IL NiiOVO C1MENTO VOL. XX, N. 4 16 Maggio 196I

Solution of the Equations for the Green's Functions

of a two Dimensional Relativistic Field Theory(').

K. JOHNSON"

Deparlme~t o/ Physics a~d Laboratory o] Nuclear Science, Massachusetts I~stitute o] Tecb~wlogy - Cambridge, Mass.

(ricevuto l ' l l Marzo 1961)

S u m m a r y . - - - T h e explicit solution of the coupled set of equations for the Green's functions for the self-coupled field theory model of Thirr ing is given. I t is found that the infra-red problem causes no special difficult)'. The question of how lo define products of singular field operators at coincident space-time points arises and it is shown that the commu- tat ion relations for such products are not consistently given by making use of the formal expressions and the canonical commutation relations. The problem of gauge inval'iance in an external field is studied and i t is shown tha t : a) the current and charge density do not commute at equal times, and b) this is necessary for and consistent with the gauge invariance of the field equations.

1. - I n t r o d u c t i o n .

THIRRING (1) has p r o p o s e d a two d i m e n s i o n a l (one space , one t ime) self-

coup led field t h e o r y m o d e l which is of some i n t e r e s t because i ts e x a c t solu-

b i l i t y enab les one to s t u d y some of the gene ra l con jec tu re s which h a v e been

p r o p o s e d in r e g a r d to t he b e h a v i o r of local r e l a t i v i s t i c fields. I n sp i te of the

mode l , no gene ra l so lu t ions h a v e been p r o p o s e d which a re free f rom po,ssible

c r i t i c i sm because of t he r a t h e r f o r m a l m a n n e r in which t h e y h a v e been ob-

(') This work is supported in part through AEC Contract AT(30-1)-2 98, by !funds provided by the U.S. Atomic Energy Commission, the ()ffice of Naval Research and the Air Force Office of Scientific Research.

(1) W. Tt~RRINt~: A'~ttt. O/ Pbys., 3, 91 (1958).

774 K. JOHNSON

rained (2). The model is exact ly soluble only when the mass pa rame te r of the

field vanishes, and in this case no dimensional pa rame te r remains (the coupling constant is dimensionless). Thus, the use of a formal construct such as an

a sympto t i c field which p resumably has meaning in a theory where there are

dimensional quant i t ies is in this c~se a ra ther dubious procedure since no para- mete r exists in the theory to characterize the (( a sympto t i c ~) domain. Even

worse, since the theory describes a field with no mass, p resumably all real processes involve infinite numbers of infra-red pairs and thus even the concept

of particle scat ter ing states with finite occupat ion number is meaningless. In this note we shall show how the vacu um expecta t ion values of all t ime-

ordered products of the fields, (i.e., Green's functions) m a y be calculated. With

the aid of these, all proper ly put questions about transit ions can p resumably

be answered. The definition of these Green's functions however does not re- quire the existence of an a sympto t i c field operator , r a ther the la t ter construct

would follow if the Green's functions exhibi t a s t ructure consistent with the

existence of mult iple particle scat ter ing states. However , the absence of this

type of s tate does not preclude the existence of Green's functions. We shall calculate the solutions of the model by making use of the invariance of the theory under two continuous gauge groups. The solutions will enable us to s tudy such groups and the operators which generate t hem in sufficient detail t ha t we can also apply the results to the s tudy of more realistic field theories

which are invar ian t under such groups. In part icular , we sh~ll settle the ques-

tions associated with the lack of commuta t ion of the charge and current den-

s i ty operators a t equal times.

2. - Solut ion of the model .

The field equat ion of the model is

(1) ~,,,1_ ~.~(x) = ~/'(x) ~,,~ ~(x) [ i

where the current j ' (x) is the (~ def ined , by the formal expression

(2) j~'(x) = 1 [~(x)~" , ~ ( x ) ] .

). is the dimensionless coupling constant . The Dirac matr ices y" can be t aken as the 2 x 2 set, (~o, yl)_~ (a2, ial) which is completed by y h = y ° y l = o a . We

(~) V. GLASER: Nuovo ('ime~to, 9, 990 (1958). F. SCARf': Phys. Rev., t17, 868 (1960). T. PI(ADHAX: Nucl. Phys., 9, 124 (1958-59).

5OLUTI()_N I)F TIlE EQUATIONS FOR TI~E GREEN'S FUNC]'ION. < ETC. 775

shall solve for the Green's functions of the theory by way of the so-called

~, generalized ,~ Ward ident i ty ('~). Thus, since the field equations are invar ian t

under the group

(3) b~v(x) = i~p (x ) ,

the current is conserved

(4) ~"),~(x) = 0 .

