Volume 258, number 1,2 PHYSICS LETTERS B 4 April 1991

On topological symmetries and the Goldstone theorem

Yigal Shamir and Seon H. Park International Center for Theoretical Physics, P.O. Box 586, 1-34100 Trieste, Italy

Received 27 November 1990

We show that one cannot achieve the symmetry breaking condition limR ,~ ( [Qn, -Q] ) # 0 when Qn is the (finite volume) integral of a topological charge density and ,(2 has a compact support. This implies that topological symmetries are never broken spontaneously. If one attempts to use an operator I2' whose support extends to spatial infinity as an order parameter the resulting symmetry breaking condition can be formally satisfied, but the Goldstone theorem does not apply because in general the topolog- ical chargc is no longer conserved. This (wrong) symmetry breaking condition need not contain any dynamical information and merely reflects the effect of£2' on the boundary conditions at spatial infinity.

The characterization of different phases of a the- ory according to the modes in which their symme- tries are realized is always desirable in field theory and in many-body systems. In certain cases this char- acterization provides direct information about the excitation spectrum. When a continuous symmetry is broken in a sufficiently local theory there must exist a masslcss mode - the Goldstone boson [ 1 ].

In gauge theories one anticipates difficulties with the Goldstone theorem. For example, four-dimen- sional asymptotically free theories have no massless particles in both the confinement phase and the Higgs phase. The absence of a masslcss particle associated to the "spontaneous breaking of the gauge symme- try" - the Higgs mechanism - is by now well under- stood. Furthermore, a completely gauge invariant dc- scription of the physical states of the Higgs sector is always possible [2 ].

The situation is different in (2+ 1)-dimensional QED. Let us consider scalar QED for definiteness. In the Higgs phase there are only massive modes, whereas in the Coulomb phase there is a massless particle - the photon. (Electric charge is screened in both phases.) This structure suggests that the two phases could correspond to different realizations of some continuous symmetry. In the Higgs phase that symmetry is unbroken, whereas in the Coulomb phase it is broken, giving rise to a Goldstone boson - the

photon. (In ( 2 + 1 ) dimensions the photon is a scalar. )

Recently, it has been proposed that the above role is played by the topological symmetry, generated by the magnetic flux cp [ 3 ]. A topological symmetry is a symmet~ whose associated current is identically conserved (regardless of the equations of motion), and whose generator can be written as a surface inte- gral. In the present case, the topological current is

= ~ ~,,,,pF~,,, (1)

and the conservation equation 0uPu= 0 is recognized as the Bianchi identity.

In order to support their claim, the authors of ref. [3] construct an operator V(x) which crcates a sin- gular fluxon at x. It should be emphasized that while V(x) is an operator valued function o f x it is by no means a local operator. The definition of V(x) in- volves integration over all space and so it is actually a highly non-local operator. Overlooking this fact, the authors ofref. [3] proceed by noting that I/(x) sat- isfies the commutation relation [q~, V(x) ] = (2re/ e) V(x) and consider it as an order parameter for the spontaneous breaking of the flux symmetry. They find that (V(x ) ) = 0 in the Higgs phase while (V(x ) ) # 0 in the Coulomb phase, and conclude that the photon is thc corresponding Goldstone boson.

In this note we show that this identification is wrong. When a non-topological continuous symme-

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Volume 258, number 1,2 PHYSICS LETTERS B 4 April 1991

try is broken spontaneously, the physical picture be- hind the Goldstone theorem is that localized distor- tions of the ground state in the direction of an infinitesimal symmetry transformation cost only ki- netic energy. In the long-wavelength limit these dis- tortions correspond to quanta of the Goldstone par- ticle. However, topological charges commute w4th all local observables, and it is impossible to perform lo- calized distortions of the ground state "in the direc- tion of an infinitesimal topological transformation".

More technically, the symmetry breaking condi- tion should be properly stated in terms of the com- mutator of the (finite volume) charge

QR= f daxjo(x) ' I x l < R

and an operator £2 with a compact support ~. We show that when Jo is a topological charge density the symmetry breaking condition

lira ( [ Q u , ( 2 ] ) ~ 0 , supp((2) iscompact , (2)

is never satisfied and hence topological symmetries are never spontaneously broken.

If one attempts to use an operator (2' whose sup- port extends to spatial infinity (a delocalized opera- tor) as an order parameter one can in general obtain

lim ( [ Q R , ( 2 ' ] ) ¢ 0 , supp((2 ' )=al l space . (3) g ~

However, (even in a strictly local theory,) current conservation is no longer sufficient to prove the time independence of the Goldstone commutator. Indeed, in general

lim ( [dQR/dt, (2 ' ] ) ~ 0 . (4) R *cc

Eq. (4) means that, despite the local conservation law, Q(t) is in general time dependent because -(2' in- duces a non-zero topological flux at spatial infinity. Thus, even if massless excitations exist in some phase they are not Goldstone bosons of a topological sym- metry as their existence does not follow from the (wrong) symmetry breaking condition (3). The lat-

*~ In fact, it suffices that .(2 has rapidly (say, exponentially) de- creasing weights for large values of coordinates. The following discussion can be easily extended to allow for such operators in the symmetry breaking condition (2).

ter need not contain any dynamical information and merely reflects the effect of(2' on the bounda~ con- ditions at spatial infinity. In fact, we construct oper- ators which satisfy eq. (3) identically using only the canonical commutation relations and without any dynamical information about the phase structure of the theory. We show that the Goldstone commutator of these operators is time dependent except for the trivial ease of a free massless theory.

