LETTERE AL NU0VO CIMENTO VOL. 28, ~. 4 24 Maggio 1980

Broken Symmetry and Bundle Representations.

S. K. Bos~

Depar tmen t o] Physics , Univers i ty o] Notre Dame - Notre Dame, Ind . 46556

(ricevuto il 27 Fcbbraio 1980)

By (spontaneously) broken symmetry is meant a symmetry transformation tha t cannot be implemented in the Hflbert space of states of the physical system (~). Since symmetry operations in physics generally form groups, i t is natural to ask if uni tary nonimplementabil i ty could be phrased in the language of group representations. Within the specific context of relat ivi ty groups in an infinite medium this problem was solved by BORCHERS and SEN (2) via the construction of canonical bundle representations. In this note we do two things. First, we provide two more examples of the appearance of bundle representations of relat ivi ty groups. These involve the description of pho- nons and of photons in a nonrelativistic medium in which long-range forces are present, and are, therefore, of obvious physical interest. Secondly, we show that the method of Borchers and Sen (2) is readily adapted to provide a description of the breakdown of an in terna l symmetry in (a conventional) quantum field theory.

Before proceeding further, we wish to summarize, very briefly, the main idea of Borchers and Sen (3). For an infinite medium, the boost operations cannot be imple- mented via continuous, unitary operators in a Hilbert space. Accordingly, the state space of the physical system is generalized to a Hflbert bundle - -a bundle of Hilbert spaces based on the homogeneous space of boosts. A bundle representation of the rela- t iv i ty group G is then constructed in such a fashion tha t the unitarily implementable portion of G is unitarily implemented on the Hilbert space, while the boost operations are implemented on the base space as bundle maps. More precisely, let G be a Lie group, H a closed subgroup of G and M = G / H the homogeneous space of left cosets. Let B = M • ~ be the product space--viewed as a product bundle with fibre ~ and base space M, where ~f is the Hilbert space. A canonical bundle representation of G on B is one with the properties that i) H is mapped into the group U of unitaries in and ii) the remaining operations in G are implemented on M via their natural action (left translations on left cosets). This procedure guarantees the desired continuity properties of the representation whenever the base space M is contractible, since in the lat ter event the principal fibre bundle G has a global continuous cross-section. This concludes our summary.

(1) See, for instance, J. A. SWIECA: Cargese Lectures in Physics, Vol. 4, edited by D. KASTLER (New York, N.Y., 1970); H. REEH: Statistical Mechanics and Field Theory, edited by R. N. SEN and C. WEIL (Jerusalem, 1972).

146

B R O K E N SYMMETRY AND BUNDLE REPRESENTATIONS 147

In this note, we concentrate on the intui t ive aspects of the problems considered. Details concerning the construction of bundle representations may be found in original memoirs (2.3) and will not be repeated here.

We provide two additional examples of the appearance of bundle representations of relativity groups.

In a translationally invariant (nourelativistic) medium in which long-range Coulomb forces are present, there exists collective excitations possessing the dispersion law (in the rest frame of the medium), as p - + 0,

4 ~ e 2 ~ 2 2 ~ . _ _ (1) E ~ = C~p 2 + e%, o~p ,

where E is the energy, p the momentum of the phonon, C 1 some constant whose precise value does not concern us here and 0% the plasma frequency, e (m) being the charge (mass) of the carriers and n their number density. Now, for a medium without Coulomb forces, the term involving ~% would be absent in eq. ( 1 ) a n d the resulting dispersion relation would be the statement of the nonrelativistie version of the Goldstone (1) theorem. Thus long-range forces alter the Goldstone spectrum by introducing an energy gap. The uti l i ty of this manner of viewing the dispersion law (1) was explained by A N D E R S O N (4) and by H I G G S (5).