Fur ther , because the mass of the particle has been t aken as zero, (1) is also invar ian t under the group

(5) @(x) = i~7~(.r)

and thus, the pseudovector current which is (again formally) defined by

(6) j " = i[~(x)7"r~, ~(x)] = ~""L(x),

where ~"~ is the an t i symmetr ic symbol, is also conserved,

(7)

Thus, bo th curl and divergence of the vector field j ' (x) vanish. (4) and (7)

lead to a set of two <( Ward identities )~ for an a rb i t ra ry ver tex which can be in tegra ted to give the ver tex in terms of the corresponding Green's function. This is the reason which accounts for the solubility of the model in closed form.

For clarity, we shall confine the discussion to the lowest order Green's func- tions, defined by

(S) V÷(xy) = i(OIT(~v(x) ~ (y ) ) 10),

(9) e+(xx ' yy ' ) = - - (0 ]T(~v(x) p(x ' ) f ( y ' ) Vp(y)) t 0 ) .

The equat ion for (8) implied by the field equations and canonical com- muta t ion relations is

(10) 7" ~- ~,,G+(.ty) = ~(xy) + )~7"i Co i T(L(x) ~(x) ~(y)) Io>. i

The ver tex which enters this equat ion is connected to the next higher order

Green's function (9) by the formal expression for the current (2). However ,

('~) Y. TAK,~.JlASHI: ffuoVo Cimen,to, 6, 371 (1957) .

776 K. JOHNSON

because of the conservation eq. (4) and (7) satisfied by the current there are

two identities satisfied by ~he general form of the ver tex

(11) i<0 IT(j.(~) ~o(x) ~(y)) 10)

which are sufficient to calculate the form of (11) wi thout making use of (2).

To derive these identities we mus t know the equal t ime commuta to r s

(12) [~o(x), jO(y)], [~(x), j l (x)] .

I t is a t this point where it is necessary to explicitly take note tha t the

formal expressions (2) and (6) are meaningless because of the singular na ture of the Green's functions on the light cone. Thus the operator

(13) ~(x -~ e)7 ~ y)(x)

has singular mat r ix elements in the l imit e -> 0, and in order to give a proper definition of the current a well-defined l imiting prescript ion mus t be given

in te rms of the non-local opera tor (13). We shall discover what tha t definition is below. I t should and will be such tha t the opera tor j"(x) is covar iant and is conserved. The same remarks hold for e~vjv(x ) = j ~ ( x ) . Once it is explic- i t ly recognized tha t j~' mus t be defined as a l imit applied to a non-local oper- a tor it is also evident t ha t the commuta to r s (12) need not be correctly given

by making use of the formal expressions (2) and (6) and the canonical com- muta t ion relations. In fact only if j ' (x) can be defined in te rms of (13) when

only spacelike displacements s" are involved will (12) follow with the use of the canonical commuta t ion relations alone. However , it is clear tha t if j ' (x)

can be consistently defined as a conserved operator , then the equal t ime com- muta t ion relation mus t have the form

(14) [y)(x), jO(y)] - - _ a b ( x - - y)~p(x)

since when j~ is conserved, jo generates the local form of the gauge group (3). The constant a however, is not specified by such a group argument . Of course, a mus t turn out to be 1 if ju is definable in te rms of (13) when e is always space- like, since then the canonical commuta t ion rules yield (14) with a--~ 1. Like- wise, if e~vjv(x) : f ' (x) is conserved we mus t have

(15) [ ~o(x), jO(y) ] -~ ~i 5(x - y)y5 ~o(x).

With these expressions for the equal t ime commuta to r s we obtain the iden-

SOLUTION OF THE EQUATIONS FOR THE GREEN'S FUNCTIONS ETC. 777

t i t les

(16)

(17)

~, [i(01T(j."(~) Ip(x) ~(y)) /0}] = a[5(~ - - x) - - 5(~ - - y) ] i (0 I T(~(x) ~(y)) I 0 ) ,

~ ~,[ i(O I T( j ' (~) ,f(x) ~(y) ) I0} ----

~-- ~ [5 (~- - x)ys~ ÷ b(~- - y ) T j i ( O t T( ~f(x) ~(y) )10 } ,

where Ys~ acts on y~(x), ~5~ on ~(y). Because of the invariance (5)~

r5 G(x - - y) = - - G(x - - y)y~ ,

so (17) can also be wri t ten in the form

(18) ~ [i(O [T(f~($) y~(x) y (y ) ) 10}----~(5(~ - - x) - - 5(~ - - y ) ) ~ i(O (T(v;(x) y(y)), I 0 ) .

Eq. (16) and (17) specify completely the divergence and curl of a vector field.

Fur ther , the v a c u u m ma t r ix elements impose bounda ry conditions. Thus, we

can integrate (16) and (17) to find

(19) i (OIT(i"(~ ) yJ(x) ~(y) lO) ----

-- (g"~ a + s "~ ~ 7~) c~ [ +($ - - x) - - D+ (~ - - y)] G+(xy) where

__ ~ 2 D + ( x ) = ~ ( x )

and is the function with positive frequencies for x ° ~ 0, negat ive frequencies

for x ° ~ 0 ( 4 ) . The constants a and d are so far undetermined, but the impor tan t thing

is t ha t the form of the ver tex function is given b y the conservat ion laws alone.