Let us begin with a brief review of the Goldstone theorem [1 ]. One starts with a local conservation equation 0~j~,= 0 and the symmetry breaking condi- tion (2). Eq. (2) implies that QR creates out of the vacuum a state withp -~0 as R ~ . In order to prove the existence of the Goldstone mode one has to show that the energy of this state tends to zero as R - , ~ . This is true if and only if the LHS ofeq. (2) is time independent. Using current conservation one finds

[dQR/dt, _(2.] = _~ da,[j',, (2] . (5) R

In a strictly local theory, commutativity at spacelike separations implies that for any t there is an R, such that the RHS ofeq. (5) vanishes fo rR> R,. Hence

lim ( [dQ~/d t , ( 2 ] ) = 0 , (6) R ~ o c

and the Goldstone theorem follows. In gauge theories one has to proceed more cautiously. When quantized in the Coulomb gauge the commutation relations contain a non-local term and the RHS ofeq. ( 5 ) may not vanish even for R--,~. Mternativcly, in a covari- ant gauge the formulation of the thco~, is strictly lo- cal but the Goldstone mode may be unphysical.

It is now easy to see why topological symmetries cannot be broken spontaneously. The topological charge density is a total divcrgence jo=0~K, (addi- tional Lorentz indices of K~ have been suppressed). As a result

[QR, (2] = ~_ da,[Ki, (2] . (7) R

Using the same locality argument as above we find that the RHS ofeq. (7) vanishes in the large R limit. Thus, in a strictly local theory the symmetry breaking condition (2) cannot be satisfied for a topological symmetry. This result extends to gauge theories, pro-

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vided they are quantized in a gauge in which their formulation is strictly local. In the examples below the topological charges are in fact gauge invariant. The conclusion that it is impossible to break the corre- sponding topological symmetries spontaneously is therefore gauge invariant as well.

What happens if (see eq. (3)) one allows for de- localized operators in the symmetry breaking condi- tion? In order that thc RHS ofcq. (7) will not vanish for R--,vo a substantial part of the norm of£2' must come from spatial infinity. One can then no longer use locality to prove the vanishing of the RHS ofcq. (5). Indced, as we shall see below the Goldstone commutator in eq. (3) is in general time dependent. Even if there is a massless particle in some phase, its existence does not follow from the (wrong) symme- try breaking condition (3) and hence it is not a Gold- stone boson of the topological symmetry.

It is illuminating to investigate what happens in the case of the flux symmetry of QED3 (examples in other dimensions are given in thc appendix). We are seek- ing a dclocalizcd operator $2. '(x) that satisfies eq. (3). We will assume that £2'(x) is linear in thc transversal clectric field

$2'(x) = ~ d2y a , (x-y)E~(y) ,

E~(x) = - i l i (x) =E, (x) + OiAo(x) • (8)

Quantizing in the Coulomb gauge and using a non- relativistic normalization, the electromagnetic po- tential (at t = 0 ) is given by ~2

A t ( x ) = f d2p toPs a(p) exp(ipx)+h.c . , (2z02 (2p) I/2 p

(9)

and the magnetic field is

B(x) =f.tjoimj(x)

f d2p I ) = - - i ~ (~p '/2a(p) cxp(ipx)+h.c. (10)

The Fourier transform of

O~R = _[ d2x B(X) 'xl <R

~2 In non-Fock representationsA,(x) may have a c-number part. Such a term plays no role in the present context and is hence- forth being ignored.

is concentrated in the momentum range 0 <~p<R-i. Roughly speaking, one has

(0l q)~ Ip) ~ 0 , p>R -~ ,

~pl/2, p < R - i (11)

The Fourier transform of E;(x) behaves like p 1/2 for p--,0 too. Thus, in order to satisfy eq. (3) the Fourier transform of the c-number function a, (x) must have a p -~ singularity at p- , 0 (and an appropriate angular dependence). A possible candidate for a, (x) is [3]

1 x s p~ (12) a , ( x ) = - ~ % ~ 5 , a , (p )= i% .

