these two equations, we can easily prove it by making use of (4.11)b and (4.11)c.

The field equations (4.4), the condition (4.8) and [c.Rrs are invariant for the transformation (4.16).

Our next task lies in the verification of the compatibility between the condition (4.8) and the equation (4.7)". Eq. (4.10) can be written as follows,

a*G-i [50 (ma~O)d( G J= 50 {~' :~o + ~'A}df,

by virtue of (3.3). From this equation we infer

~'=O, ~'=O, (4.18)

where

~!' =a*~o _~o aa~o + ~(~o a~ji) +~a~o afo af1'-

6) We can give the alternative method to obtain the transfonnation character of U (or F(r» under the gauge transformation. From the eq. (4.5), the following integral equation is derived (we shall

write t instead of {o, for simplicity) ;

U(t) = l-i[!i(t')U(tl )dt'.

where we assume U(-oo)=l. From this equation we get;

8GU(t) = ir ) D(t)j1'-(tl) 8:(~) i[t,U(f/)dt'-J fi3(II)8GU(t')dt' "=_00 't-const e -ex)

-= -iG1(t) .U(t) -£ r ~(fI){8GU(t') +tGJ (I") .U(t')}dt',

by making use of the equation of continuity. If we put

X(t) =8GU(t) + iGJ (t). U(t),

then the above equation appears as follow;

X(t) = -i) I ijj(t/) ''l.(fI)dtl.

Since it holds x( -00) =0, and

dX(t) . -~-=-t!B(t), x(t)

in consequence of (a), we confirm that identically

so that

or 8GF(r)= -iG1(t). F(r).

This is just the second equation of (4.17).

(a)

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~I=-a*~+ O_(G;o Od~~)_~ma~~) O~~ O~o O~~

+ !2 _( r~.-;G;O a~O)-~[(~~ +~r~Or~O,prPIA) ~~ MO] a~~ o~~ rOo a~~ 2 ' rOo

(4.19)

If the e:-transformation is restricted to the special case (3.9), we obtain the following relations, from (4.18) with (4.19) on integration over a surface q;

~n;(p) +~(ro~+~r~or rOP) G;O(P) D u 00 ~ 2 ~O,P D

p r p

-i[V(P) , G;0[q]],

amq] =-i[V(P), ~[qJ], iJqp

(4.20) 7)

where

Now, from the condition (4.8), we obtain

~(G;O[q], qJ"[q]) =iJG;°[q] .qJ"[q]-iG;0[qJ- V(P) .qJ"[q]. dqp dqp

If we substitute from the first equation of (4.20) into the above, and make use of the condition (4.8), then we arrive at

In quite the same way, we can prove

~(mq]·qJ"[q])=O. dqp

Thus the condition (4.8) is compatible with the eq. (4.7)". From now on, we shall avail ourselves exclusively with an rectilinear or

thogonal e:-system, the transformation of which being restricted to the Lorentz group. This restriction, however, does not destroy the generality of our theory because we have already shown the covariance of our formalism under the e:transformation. In this simple case we can easily solve the equations of field, the solutions of which will be given below. In particular we will express any

7) cf. (I). p. 453.

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82 R. UTIYAMA

field quantity rp at an arbitrary world point (,;) in terms of the canonical quantities on the surface u. As to E, for example, we get

( 4.21)

Making (4.21) operate -on W[ u], we obtain the following condition on account of (4.8) j

(4.22)

This is more general than the first of the conditions (4.8) to the effect that the point (,;) may well lie outside of u. On differentiation of (4.22) with respect to ';0, we get the generalized second condition

(4.22),

By means of these solutions, we can also generalize the [C.R]'s as follows;

[E(~), Av.(;')]= _iaD(~~~') a~v.

[Av. (~), A~ (e)]= -irv.~D(';-~') + iJ .. J" aD(~ -r;) aD~~-:';') (dfj )4, (4.23) "f arl

where u' is a hyper plane on which ~o is a constant. As for the electron field the [C.R]'s remain the same form with that of the usual ones.

In concluding this paragraph we want to write down several expressions with reference to the original Lorentz frame.

In the first place the eq. (4.7) mns as follows

idW(-r) =mW(r), d-r

where

-r=ltixk-:- -nkx",

~= -nkJpA1dl1k ,

and the more general eq. (4.7)" becomes,

or

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On the Covariant Formalism of the Quantum Theory of Fields, 1l 83

where we have in mind the infinitesimal Lorentz transformation of the orthogonal rectilinear ~-systcm, i.e.

with

and the hyper plane u' is that on which the new time coordinate r' has the same value as that of r on the plane u.