In order to find the equat ion for the Green's funct ion G+(xy) we mus t allow the space-t ime point ~ coincide with x. T h a t is, we mus t define the product of the singular operators j ' (x) and ~0(x). We shall define the produc t by the

l imit

(20) / ( x ) ~ y . j " ( x ) ~ p ( x ) - ~ l i m ½ [ j " ( x +~)y.~p(w) + r .~p (x ) j " ( x - - s ) ] . e--~ 0

We shall find tha t our general definition of the products of singular operators to be given below is in this case equivalent to (20). Thus, we obtain (5)

(21) i7"(0 I T( j , (x ) y~(x) ~ ( y ) ) / 0 } ---- - - (a - - ~)7"~ D+(x - - y) G+(x- - y ) .

(4) See appendix. (5) We note t,he difficulty which would occur if a=~, namely, the vertex would

vanish. This would imply that the operator ](x)= yt'~l/i)~t,~f(x) vanishes if the theory is local and the metric in Hilbert space positive (6). However, the vanishing of ] would be in contradiction to (25) with a=~.

(6) p. FEDERBUSII and K. JOI~NSON: Phys. Rev., 120, 1926 (196.).

778 K. JOHNSON

The equa t ion for G+(xy), (10), m a y then be in tegra ted and yields

(22) G+(xy) = exp [ - - i;t(a - - ~)[D+(x - - y) - - D+(0)]] G(°)'x+tyJ',

where G~)(xy) is the free field funct ion,

(23) y " 1 ~;,G(~'(xy) = 6 (xy) .

W e shall discuss (22) in Sect ion 3.

Le t us now consider the equa t ion for the Green 's func t ion G+(xx'yy ' ) ,

(24) y~ l ~ G + ( x x ' y y ' ) =- ~(xy)G+(x 'y ' ) - - 6 (xy ' )G+(x 'y ) - -

-- ,ty~ <01 I' [j.(x) y~(x) F(x') ~(y') ~(y)] I 0> .

W i t h the use of the same m e t h o d t h a t was appl ied to the ve r t ex to give a

closed equa t ion for G+(xy) we can also calculate the fo rm of the Green 's func-

t ion G+(xx'yy ' ) un ique ly in te rms of the two cons tan ts a, ~. I f we then require

t h a t G+(xx'yy ' ) yield the ve r t ex (19) as x'--y'---->O, we will enforce the coup-

l ing be tween the var ious Green 's funct ions impl ied b y the field equat ions . This

r equ i r emen t yields two equat ions for the cons tan t s a and ~. The general fo rm

of the ve r t ex which enters (24) is ob t a ined as

(25) - - <0 IT[j"(~) yJ(x) ~0(x') ~ (y ' ) ~(y)] ] 0> =

= [a~ '(D+(~ - - x) ÷ D+(~ - - x') - - D+(~ - - y) - - D÷(~ - - y')) +

+ ~ s " ~ ( D + ( ~ - - x)y5~ + D+($ - - x ' ) y s . - D+(~ - - y)y~, -~ D+(~ - - Y'))Ys¢] G+(xx ' yy ' ) ,

b y m a k i n g use of the conserva t ion equa t ions and equal t ime c o m m u t a t i o n

rules. W i t h the use of (25), (24) can be in teg ra ted and gives

(26) G+(xx'yy ' ) =-

= exp [i2(a - - ~ ~5,ys,,)[D+(x - - x ' ) - - D+(x - - y ' ) + D+(y - - y') - - D+(y - - x')]] •

• G+(xy) G+(x'y') - -

- - exp [i2(a - - ~ Ys~Y~')[n+ (x - - x ' ) - - D ( x - - y ) -~ D+ (y - - y ') - - D+ ( y ' - - x')]] "

• G+(xy)' G+(x'y) .

I n order to calcula te the ve r t ex (19), we allow x ' - - y ' ~ s - + 0.

the Green 's func t ion G ( x ' y ' ) = G(e) becomes singular,

7s 1 (27) G(e) --> s G ' 2 ~ .

I n this l imit

SOLUTION OF TIIE EQUAT[(~NS FOR TIIE GI~EENSS FUNCTIONS ETC. 7 7 ~

We also note t h a t the exponent ia l fac tor in the first t e rm of (26) approaches

exp [ i ;~ (a- ~ r~. y~ ) e ~ [D+(x-- x') - - D+(y - - x ' )]] .

