LETTERE AL NUOVO CIMENTO VOL. 29, N. 13 29 Novembre 1980

Evasion of Coleman's Theorem and Goldstone Bosons in Two Dimensions (*).

A. Z. CAFRI (**) and R. FEa, RAnI (***)

Max.Plane]c-Institut ]i~r Physik und AstrophysiIc - Munich (Fed. Rep. Germany)

(ricevuto il 24 Settembre 1980)

Coleman's theorem (1) states that no Goldstone bosons exist associated with the spontaneous breakdown of internal symmetries in standard quantum field thcory of two-dimensional space-time. In this paper we present models where Goldstone bosons occur in connection with the evasion of some assumptions of Coleman; in particular we relinquish positivity or the space independence of the symmetry. We also presen~ a model in which the Lorentz-symmetry is spontaneously broken, but no Goldstone mode arises.

This model is included mMnly for paedagogical resons, since most of the results are available in Wightman's Carg~se lectures (2).

The algebra of the scalar field ~(]) with ] e 5z(R 2) satisfies

(1)

and

(2)

with

(3 )

V79(x) = 0

[ ~ ( x ) , ~ ( 0 ) ] = - - iD(x)

D(x) = �89

The algebra is invar iant under the transformation

(4) ~(x) -~ ~(x) + cons~.

(*) Research supported in pa r t by NSERC. (**) Permanent addrcss: Theoretical Physics Ins t i tu te , Universi ty of Alberta, Edmonton, Alberta, Canada T6G 251. (***) On leave from I s t i tu to di Fisiea dell 'Universith, Pisa, I ta l ia . (0 S. COLEMan: Commun. Math. Phys., 31, 259 (1973). (~) i . S . W~IGRT~IAN: Garg~se Lectures in Theoretical Physics, 1964, edi ted by :tel. L~vY (New York, N .Y . , 1967).

423

424 A.Z. CAPRI and R. F~.RRARX

This symmetry is generated locally by the conserved current

(~) j~(x) = a ~ q ( x ) ,

i .e. for any local operator B of the algebra (a)

(6) Z

--J5

where 8B is the transformation induced on B by (4). The Foek representation of this algebra, given by

(7)

(8)

<olq~(x)lo> = o ,

<o[9(x)(p(o) Io) = - iD(+)(x),

where

(9) 1

D(+)(x) = ~ In ( - - x ~" -b iexo) ,

displays a spontaneous breakdown (4) of the symmetry (4) and the associated Gold- stone field is ~ itself.

I n this case Coleman's theorem is evaded by relinquishing positivity in the scalar product induced by the two-point function in eq. (8).

In the following model we consider symmetry transformations that arc space-time dependent. This case has been excluded by COLEMAN with the tacit assumption that the symmetry is internal, and therefore invariant under translations.

There is a semantic problem about the meaning of spontaneous breakdown of a local symmetry. In the presentation of the model this point will be clarified.

Let ~( f ) , ] e 5P(R~), be a free Fermi field (5), i .e .

(lo)

iT~'~, ~,(x) = O,

{~a(x), v~(0)} = y~O~tD(x ) -= (x" r)a~e(Xo)~(x*),

{w(x), v~(0)} = { ~ ( x ) , ~ (o )} = o .

This algebra is invariant under the (~ gauge ~ transformation

(11)

(~) J.A. SWIEOA: Garg~sv Lectures in Physics, Vol. 4, edited by D. KASTLER (New York, N. Y., 1970); ~[. REEH: Fortschr. Phys. , 18, 687 (1968); C. A. ORZALESI: Rev. Mod. Phys, , 42, 381 (1970). (') R. F. STREATER: Proc. R. Soc. London Ser. A , 287, 510 (1965). (~) A.Z. CAPRI and R. FERRARI : Sehwinyer model, chiral symmetry, anomaly and O-vacuums, University of Alberta preprint (1978).

EVASION OF COL~MAN'S THEOREM ETC. ~2~

where

(12) ~g2(x) = e~v~(x) �9

The current generating this transformation is

(13)

where

(14) i'(~a) = : ~ , ' ~' : (z).

The Fock representation of this algebra is fixed by the two-point function

(15) <olr~(x) ~(o) Io> = (r. ~ ) ~ D ( + ~ ( z ) - (x.?)~

2zi(x2--isXo)

This representation is not invariant under the tramsformation (11). We describe this property as a spontaneous breakdown of the symmetry. This situation can be ascribed to the fact that either the charge does not exist or, if it exists, i t does not annihilate the vacuum. However, in both cases there exists at least one element B of the local algebra such that

L

(16) <0[~B[O> = l i m i<O[ / i d x X k ~ 5 0 , /,--~-t- co / J J --Z

From eq. (11) and (15) we see that B can be :~a(y) t~(z):. In the representation we are considering the current Jr may be writ ten as (2)

1 (17) i.(z) = ~ ~.Q(x),

where

(18)

and

(19)

[ DQ(x) = o ,

[~(x), Q(0)] = --iD(x),

{ <oIq(x)[o> = o,

<olo(x ) o(o)]o> = - - iD(+)(x),

so that Q(x) is a field such as we considered in the first part. The only important differ- ence is that Q is given only for test functions of the type ~]~,(x) with ]~ eS~(R,). With this restriction the metric of the Fock space is positive definite. The field ~(x) describes the Goldstone mode and it is a collective motion of a fermion and an antifermion.