The condition (4.22) now is expressed by

{E(X) + LD(X-x')/"(X')du,/}W (.) =0, (4.24)

and the [C.R],s are rewritten as follows;

[E(x), Ak(x')J=-i aD(;;x') ,

[Ak(x),At(x')]=-igktD(Z-x')+i .. - (dz")4. f f" aD (x - x") aD (x" x') ,,/ axk " ar"

• (4.25)

Here it should be noticed that the condition of the flatness of u and a' in (4.24), (4.25) is not necessary, but we can rather adopt any space-like hypersurface as easily seen from the nature of these equations.

§ 5. Elimination of the longitudinal component

In this paragraph we deal with the special case of an orthogonal rectilinear ~-system.

In the first place, let us separate Ali into two parts ;

where we assume

A-=.B-+ aA '" '" a;ji: ,

aB'" --_-=0, M'"

BO-B·-O - 0- .

Then from (2.17) b, A £\lust satisfy the following equation

-.

(5.1)

(5.2)

Introducing the Green function G(~) of the Poisson equation, (5.2) can be solved as follows;

(5.3)

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84 R. UTIYAMA

where .. .. dG (~) = a(~).

The [C.RJ's between A and other quantities are given by;

[E(e), A(~')]=O,

[AI£' A]=O, [A (;), A(~') ]=0, [Bji, AJ=O, (5.4)

and further

(5.5)

Making D' Alembertian operate on both sides of (5.3), we get

(5.6)

by virtue of the field equ.ation .• Hence from (5.1) we obtain the following equation;

OB",=O. (5.7)

The [C.RJ's between Bji and other quantities are given by; .. .. [ttii, B-vJ= -£a"F:a(~-{') + £ a2G(~-e) ,

" a~I£a,"

[E, B"J= O,[Aji, Bii]=O,

[ att~, Bv]=O. a~p.

(5.8)

Now we can give the more general [C.RJ's by using the field equations and [C.R]'s (5.4), (5.8). To do this let us introduce a new function (I, which is defined by;

D(~), (5.9)

In terms of this new function, the generalized [C.RJ's are given as follows;

[A(~). A (;') ]=£(I(e=-~') + £J;,. S D(~-~)D(7)-e) (d~)4. [Bi" A]=O,

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(hz the Covariant Formalism of the Quantum Theory of Fields, II 85

If we put

then A' satisfies the following equation which is derived from the eq. (2.17)' (where jll- should be put equal to zero),

[JA'= -E. (lUI)

Therefore the [c'RJ's between A' and other quantities are

[A' (~), A(~') ]=i S:P(~-7)D(~-e) (d?)4,

lA' (~), A'(e)]= -i(I(~-e) + iJ" D(;-~)D(7)-:') (d~)4, 0'

[A'(~, Bv:(e)]=O. (5.12)

Now let us try to eliminate the charge density i) from the second condition -of (4.8) so as to obtain the same one as in case of the free field.

Consider the unitary transformation;

lJ!(r)-.Ij!'(r) =S(r) .1J!(r),

where S( r) is a unitary operator defined by

S=eiR(~) Rer) = - Lt:1,s~f. .-)dt

From the [C.R]'s (5.4), (5.5), it results:

[R,jO]=O, [R, E]=O,

so that it holds

iLr R, a~: ]+jb=O, a~1I-

SiYS-1= a~:, SES-l=E. a~1I-

Thus the condition (4.8) is transformed in to

as anticipated.

(5.13)

(5.14)

Corresponding to the transformation (5.13), the Tomonaga-Schwinger equation is now written as follows;

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86

where

R. UTIYAMA

V'(.) =idS S-l+SV'S-t, tI.

and this is expressed by

V'= -J jP:B~d~ + ~JJ i'(~) G([_{')jilW)d~dr 't=const 2 't=CQIIst

(5.15)

(5.16)

by making use of the definition of R(I:) and the eq. (5.1). Substituting from (5.16) into (5.15), the last integral of (5.16) disappears because of the condition (5.14).

Next let us consider the gauge transformation. The transform1tion of B"" A and A' are given by

aoBii=i[Go, Bji]=O,

aGA=J., aGA'=A, (5.17)

where the following [C.R]'s are used;

-+ -. [Ail (~), A' (e) ] = -iG (~-e),

[E, A']=O, ~il=~o,.