We wish to consider the problem of bundle representation description of the exci- tation obeying dispersion law (1). We first take note of the following properties of the excitation, i) The transformation property under Galilean boosts of the excitation is that of a nonrelativistie zero-mass system, that is p ' = p , E ' = E + p . V, where V is the boost. This follows directly from the meaning of a collective excitation, as ex- plained by LANDAU (e): the energy and momentum (but not the mass!) of a weakly excited medium can be ascribed to collective excitations, ii) The dispersion law (1) does not change the above conclusion regarding the inertial property of the excitation. Since the concept o f , inertia of energy ~ is nonexistent at the nonrelativistie level (where mass corresponds to an absolutely conserved superselection rule), the dispersion law (1) cannot be used to impart mass to the excitation. Stated differently, w, does not have inertial properties of mass. Notice, however, one oddity; the dispersion law (1) permits the excitation to travel with a nonconstant, arbitrarily small group velocity. These two observations enable us to solve our problem when we recall the following features of the Botchers-Sen (2) t reatment of the Landau excitations in the superfluid helium: i) if G is the Galilean group, H = E 3 • T , where E 2 is the Euclidean group in three dimensions and T the group of time translations, the homogeneous space M ~ G / H

is parametrizable via the boosts V, ii) the excitation has the transformation property of a zero-mass nonrelativistic system, iii) the bundle representation for the excitation is highly reducible and is such that E 3 • T is implemented on the fibre which is a Hilbert space and boosts on the base space M, iv) the symmetry property does not dictate a dispersion law, which has to be imposed separately, the possibility of doing this arises from the reducibility of the representation of E 3 • T. Collecting all the foregoing remarks, we thus arrive at the following conclusion:

(i) H . J . BORCHERS a n d It . N. SEN: Commun. Math. Phys., 42, 1Ol (1975). (3) R . N. SEN: Physica A (The Hague), 94, 39, 55 (1978). ( ' ) P . W . ANDERSON: Phys. Rev., 130, 4:39 (1962). (5) P . W . HICtGS: Phys. Lelt., 12, 132 (1964). (6) L. D. LANDAU: J. Phys. Moscow, 5, 71 (1941), r e p r i n t e d in I . M. KHALATNIKOV: Introduclion to the Theory ol ~uper]luidity, t r a n s l a t e d b y P . C. HOHENBERG (New Y o r k , N . Y . , 1965).

1 4 8 s . x . BOS~.

The bundle representation o] the Landau excitations developed by BORCHERS and SE~ (2) describes the present situation as well, provided only that we replace the ~andau phonon- roton dispersion law by the dispersion law (1).

A second example of the use of bundle representations is provided by the following. In a conducting medium, as in a tenuous plasma, there exists purely transverse electro- magnetic waves with the dispersion relation (~)

(2) E 2 = pS + o)~,

where wp is, as before, the plasma frequency. The relativity group in the present instance is Poincar~, since Maxwell's equations in the medium are formally eovariant. Thus we may take E and p to constitute a 4-vector. The dispersion law (2) now gives us a massive vector meson (notice, in contrast to the nonrelativistic case, cop now does have the inertial properties of mass; note also that ~%, as given by eq. (1), is boost invariant) which is transversely polarized. As such, this object cannot be described by the Wigner method based on irreducible, uni tary representations of the Poincar6 group. This conclusion is not surprising, since we know that the boost operations are not unitari ly implementable and hence we should not try to describe the situation via the Wigner method. A mathematical description of the system is possible in the framework of bundle representations developed in ref. (2) The bundle representation has the following features: i) the group E3 • T is unitarily implemented on the fibre which is a Hilbert space; ii) the boost operations are implemented on the base space M = P I E a • T (P = Poincar~) which is a homogeneous space; iii) the representation of E 3 contains as a label the helicity, which takes up the values • 1 in the present instance; iv) the rep- resentations are reducible; v) the electric and magnetic fields, in the zero-boost frame, are given explicitly by

I (F+ + F_) , B i]P] (F+-- F_) (3) E = ~ = 2E

in terms of the measurable vector fields F~ introduced in ref. (2). From the properties (2) of F~ it is easy to recover from eq. (3) Maxwell's equations for the present prob- lem (~). We now proceed to discuss the breakdown of an internal symmetry in a con- ventional quantum field theory.