Thus, if we average the l imit ing forms for e and - - s we find a contr i -

)u t ion f rom the first t e rm of (26) mul t ip ly ing the q u a n t i t y

8 ~ 6 8

g2

i or small e.

pe ra to r

(.,8)

I t is thus clear t h a t if the cur rent is defined in terms of t h e

/"(. : e) = ½ [W(x + e)y"w(x) - w(x) ~(x - - e)r"]

a~ e-->0, the l imi t ing opera to r will be finite bu t no t covar ian t , since e%a/e -~

a pproaches a finite l imit as in any f rame, b u t <~ remembers ,~ the f r ame in which

t m componen t s of e vanish. I t is no t possible to m a k e a covar i an t l imit b y

a eeraging over all direct ions d' in a hyperbo l ic space because the vo lume of

the surface e 2 = const is d ivergent . We shall do the nex t bes t thing'. Thus ,

t f k e e ' to be t imel ike and define a reciprocal spacelike vee tor b y

\'~ e shall then let

In this case, since

E' 8t¢

j ' (x) =: }[j ' (x; ~) + ?(x ; ~)]~+o. 7-+o

e ~ s¢ ~ ~

we ob ta in a covar i an t l imit ing operator . Consequent ly , the first t e rm of (26)

as well as the second gives a con t r ibu t ion to the ver tex and we ob ta in

(3() i (04 I ' [i"(~)~(.r) ~(y)] I 0) =

I f we compare wi th (19) we find

1 1 (31 , (~ -- d -

7 8 0 K. JOflNSON

In summary , we find t h a t the definition (29) of the current operator leads

to a conserved, covar iant opera tor whose commuta t ion relations with the field

are given by (14) and (15) with constants a, ~ which differ f rom unity. This

conclusion is not in contradict ion with the canonical commuta t ion rules since,

the definition (29) involves infinitesimal displacements in bo th t imelike and spacelike directions. Finally, it m a y be remarked t h a t the above procedure has been applied to the general Green's functions with the result t ha t (31)

emerges consistently in the general case. Hence, we have a complete and

consistent solution of the model.

3 . - R e n o r m a l i z a t i o n .

I f we replace a and ~ by (31) in the expression for G+(xy) we have

(32) G+(xy) = exp - - i ~ _ (~/2~) ~ [D+(x -- y) -- D+(0)] G(°+)(x -- y).

The Green's function D+(x) is singular at x ~ = 0. In fact, the solution is

(33) D+(x) = (-2~)~ log tx2/ ,

where (1/~) i is an a rb i t ra ry constant with the dimensions of a mass. Thus, the posi t ive-negat ive f requency boundary condition does not specify the con- s tant , zero f requency t e rm in D+(x). ~Naturally, since only differences of D+

enter the equations, the constant t e rm is a rb i t rary . However , since D+(0)

diverges, the solution (32) is only formal, t ha t is, the Green's functions require a renormal izat ion since the coupled fields do not admi t a t rue canonical com- muta to r . We m a y renormalize by making the ma t r ix element of the oper- a tor y~[(Oly~[p~'}, p2=--m2] take on a chosen value at an a rb i t ra ry state.

:Naturally, no physical question will depend upon either the normalizat ion

chosen, or the mass of the state. In this way we define the (( renormaliza-

t ion ~) constant

I ).2/7~ ~ , 0 , ] [ (/t]2~) ~ xo 2 ] (3 4) Z2 = exp i 1 - - - ( ~ 2 - ~ i)+ ( )] = exp -- 1 -- (;t/2zc) 2 log x.~]~,_~o,

= 0 ,

so the ~ renormalized ~) Green's function is

G(xy) log 1 G,0)(xy) _--= (35) G+(xy)= exp 1 - - (~t/2~) 2 ( x - y)q Z2 '

SOLUTION O~ THE EQUATIONS FOR THE GR]~:EN~S ~UNC~IONS ETC. 781

G(xy) admits the spectral representat ion in m o m e n t u m space

(36)

where

¢o

sin zt~ f dk ( k / ~ - - 2yp 1 G(p) = - - ~ - - j ~ x~,, ~ x ; ~ - ~ (P:/m~)~"

. . . . . a n d Z- (~12~) ~

~ = ~ \ - ~ . ' / ,

(we assume ~ 1). We notice t ha t an a t temp~ to expand the spectral func- t ion in powers of ~ (or 4) would lead to a sequence of infra-red divergent in- tegrals. The cause is the zero mass of the field and the confluence of an infinite sequence of thresholds a t k = 0. Because G has no pole, a single par- t icle s tate does not exist in the theory, and hence also, no a sympto t i c field. The theory provides a perfect model of the infra-red s t ructure of massless fields. Thus, the particle Green's funct ion in q u a n t u m eleetrodynamics has precisely the singulari ty described by (36) when p2_+ p2 d- m~ ~ 0 (7).

4 . - Coupl ing to a n e x t e r n a l field.

I t is interest ing to discuss the coupling of the current to an external field because such a coupling presents problems quite analogous to those regarding the questions of gauge invar iance in quan tum electrodynamics.

Suppose the equat ion in an external field is

(37) ]

~"-~ ~s~)(x)= aTS~s(x)v;(x) + 2j"(x)"y~,Vz(x).