[ ~~: ' A'J=iJCt-h

The state vector 1Jf' (.) is transformed in the following way:

oGIJf' = (aGs· S-1-iSG1S-1) 1Jf',

The right-hand side of this equation turns out to vanish, i.e. 1Jf' reveals itself to be gauge-invariant, if we take into account the eqs. (4.17) and (5.17). From this fact, we can see the gauge-invariance of eq. (5.15). In reference to the original Lorentz frame the eq. C5.1) runs--

where Bk is defined by

Bk=h~B .... ,

and satisfies the following condition

aRk - =0 1zk Bk =0. axk '

(5.18)

(5.19)

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The last equation of (5.10) is replaced by

[B,,(x) , Bl(x') J=-i( gkl+71,,11l) D(x-x')

+i (Y- + ll,,(ni ~ )).(~ + 11l(nh ~))(J(X-X')' a.1:k axi azl axh

(0.20)

and the definition of a is now

( a )2 1Zk_ (J=D axi

lim (I=O, lim (nk ~)(J = v 1 ) (n .. )-..O ( .... )-..0 ax" x,.xk (nz)-O

Finally the eq. (5.15) is represented by

id'F'(,) = V"'F'(r) , d,

(5.21)

(5.22)

All the expressions from (15.18) to (5.22) agree with those given by Schwinger.

§ 6. Proof of the equivalence of the present theory with the ordinary one (Fermi's theory)

In this paragraph we shall refer all quantities to a rectilinear coordinate system and also to the Heisenberg representation.

Let us recall the main features of the current theory due to Fermi. The Lagrangian used in Fermi's theory is

where Lm and V are given in (1.1). The canonically conjugate quantity to the electromagnetic potential is accordingly defined by

Ek=r~_g~(~~l),

the Hamiltonian density:

where H is the Hamiltonian density of our own theory. The field equations are;

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88 R. UTIYAMA

And finally the supplementary condition is

or

by virtue of the field equations.

Now, within the framework of our theory, consider the following unitary transformation

where (j) stands for any field quantity of our theory, From our eq. (2.17)"c, we obtain the foHowing [C.R) in

representation;

(6.1)

the Heisenberg

[E(.x), E(.x')] =0, (6.2)

where the two world points need not be simultaneous. By virtue of this [C.R] we have evidently

E'=E and [R, dR J=o. d.x°

(6.3)

The new field equation of the transformed quantity ~, turns out to be

d(j)' = i [(SitS -1 - idS . S-I), (j),]. d.x° d.x°

(6.4)

On the other hand, in consequence of (6.3), it holds;

dS S-I- .dR _ i J(E')2J ~ _. -t-_-- u.x, d.x° d.x° 2

so that the eq. (6.4) becomes

-=t HFI drjY .[ -

d.x° (6.5)

with Fermi's Hamiltonian H F • In particular, for (/J'=Ao', we obtain the following equation,

dAo' (JAk' , --=--_ -E, i.e. E'=E=(Fermi's EO). d.xo (J.xk

This is nothing but Fermi's equation. According to the transformation rule (6.1), our supplementary condition (2.20) becomes

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( 8Akl ).'11" ,_ 0 {8Elk-+ .,O} '11" 1-0 -- :1'0 -, --_- J ·:1'0 - •

8Xk 8Xk

Thus the required proof of equivalence has been completed. In concluding this paragraph let us return to the second case of §2 (page 90).

In this case we must express S as a linear combination of the canonical quantities, where S is defined by

(2.21)

The only possible form of S is

S=cE (c is an arbitrary constant).

If c is put equal to 1, then we arrive at Fermi's theory, whereas if c equal to zero, then our theory, treated as the first case, is derived. Among different choices of the value of the constant c, the case c=O is distinguished from other cases in the fact that the Lagrangian in its original form is restored on the inverse Legendre transformation from the Hamilto.nian obtained, only in that case of c=O.

§ 7. Application to the vector meson fieldS)

We encounter with the same kind of difficulty in case of the vector meson field as that of the radiation field. The Lagrange function of the system of the vector-meson and nucleon fields is taken, as usual, to be

L=L".+L .. -V

where L.. belongs to the nucleon-field in free case, whereas Lm and V are given by,

and

(7.1)9)

Here we describe the meson field with a vector All- and a redundant scalar field -<po The canonically conjugate quantities to AI£ and 'P are defined by

8) In this paragraph, we shall refer all the quantities to an orthogonal rectilinear ~-system. 9) Y. Miyamoto, Prog. Theor. Phys. II (1948), 124.