426 A . z . cAPri and R. FERRARI

This is i l lus t ra ted by the equa t ion

(20)

L

- - L

Z

_ i , _ l im ~dx ~ (O[[(A(x) ~OO(x) - - ~OZ(x) O(x)), : ~a(y) ~ ( z ) : ] [0) , V ~ x.-~+ = j

which fu r the rmore expl ic i t ly displays the role of 0 as a Goldstone field.

I n t he const ruct ion of expl ic i t opera tor solutions for solvable models in two dimen- sions t he concept of the dual of a massless scalar field is f requen t ly used. Tha t is, g iven a field ~0(x) as described in section one, the dual field ~(x) is in t roduced v ia the equa- t ion

(21) Ou~(x) = s~v~ ~ ~(x) .

F o r m a l l y one gets the fol lowing algebra:

(22)

(23)

(24)

where

(25)

E3~(x) = 0 ,

[qS(x), ~(0)] = - - iD(x),

[~(x), ~(0)] = - - i D ( x ) ,

D(x) = �89 e(x 1) 0(-- x~).

I t should be stressed tha t ~ is defined on test funct ions of the form az]z wi th ] e 5P(R2) and therefore ~ is stil l local w i th respect to ~, a l though the commuta to r g iven in (24) and (25) has suppor t in the spacelike regions. The reason is t ha t

(26) ~ , D ( x ) = ~ , , ~ D ( x ) .

J u s t as the ~-field is defined for tes t funct ions in 5: one wants to ex tend ~ also to the whole 5:-space. This ex tens ion causes some difficulties, in par t icu lar the algebra is no longer local and moreover the separat ion of the fields into posit ive- and nega- t ive - f requency par ts can no longer be accomplished in a Loren tz - invar ian t fashion. Thus we get, for example,

(27)

(28)

;o[~(x) ~(o) 1o7 = (o[ ~(x)~(O)lO) = - - iD(+)(x),

<O[q~(x) r = - - iD(+)(x),

which is g iven by

(29)

where x* =- x ~ • ~ .

EVASION OF COLEMAN'S THEOREM ETC. 427

To diagonalize the representation of the algebra we introduce the two independent fields

1 (~(x) = ~-~ (~(x) + ~ ( z ) ) ,

(3o) 1 v(~) = ~-~ (~(x) - ~ ( z ) ) .

These fields satisfy the following algebra:

(31)

i [o(x), a(O)] = - - ~ ~(z+),

i Iv(x), ~(0)] = -- ~ ~(z-),

[a(x), T(O)] = O,

and they are represented on a Fock space induced by the two-point function

(32)

where

(33)

{ <ola(x) a(o)lo> = -- ia(z+),

<o[~(z).(o)lo> = --i(~(x-),

1 " 2 G(x) ~-- ~ In [-- (x - - me) ] .

Since the Fourier transform of G is

(34) dx exp lips] q(z) = i ~-p{0(P)[ln IPi + V]},

where C is Euler 's constant, one can easily see that the a-field has support in momentum space on the branch of the light-cone

PQ> O, po = _ _ p 1

and the v-field on

P o > O , p o = p i .

The Lorentz transformation on the fields a and ~ are generated locally by the con- served current M~ given by

(35) M o = � 8 9 M_),

= �89 ~ + + ~ _ ) ,

~28 A.Z. CAP~I and ~. F ~ A ~

where

(36)

in fact

M + = x+:(O+a)2:,

M_ = x-:(O_~)~:,

(37) s 1 7 6

--L

and a similar expression for 3. However, the representation given by eq. (32) is not Lorentz invariant. This can be seen from the explicit form of G in eq. (33). I t there- fore follows that

(38)

L

s co --/,

and consequently the Lorentz symmetry is spontaneously broken. No Goldstone boson is associated with this spontaneous breakdown as can be seen

from the following argument:

(39)

where

(40)

J5 f l l im dxl~x+(Ol[:(O+a)~:(x+), :a(y) o(z):]]O) =

s o~

- - L

L

= lim rdx ~ - x + (exp [-- iPx § dp W(P, y+, z +) --= - - ~i dP ~(P) = W(P, y+, z+), ~-*+~J 2 J ~-

- -Z

W(P,y+,z+)=-~ x+exp[iPx+](Ol[:(~+a)~:(x+),:a(y+)a(z+):][O}.

I t follows from (39) that no massless field singularity, i.e. 1/P, is required for k~(P, y+, z +) in order to satisfy eq. (38).

We would like to thank the kind hospitality of the Max-Planck-Insti tut fiir Physik where this work has been carried out.