The renormallzing factor a is included for convenience since we have al- r e ady discovered tha t the local group genera ted b y jo includes such a factor. Thus, (32) is invar ian t under the group

(3s) { ~yJ(x) = iac¢(x)~(x) ,

~%(x) = ~.o~(x),

if we assume tha t the current operator is invar ian t under this group. Again, jU(x) is a conserved vector field with this assumption. To insure the invar iance of the local current under (38), we shall define it as the local l imit of the gauge

(7) K . JOtINSON a n d B . ZUMINO: Phys. Rev: Lett., 3, 351 ( 1 9 5 9 ) .

50 - II Nuovo Cimento.

752 K. JOHNSON

invar ian t (s) noa-lo¢al ol~rago~

(39) 7(x, ~) = x + s x 'I [+1 [ = -5 -~(x ÷ c)y'V~(x) exp ~"q~, -- y,(x) ~ (x -- e)y" exp - - iafd~'q~,]l J . . ' / l '

X--8

in the sense of the ",li~itiag procedure (29). I t might be supposed tha t (37) is also invar ian t under th~ group

(~o) I a~(z) = i~a(x)7~(x) ,

a%(x) = ~,,, ~ 'a(x) .

this is to be true, we should also be able to write (37) in the form

(~) 1 D

In this case j~ mus t be defined in te rms of the non-local opera tor i n v a d a n t under (40), namely

(42...} ~"(x; e) = :i ~ ( x + e)~'y5 exp ig~ d~"e.,~ v

z + e

v;(x) - x

x

in the sense of (29). I t w o ~ d uot be possible, for (37) and (41) to be mut t ta l ly consistent if the relat ion j " = g'~j~ held for the current operators in the pres-

ence of an external field. We see t ha t the operators j" and f ' axe no longer necessarily connected in this way since they are defined by a l imiting pl~)ce- dure applied to non-local operators no longer related by e "~. In fact, if eq. (37) and (41) are consistent, the connection between the operators j" and j " mus t be

(43) 1 1 - - ] l~V E P v . . " - - ll~'~,,) f ' = et"j, -I- (a - - a) ~. e F~ = 1~ t ~ ] - - t'./,~U, ~ ,2--, 2 e T v .

W e shall fro4 t ha t this is indeed the re la t ion between the opera tors implied by the definitions (39.') and (42).

(s) j . SCtI~VINGER: Phys. Rev., 82, 664 (1951).

SOl,UTION O1-' TIlE EQUATI()NS FOI~, TI lE ( ;REEN'S FUN( ' i ' IONS ETC. 7~3

To show this , le t us accep t the cons i s tency of (37) and (41}, or, equ i .

, ' a l en t ly , (43). W e m a y then calcula te the v a c u u m cur ren t i nduced b y the ( x t e r n a l field b y m a k i n g use of the conse rva t ion laws for jr, a n d j C W e con>

pa re this wi th the cu r r en t ca lcu la ted b y first d e t e r m i n i n g the Greer/ ' s functioi~

in the externa.1 field a n d t h e n m a k i n g use of the defitfit ion of the cu r r en t oper - a to r in t e r m s of (39). A s imilar c o m p a r i s o n can l ikewise be m a d e work ing

, r i th the genera l Green ' s func t ion in which case it shows t h a t the defini t ions (39) and (42) lead to operators re la ted b y (4;3).

L e t @utl= <0, t -+ + co I, ] in) = ]0, t -~ - - oo>, where the ex t e rna l pe r tu r - b a t i o n is t u r n e d off in the d i s t an t fu tu re and pas t . h i v i r t u e of the conser- va t ion of cu r ren t

(44) , (ou t i j"(~) i in> = 0

and because of the conse rva t ion of j " and (43)

(45) ~. <out ! j"(~) l in) -

H e n c e

(4~) ( o u t ! j"(~)lin), gout !in.)

1 1 s Fv(~) ( o u t iin3 .-tl - - (2/2n)" 7/, ' ' , "

1 1 i - - (~/..,~)'- e ''~ ~ e "~ GfD~(~ - ~')~.(~')(d¢').

N e x t , we calcula te G~(xy: el) and then show with (39) t h a t (46) resul ts . T h u s

(47)

a a d

G+(xy; ~ v ) i ( ° u t i T ( W ( x ) ~ ( Y ) ) bin} , :{out [in )

(4s) yt'~ - %G+(xy; q~) = b,(xy) + a7%f~,(x)G+(xy; cp) +

+ 2 7 " / ( o u t !T(j~,{X!V(x ) -~(y) ) i in) <out l in )

I f we again use (13) and tile conse rva t ion equa t ions for j~ and j ' tile, v e r t e x func t iou is

(49) i ( o u t I T(f '~p~)[in> " T o u t ]in> - - - [a~ *'~ -'- ag '"ya] ~, [D+(~ -- x) -- D+(~ - - y)].

1 1. • ¢4.(xy: ~) q;'(~)6~(x~l; el) +

' = 1 - (2 /2n)" "

i I f . , .~ n 1 - (),/2n) 2 |eJ ~ n+(~ - - ~')q~v(~')(d~')'G+(xy: of) .