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90 R. UTIYAMA

EI'-= Fl'-O + g J~o EO=O, x

(7.2)

Let us introduce as was done in § 3. following solution;

a new quantity EO which is canonically conjugate to Ao Solving (7.2) wIth respect to AI'- 0, q;,o, we obtain the

A- . =E-+A- -_K J'ii.ii 1'-.0 I'- 0, I'-X

Ao,o=A=indeterminate,

q;,o=xAiJ+ 7!+.Li. x

If we put A equal to, with two arbitrary constants a and b,

A A b EO = iL,jL-aq;--2

(7.3)

as in case of the radiation field, then we obtain the following Hamilonian density

H=Ho+V',

H,.o=~EjiEji+E(i..A-+EO.A- --alfJ. EO_~(EO)2 2 I'- 1'-'1'- T 2

f o 0 0

g EjL ·ji.o + -'0 -0 + g- 'iio'iio, g-. 'I'-> ---;- J 2x2J J 2;-J J T 2x211'->1 - (7.4)

The nonvanishing [C.R]'s related to the meson field are

(7.5)

Thee quations of motion of the meson field derived from the Hamiltonian thus. estau1ished, run as follows;

A- - -Eii+A' _-KJ'ii,O ~'o - o,~ ,

x

Ao,o=Av:.v:-aq;-bEO,

~'o =xAo+ 7!+ f i, x

(7.6)a

(7.6)b

(7.6)c

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On the Covariant Formalism of the Quantum Theoy)' of Fields, II 91

. a ja'V: 7r,o=aEO-~(xA--tp,-) +-~. (7.6)f

a~1'- I'- I'- X a~1'-

Combinations of these equations give

(O-x2)AI'- + (I-b) aEo + (x-a) a! + jJI'-- g all'-II =0, a~1'- a~1'- x a~II

(O-ax)tp+ (a-bx)Eo=o,

(O-ax)Eo=O

provided that the equation of continuity

a'l'--'L =0 a~1'-

(7.7) a

(7.7)b

(7.7)c

is taken into account. As one easily sees, the eg. (7.7) b turns out to be a consequence of the remaining eqs. (7.7)a (7.7)c and (7.6)b. Thus the eqs. (7.7) are not mu.tually independent.

Now our Lagrangian is invariant under the gauge transformation

(JG¢=(JG1} =0,

oGAI'- = :;1'-' (JG'!'=x).. (7.8)

The generating operator of this infinitesimal transformation is

(7.9)

and satisfies

(7.10)

as in case of the electromagnetic field.

The arbitrary function A which is a scalar, must satisfy, in the present case, the following equation;

(0 -ax)A=() (7.11)

by virtue of the gauge invariance of (7.6) b, EO being assumed as gauge invariant. Actually the eq. (7.10) is derived by using (7.11) and the field equations.

Now we have to propose the following supplementary conditions

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92

where

R. UTIYAMA

to be valid on q,

aE~ F=---1t7r.

a~ji:

(7.12)

From the eq. (7.10) we can easily show that the conditions (7.12) are mutually compatible and, moreover, consistent with the field equations. This amusing circumstance depends on the clever introduction of the redundant field ¥,.10)

Let us introduce a new quantity Uv. defined by

_ 1 a¥, 7 Uv.=Av.-- -. , ( .13) 1t a;v.

then from (7.7)a and (7.7)b, we conclude

(O-1t)Uv.+(l-~) a:o + iiv.- g a.,(v." =0, 1t a~v. 1t af"

and the (7.6)b becomes

(7.14)

Hence, provided that the indeterminate const. a is not equal to zero, we can replace (7.12) by

(7.12)'

In particular, if we put a and b equal to 1t and zerO respectively, then from (7.14) (7.15) we obtain,

(O-X2)Uv.+/lv.- g ajJl.lI=O, 1t a~lI

auv. -Eo afJl. - J •

The eqs. (7.7)b and (7.7)c, are, now, replaced by

(O-1t2)¥,=0, (O-.~2)Eo=0,

and (7.11) runs as follows;

(O-1t2)A=0.

The eq. (7.14)' is tbe usual eq. of the me.:on field.