By internal symmetry we mean a symmetry operation that commutes with the Poincar6 group P. To bring out the essential features of the problem let us first consider the simple case in which the internal symmetry corresponds to the uni tary group U 1. Thus, the direct product G = U1 • P is a group of automorphism of the field theory. We seek a bundle representation of G on a Hflbert bundle such that P is unitari ly im- plemented and U1 is not. Physically, this would correspond to the situation in which there exists a local but not a global conservation law associated with U 1.

We look upon G : /71 • P as a product bundle based on the coset space M = G/P. M is homeomorphie to the 1-sphere S 1, which can be parametrized by an angle 0, with 0< 0 < 2~. Let B = S 1 x ~ f be the product space, where ,~f is the Hilbert space whose existence is assured in field theory via the Gel'land construction. Then, B is a product bundle based on S 1 with fibre ~ . The state space of the theory is the bundle space B. We proceed to construct canonical bundle representations of G on B.

(') See, for instance, J. D. JACKSON: Classical Electrodynamics (New York, N . Y . , 1962), p. 339.

B R O K E N SYMMETRY AND B U N D L E REPRI~SENTATIONS 1 4 9

Denote by (a, A) an element of P and by (0; a, A) that of G = U~• Let 0 '~ S 1. The action of the group G on the manifold is

(4) (0; a , A ) O ' = 0 + O' .

Since G is a trivial bundle, a continuous global cross-section exists and is %

(5) ~(0) = (0; O, 1).

The canonical (G, S 1, P) co-cycle K is defined as (3)

(6) K(g, O) = ~(gO)-~g~(O) ,

where g e G and 0 e S 1. Using eq. (4) it is easily shown that

(7) K(g, 0) = (0; a, A).

A continuous uni tary representation U(g) of P on ~ yields a bundle representation of G on B by the identification

(8) u(g, 0) = U(K(g, 0))

as shown in ref. (3). This completes our construction of bundle representations. We note the following features.

1) According to eq. (7), K(g, O) is independent of 0 for all g e G. This independence is the statement, in the bundle representation language, of a theorem due to STREATER (8) :

inequivalent sectors of the field theory are unitarily equivalent as far as the representa- tion of the Poincar4 group is concerned. As a consequence, the spectrum of the theory does not distinguish one fibre from another and it becomes permissible to speak of the spectrum.

2) The theory of bundle representations, by itself, does not yield any further in- formation regarding the spectrum. To learn more about it, that is to see the Goldstone and the Maison-Reeh (9) theorems, one has to invoke more detailed properties, e.g., local commutat ivi ty of the field. However, in view of our preceding result (1), it suffices to work in one fibre and then the t reatment of the spectrum collapses into the conven- tional t reatment (1).

3) Each fibre has a unique Poinear~-invariant ground state. Thus the uniqueness of the ground state (which is part of the axioms of the theory) is naturally reconciled with the appearance of the degeneracy of the ground state.

4) Fields (suitably smeared fields) appear as bundle maps rather than as operators (operator-vahed distributions).

5) The foregoing discussion is readily adapted to treat other instances of broken symmetry. For example where the symmetry corresponds to the additive group R 1

(s) R . F . STREATER" PrOC. R. SOC. Londo~ • e r . . 4 , 287, 510 (1965). ( q D. MAXSON a n d H . REEH: Commun. Math . Phys . , 24, 67 (1971).

1 5 0 s . K . BOSE

of reals. A free massless real scalar field is a concrete example of a physical system possessing this symmetry. The base space is now homeomorphic to the real line and can be parametrized by a real number x. Then all conclusions reached above for U1 continue to be true for the present case, if we replace 0 by x. In the general case, one has to construct bundle representations of the group G = I • P on a Hilbert bundle B = M x ~ , where M = I / H is the homogeneous space, I the internal-symmetry group (assumed Lie) and H a closed subgroup of I . Then H x P will be 'kunitarily implemented on ~ and the remaining operations in G implemented on M.

This work was completed during a visit to the Ben Gurion University of the Negev, Beersheva. The author would like to thank the members of the Department of Mathe- matics and the Department of Physics at Ben Gurion for warm hospitality. He would like to acknowledge his indebtedness to R. N. SE~ for many helpful conversations and for reading the manuscript.