7 8 4 K. JO~[NSON

If we let ~ -+ x and use this expression in (48) we can integrate the resulting equation to find

(50) G(x~y; ~c) = exp [-- ift(a -- ~ ) [ D + ( x - y ) - D+(0)]] •

• e x p [ - - i a f [ D + ( x - - ~ ) - - O + ( y - - ~ ) ] 5~q~, (~)(d~)] •

.We t h e n let x - - y - + 0 and calculate the current. With the impor tan t fac- tors which make the operator j"(x; e) gauge invari~nt, the result agrees exact ly with (46). B y making use of the techniques used in this section explicit for- mulas for the general Green's function in an external field m ay be given.

5. - The current-charge density commutator.

The gauge problem similar to one in quan tum electrodynamics may now be raised. The question arises from the lack of commuta t ion at equal times of the current and charge density operators (2,~). In quan tum electrodynamics the commuta to r is proport ional to a positive divergent integraI also identified as a (~ photon mass )>. I t hence has been argued tha t (~ since in a gauge invariant theory the photon mass must be zero, this commuta tor must vanish )>. We wish to point out tha t the commuta to r in fact cannot vanish, and secondly, tha t this entails no contradict ion to gauge invariance, bu t indeed is required by gauge invariance. We shall investigate the question of this commuta tor in the context of the model studied in this paper but exact ly the same general remarks apply also in quan tum electrodynamics (10).

The current operator j~(x) has been defined so tha t it is invariant under the group (38). The generator of (38) on the field operators ~f is the opera- tor jO(y). Yet, it cannot be t r u e : t h a t (10,~)

(51) [ j l (x) , jO(y)]~°=,. -- 0

for this would raise a contradiction to the conservation of charge. Let us review briefly tha t contradiction. I f (51) is assumed, then f rom ~l,j "=- 0 i t

follows tha t 0 ---- [~111(x), jO(y)] = _ [3ojO(x), jO(x)]

o r

o = [[jo(x), HI, ? ( y ) ] ,

(9) F. L. SCA~: Nucl. Phys., 11, 475 (1959). (lo) K. JOHnSOn: to be published. (~1) j . SCHWI~R: Phys. Rev. Lett., 3, 296 (1959).

SOLUTION OF T t IE EQUATION, < FOR TIIE /t~EEN',~ I'~UXCTiONS ETC. 7 8 ~

where H is the Hamil tonian operator. eigenstate (HI0) = 0)

(52)

which implies

But since (O I is the lowest energy

(OIj°(x)]E, y} E(E, ylj°(y)[O} : 0

(53) < o l ? ( x ) = o or <0I?*(x) = o .

This, in a local, relativistic field theory, means tha t i t ' (x)= 0 (6). Hence (51) contradicts the conservation of charge, and the non-vanishing of j'(x). Yet, if (51) does not hold, how is it tha t the current operator is invariant under the local gauge group which is necessary for the law of charge conservation? This dilemma can also be sharply put by observing tha t the usual formula for the current induced in the vacuum by an external perturbation (for simplicity, to lowest order in the external field) is

= i f (OIT( f ' (x ) j~(y) ) 10}F~(Y)(dy) (54) ~out ] j'(x) i in ~

and h'om (54) it would appear tha t the induced current is not conserved unless [j°(x), ~(y)] ~ 0, i.e., unless j~ is invariant under the group generated by jo.

The operator which generates the gauge group (38) is

= -f(dy)~ ~(y)),,(:,/), G (55)

thus

(56) ]

b~(x) = ~- [V~(x), G] = ia~(x)~(x).

But the /ull invariancc for the theory means we must also perform the com- pensating change in the external potential

(57) bqJ'(x) ~_ ~ '~(x) .

Thus, the change in the current operator produced by the generator (55) is

6j~(x) = i tdy~(y)[ j ' (x ) , j°(y)], ,/ l

and because j'(x) is invariant if we also perform (57), the change in the current produced by (57) alone must be

(58) -Si~(x) = -- i f [f'(x), j°(y)~ •

786 x. JOHNSON

The inv~riance of jr(x) under the combined t ransformat ion is guaranteed

since the non-local form of the current is invar ian t under this full group a t every stage, hence the l imiting opera tor will also be character ized b y this Jnvariance. Accordingly, if [~'{x), jO(y)] does not vanish, there is no contr~-

diction to the conservation law, because (54) is not a correct equat ion for the induced v~cuum current. This is because (54) is derived with the (implicitly

made) assumpt ion t ha t the current opera tor does not explicitly depend upon the external pe r tu rba t ion ¢ , and this assumpt ion is not consistent with (58).

In fact, if we take into a~count the explicit dependence of j~'(~) on ~ ( ~ ) de-

noting it b y

(59) ~j"($) = S~"~,(~)

then the correct version of (54) is

(60) (ou t 17"(~) l in) =f(i (0 1T(j"(~) j"(~')) [0} -i- S ''~ 6 (~ - - ~')) ~,(~')(d~') ,

(in this model, 8 ~'~ turns out not to be an operator) .