10) E. Stilckelberg, Helv. Phys. Acta 11 (1938), 299. K. Husimi, cr. footnote (2).

(7.14)'11)

(7.15)'

11) Of course other choices of a and b are also possible, for example~ a=b=O, or a=x, b=l. It can be shown, howev~r, by means of the suitable unitary transformations, that these cases are equivalent to our case.

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Next let us adopt the interaction representation. In this case, the equations of fields are those of the free fields, and by virtue of these we can eliminate the canonical momenta from the interaction energy density V'. Thus V' gets the following form;

(7.16)

or, describing this with regard to the original Lorentz frame, we obtain;

The normal-dependent terms, which are contrived previously through integrability conditions, emerge now quite naturally in our theory.12)

The Tomonaga-Schwinger eq. runs as follows;

.'aW (r) _ V-, 111" (_) t---- ':r " ,

'ar r=-n,.xlc,

or

i aW[o] V,,' qr [a], aa"

where V' can be expressed as follows by using the definition of U.". ;

V'= -fU jiJ.+g u /.".~+ g2j' J.""~ .". 2x""> 2x2 .".~

In the case where a and b are put equal to x and ° respectively, the [C.R]'s of the meson field are given by

[EO A']= _1'aJ(~-e) ,.". 'a~"'"

[~, A~J= -ixJo J(~_'1)'aJ(r;-e) (dr;) 4,

0 1 'a1)""

[~, EOI]=ixJ(~-e),

[EO, EO'J=O,

[~, ~'J=-iJ(f-~') +ix2fo J(~-r;)J(7)-~') (dr;)4, al

12) cf. foot· note (9) and also S. Kanesawa and Z. Koba, Prog. Theor. Phys. 4 (1949), 297.

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94 R. UTIYAMA

[U , U~J=-i(d(~-~') _~ a2d(~-e) ). Iol r Iolll X2 a~lola~lI (7.18)

]n case of other choice of a and b, there appears the invariant delta function D. Now we want to investigate an alternative type of the interaction part of

.the Lagrangian.

Let us adopt the Lagrangian of the following form, instead of (7.1) ;

L=L",+Ln-W

where L", and L,. are given by (7.1), but W is defined by

W=-fjIolA +1-F jlolll + g2p~ . Iol 2x Iolll 2x2 Iolll

The Hamiltonian density corresponding to (7.19) is

H'=Do+W',

(7.19)

(7.19)'

W=-fA jlol+_K_F--j~._ /{ EP-jli'o+ g2 lOj"it + g2 ~ jlolll (7.20) Iol 2x Iolll X 2x2 2x2 Iolll ,

and Do is the same one as in the former Case. The Lagrangian (7.19), now, admits the following gauge transformation,

instead of (7.8) :

iJG<p=iji.Sb, iJG<pt = - zlN,

iJGA .. = a)~, iJGm=x)., ~ a;1ol T

and the generating operator for (7.21) is given by

G=J {EO a~. +(X7r- aE~ - ti);'}dt a a;-o aflol

In this case, the supplementary condition is

F' aEii +"c.u =--_- JJ -X7r. a~1ol

on (1,

(7.21)

(7.22)

(7.23)

In the interaction representation, the field equations are the same as in the former case, (we have put a and b equal to x and zero respectively,) whereas the interaction energy density W' now turns into,

(7.24)

Now let us consider the elimination of the term depending on the nucleon field from the supplementary condition as was done in § o.

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On the Covariant Formalism of the Quantum Tlteory of Fields, II 95

Consider the following unitary transformation;

then we obtain

IJT (r:) -IJT' (,) = SIJT (r),

S=/R, R=_fJi.rpd"f x 0

the supplementary conditions are transformed into

EOIJf'(r:) =0, FIJT'(r:) =0,

and the Tomonaga-Schwinger eq. runs as follows;

. dlJT'(.) t W"IJT'(r:) ,

d.

where

(7.2.5)

on a (7.26)

(7'27)

Since the new expression for the interaction energy turns out to be just the same as the previous one V', the equivalence of the both theory has been proved.

In the latter theory it is an easy matter to make the meson field agree with the electromagnetic field by making x and g tend to zero, whereas in the former

case it seems somewhat difficult because of the term : rp'[J.j[J. in the interaction.

Acknowledgement

I would like to express my heartful thanks to Professor K. Husimi for his kind interest and valuable advices.

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on the covariant formalism of the quantum theory of fields, ii

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