Notice that %0;afix)=62(x). Using the canonical commutation relations

[At(x), Ej(y) ] = [Ai(x), E)r(y) ]

• lr =x6;j(x-y) , (13)

we find at t=O the operator identity

lim [q~,~, ~' l = - i . (14) R - ) ~

It should be clear from the above discussion that the RHS of eq. (14) depends only on thc asymptotic be- haviour of at(x). For instance, one can smooth thc singularity of a, (x) at x = 0 without changing the RHS of eq. (14)• (This smoothening is in fact necessary in order to create a finite cnergy configuration [4].) One can even replace ai(x) by fR.(x)a,(x) where re(x) is any smooth function that vanishes for Ix] < R' and is equal to onc for Ix] >R' +3 without affecting eq. (14). Thus, eq. (14) contains no infor- mation about the local dynamics and merely reflects the effect of£2' on the boundary conditions.

In order to calculate the lime dependence of the expectation value of the LHS of eq. (14) we use the spectral representation of the electromagnetic field

( [At(x, t),As(y, 0 ) ] )

= i f dmp(m)A~(m; x - y , t) ,

c d2p (6,,- ptpf~sin(p,x,-Et) A S(m;x,t)= i (2~) . \ ~ I==-= ~ p 2 j 1:"

(15)

A straightforward calculation gives rise to

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Volumc 258, number 1,2 PHYSICS LETTERS B 4 April 1991

lim ( [ q ~ R ( t ) , g 2 ' ] ) = - i ~drnp(m) cos(rot) . /¢. ,~o

(16)

Notice that due to the transversality of the magnetic field the longitudinal terms in cqs. ( 13 ) and ( 15 ) do not contribute to eqs. (14) and (16). Eq. (16) im- plies that the Goldstone commutator of eq. (3) is time dependent except for the trivial case of a free elcctromagnctic field (where p (m) = fi(m ) ).

Wc point out that in the case of a free electromag- netic field one can relate the masslessness of the pho- ton to the breaking of a shift symmetry A~(x) ~Ai(x) +ft, r/,= constant (see rcf. [ 1 ] ). The current conservation equation of the shift symmetry is the Maxwell equation 0~,F¢,~=0. In ref. [1 ] it is also shown how to rewrite the Maxwell equation in the form of a current conservation equation in an inter- acting theory. The corresponding symmetry is always spontaneously broken, but the Goldstone theorem is evaded again. In this case the time dependence of the Goldstone commutator (in the Coulomb gauge) arises from the non-local term in the commutat ion relations.

One can construct non-local operators which sat- isfy the formal symmetry breaking condition (3) as a consequence of an operator identity similar to eq. (14) in other dimcnsions as well. The Goldstonc commutators of these operators can again be ex- pressed as spectral integrals and, in complete analogy to the QED 3 case (see eq. (16) ) , they arc time de- pendent except for the trivial case of a free masslcss theory.

In scalar ( 1 + 1 )-dimensional theories the topolog- ical current i s ju= eu~0~¢ and the topological charge is Q = Jdx 0x0(x). The analogues of eqs. (14) and ( 16 ) hold with £2'=½fdx((x)n(x) where e ( x ) = x / I x l . Any function which behaves asymptotically as ~ (x) will also do. For instance, one can replace ~ (x) by the kink solution (normalized to + 1 at infinity).

In four-dimensional QED one can again interpret the Bianchi identities as conservation equations for

topological currents. The corresponding charges are q)i= fdx Bi(x). Here, again, it has been claimed that the masslessness of the photon is related to the break- ing of these topological symmetries [ 5 ]. Once more, a delocalized operator which satisfies the symmetry breaking condition identically can bc constructed. Define au ( x ) = ( 1/8r~)~,j~c,/x 3, one finds a t / = 0

3

lira ~ [ (q )R) , , g2 ; ]= i , (17) R *oo i = 1

where I2'i=fdxa~j(x)E~r(x). By rotational symme- try, the three terms summed on the LHS of eq. (17) are equal. Thus, every, topological charge satisfies the identity lim~_o~[(q)g),, £2}]=i /3 (no summation over i). Likewise, a relation analogous to eq. (16) holds for every q~.

In conclusion, we have shown that topological symmetries are never spontaneously broken if the symmetry breaking condition is properly defined in terms of a sufficiently local order parameter. Trying to evade this conclusion by using delocalized opera- tors in the symmetry breaking condition does not change the physics of the problem. The resulting symmetry breaking condition need not reflect any lo- cal dynamics and merely exhibits the effect of the de- localized operator on the boundary conditions. Even so, the Goldstone theorem does not apply because in general (and perhaps in all cases excepting free mass- less theories) the topological charge is no longer conserved.

References

[ 1 ] For a review sec G.S. Guralnik, (2.R. Hagen and T.W.B. Kibble, in: Advances in particle physics, Vol. 2, cxls. R.L. Cool and R.E. Marshak (Wiley, New York, 1968).

[2] J. Fr6hlich, G. Morchio and F. Strocchi, Nucl. Phys. B 190 ( 1981 ) 553.

[3] A. Kovner, B. Rosenstein and D. Eliczer, Nucl. Phys. B 350 (1991) 325.

[4] Z.F. Ezawa, Phys. Rev. D 18 (1978) 2091. [ 5 ] B. Roscnstein and A. Kovner, University of British Columbia

preprint UBCTG 9-90.

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