Since

we find

0 = ~ , (out I j " (~) l in) ,

~(~o _ ~,o)i(o ][jo(~), j"(~')] I o) + s - ~ , ~(~ - - ~') = o

or S °~= S ~°=- S ° °= 0 and

(61) <0 [[~0(~), jl(~')] 10 > = iSl1~1~( ~ __8 r) .

Now, this s t ruc ture for S ~ indicates t h a t i (0 IT( j ' (~) j ' (~ ' )}0) is not a eovar ian t

function. Indeed this is the case as we can now show b y an explicit calculation of (oly ' (~)y(~ ' )10} • This ma t r ix e lement m a y be calculated f rom (18), and

the definition of the current opera tor in te rms of (28) and (29). Thus we find

(62) (z/z~)

where --D2D(+)(x)= 0, and D (+) contains only posit ive frequencies. The t ime-

ordered funct ion is defined b y

(63) <o l T(j"{~) j~(~')) I o> - o ( ~ o - ~,o) <o l J"(~) Y(~') I o> +

+ o(~,o_ ~o) <o IY(~')j"(~) I o>.

SOLUTION O}" TI|Jd Ii~QUATIONS t.'0t~ THE G R E E N ' S F U N C T I O N S ETC,

We see immedia te ly t ha t

/ 1 1 / (64) <o I [j')(,~), i'(~")] i o> = { e, a(~ - ~') I - ~ Y - (Xi2=9/'

o S n = - - (11u)(1/(1-- (2/2u)~)). Fur ther , with

787

we find b y combinat ion of (62) and (63)

1 1 (65) <OIT(j"(}))~(}')) {0>---- ~ ] - - (z/ )'~'2~:;- ~ 8~D+(~ - - }') +

1

which explicitly shows the lack of covariance of this function. 5Tevertheless,

we find t ha t the correct Green 's function for the induced current

(66) i<O[ T ( j " ($ ) ) " ($ ' ) ) i0> + SUV S(~ - - $') =

1 1 = + 1 - (),/2:~)~

(#'" [B-" - ~" ~) D+(* -- ~') ,

is eovar iant and conserved. We also see the consistency with our previous result (16). Natura l ly , the remarks made here apply as well to the pseudo-

vector current i t . . Thus, the analogue of (59) is

/ix v (67) ~ j - = s ~=,~

o r

(6s)

1 1 b o 5J~ = -- ,~ 1--- ()./2.~)~ e,o q ,

6 j ° = O,

but in fact , (59) and (43), which give the connection between jr and i f , are equivalent to (68), again i l lustrat ing the internal consistency of the model. (This r emark also establishes the fact tha t S "~ is not a field operator.)

I u summary , if the current is defined in terms of the gauge invar ian t expres- sion (39), no difficulty arises f rom the fact tha t the charge and cm'rent oper- ators do not commute a t equal times.

788 K. JOHNSON

6. - C o n c l u s i o n s .

We have shown how it is possible to solve the two dimensional model of

Thirring b y making use of the existence of the two vector densi ty conser-

va t ion laws. In this solution the infra-red problem caused no special difficulty,

a l though the physical consequences of the coupling are somewhat complicated

because of the impossibi l i ty of a classification of the states b y means of par- ticle quan tum numbers . However , in this connection the model is interest ing because it provides a soluble theory in te rms of which the infra-red s t ructure

of a relativist ic field m a y be invest igated. I t was shown how it is possible to define the products of the singular operators yJ(x), in order to determine other

covar iant operators bu t t ha t these singular field products do not satisfy the

equal t ime commuta t ion relations with the field operators F(x), t ha t one would

obtain by means of the canonical commuta t ion relations, together with the

use of the formal expressions for the singular field products.

I n fact, it was found tha t in general it is necessary to make use of t h e gauge invar iance principle to de termine the correct expression for the current

operator . In this connection it should be noted tha t other finite operators, which are vector densities, (~ formal ly >) identical bu t in fact different, could be obtained. I t is thus quite interesting tha t an invar iance principle was needed to provide a guide to find the proper current operator .

Finally, i t was shown how the lack of commuta t ion of the charge and cur-

rent densities at equal t imes is not something which contradicts gauge inva- r iance but , on the contrary , is required b y gauge invarianee. Fur ther , if this commuta t i on relation is consistently t rea ted one is led to solutions for the

Green's functions and ampl i tudes in the presence of external fields which are invar iably consistent with the gauge invariance of the original field equations.

In this note we did not discuss the question of the construct ion of the energy m o m e n t u m operators as functdons of the local field operators. This

problem is complicated by the lack of finiteness of the renormal izat ion constant .

We also did not discuss the meaning of the singulari ty of our solutions for ~/2~ ~ - ~ 1. Presumably , this is the l imit on the magni tude of ~12~ for which there is a ground state, ( i .e . , a spect rum for the field Hami l ton ian bounded f rom below), bu t this problem is na tura l ly be t te r examined in the context of a discussion of the field Hamil tonian.

The au thor wishes to acknowledge useful discussions with Professor P. FEDERBUSH and D r . C. SO:~IMERFELD about two dimensional field theories.

SOLVTION OF TIlE EQUATIONS I-~OR+'TIIE ";REEN ~ FUNCTIONS ETC, ~80

~ k P P E N D I X

We should like to prove that , if a#(x) is an arbi t rary vector field in two dimensional space-time and if ~t'a~, and ?~'s,,~a ~ are given together with,.~the boundary condition tha t a t̀ has positive frequencies in the <¢ distant + future (i.e., for x ° > r , a ~' can be given a Fourier representation w i t h o n l y : p o s i t i v e frequency components) and negative frequencies in the <, distant + past (i.e., for x ° < ~ ', a # etc.) then a # is given uniquely by

(A.1) a~'(m) = -- ~ +(x -- .r') c a,,(x )(dx ) -- d '"~ +(. -- :c') c e~ea ( d . ) ,

where

(A.2) -- [I + D_(x) d(x),

and has positive frequencies for x ° > O, negative trequemfies for x~< O. First, it is clear tha t the expression (A.1) for a t' gives the 'divergence and

curl correctly and satisfies the mentioned boundary condition. To show the uniqueness we must ask for a function with vanishing curl and divergence which satisfies the boundary conditions. Suppose b" is such a function. I t is clear tha t we can always determine a g and { sueh that

(A.3) b"(x) = ~'+~(.r) + e ~ 9 ( x ) .

The vanishing curl and divergence are equivalent to

(A.4) (E2g , = 0 = ~ F ~ .

So the problem can be restated as showing tha t no non-constant solution of O+~v = 0 exists which satisfies the negative-positive frequency boundary con- dition. ] t is dea r from the differential equation that

q-oo --co

+i.+., :fa @ e x p [i(~Z* + !~ I x ° ) ] ~ - ( q ) ,

where we Ban remove any zero frequency term as excluded (i.e., q ; i ( O ) = 0). Further , for xO> ~ the second term must vauish, i.e.,

+co

f dq exp [i(qx - - c o

+ IqixO)i¢_(q)= o , .~o > T ,

and for x°< T' the first te rm must vanish, i.e.,

+co

exp [i(qx + iq ix°)] ~v~(q) = 0, x o < ~ ' .

790 K. JOHNSON

I f we i n t e g r a t e e i t he r exp res s ion x exp [ - - i~ t 'x ] ove r a l l x, we f ind

exp [il~'[x°]q~_(q ') : O, x ° > ~ ,

e x p [ - - i tq' lxO]~v+(q ') = 0 . x " < ~'.

H e n c e ~+ = ~v_ = 0, which p roves t h e un iquenes s of (A.1). L e t us n e x t cons ide r t h e f u n c t i o n D+. D+ o b e y s t he e q u a t i o n

(A.5) - - Y ] ~ D + ( x ) = 6 ( x ) ,

a n d has p o s i t i v e f r equenc ies in t he fu tu re , n e g a t i v e in t he p a s t . The f u n c t i o n

(A.6) D~(x; y) f (dq) exp [iffx] -- exp [iqy] ~ --+ -? O, - '

sa t i s f ies t he se b o u n d a r y cond i t ions , a n d the d i f f e ren t i a l equa t i on . The c o n s t a n t , zero f r e q u e n c y t e r m which invo lves an a r b i t r a r y v e c t o r y c a n n o t be r e m o v e d s ince t he zero f r e q u e n c y p a r t of t he f i rs t t e r m gives a d i v e r g e n t c on t r i bu t i on . Th i s , of course , p l a y s no role in (A. I ) s ince on ly

exp [ i a x ]

' j (2z)" q~ - - ie

en te r s . I f t he i n t e g r a l (A.6) is c a l c u l a t e d we f ind

{A.8) D+(x; y) =- ~ . , log ~ [~ l .

R I A S S U X T O (*)

Si dh la soluzione esplicita del gruppo accoppiato di equazioni per le funzioni di Green per il modello auto-accoppiato della teoria del campo di Thirring. Si t rova che il problema infrarosso non genera part icolare difficolth. Sorge la questione sul modo di definite i prodot t i degli operatori di camp() singolari in punti coincidenti dello spazio- tempo e si mostra the le relazioni di commutazione per questi prodot t i non vengono espresse coerentemente facendo uso delle espressioni formali e delle relazioni di com- mutazione canoniea. Si studia il problema della invarianza di gauge in un campo esterno e si mostra a) c h e l a corrente e la densit~ di carica non commutano in tempi uguali, e b) che questo b necessario per l ' invarianza di gauge delle equazioni del campo, e coerente con essa.

(*) T r a d u z l o n e a c u r a d e l l a R e d a z i o ~ e .