IL NUOV0 CIMENT0 VOL. 32 A, N. 3 l Aprile 1976

On the Covariant Description of Spontaneously Broken Symmetry in General Field Theory (*).

H. J o o s (**)

CEI~Y - Geneva

~E. WEIMAI~

tVreie Universitdt - Berlin

(ricevuto il 7 Gennaio 1976)

S u m m a r y . - - Reducible fields A(x) with degenerate vacuum which allow the unitary-symmetry transformation U-I(e)A(x) U(c) = A(x) + c are analysed. We describe the mathematical properties of the <<charge integral )> related to the conserved current of this spontaneously broken symmetry. Further we discuss the structure of the S-matrix theory in such a generalized field theory as a guide-line for the treatment of more complex examples of spontaneously broken symmetries.

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

The idea of spon taneous ly b roken s y m m e t r y p lays an i m p o r t a n t r61e in t he a t t e m p t s (1) to u n d e r s t a n d e l emen ta ry -pa r t i c l e phys ics : spon taneous ly

b roken S U 2 × S U 2 s y m m e t r y , wi th pions as Golds tone bosons, seems to be

real ized a p p r o x i m a t e l y in N a t u r e (5). Spon taneous s y m m e t r y b reak ing p lays

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. (**) On leave from DESY, Hamburg. (1) W. HEISENBERG: Ein]i~hrung in die einheitliche Feldtheorie der Elementarteilchen (Stuttgart, 1967); Y. NAMBU and G. JONA-LAsINIO: Phys. l~ev., 122, 345 (1961); 124, 246 (1961). (2) S. W]~I~IBER~: Proceedings o] the X I V International Con]erence on High-Energy Physics (Vienna, 1968), p. 253.

283

2 8 4 ~ . a o o s a n d ~ . w ~ i ~ g

an essential rble in the construct ion of renormalizable unified theories of weak

and electromagnet ic interact ions (3). Loug-range forces related to spontaneous ly

b roken gauge invar iance might be responsible for quark confinement (4). These

few physical examples should be enough to jus t i fy our in teres t in a new for-

mal ism for the t r e a t m e n t of spontaneous s y m m e t r y b reak ing in the f r amework of relat ivist ic q u a n t u m field theory.

The physics of spontaneous s y m m e t r y breaking (SSB) is mos t clear in the

case of a field theory with ~n explicit ly given Lagrangian. I t describes the s i tuat ion in which a symmet r i c L a g r a n ~ a n leads to a nonsymmet r i c v a c u u m

(ground state), whereas the symmet r i c v a c u u m is unstable. Inspect ion of the (~ classical po ten t ia l ~ of the Lagrangian (~) often gives a clue for the appea rance

of SSB. lqonvanishing v a c u u m expecta t ion values of fields indicate the non-

symmet r i c character of this state. As an example we ment ion the nonl inear

Lagrangians describing spontaneously b roken chiral s y m m e t r y (6). Since in

Lagrangian field theory the description of SSB is intrinsically connected with

the dynamica l problems, results following direct ly f rom s y m m e t r y consider-

a t i o n s - w e mean spontaneously b roken s y m m e t r y - - a r e often difficult to de- rive. But , on the other hand, we know how simply consequences of unbroken s y m m e t r y can be discussed in (( general field theory ~ (GFT) (7), a formal i sm

describing only those propert ies of quant ized fields which are independen t of

the special dynamics . Therefore, one would expect t ha t in the f r a m e w o r k of G F T also the general results of spontaneously b roken s y m m e t r y can be studied

formal ly in a simple way. Such an analysis of spontaneously b roken s y m m e t r y

is the a im of this paper . In order to explain our intentions, we have to discuss first the abs t r ac t

features of SSB (s). For this we consider A(x), a quant ized, Poincar~-covar iant ,

local field (with suppressed tensor and other indices), and assume t h a t the de-

(3) S. W]~IN]~ERG: Phys. Rev..~ett., 19, 1264 (1967); A. SALAM: Elementary Particle Theory, edited by N. SV~T~OLM (Stockholm, 1968); B. W. L ~ : Proceedings o] the X V I International Con]erence on High-Energy Physics, Vol. 4 (Kiev, 1970), p. 249. (4) S. WEIN]~ERC~: Phys. Rev. Lett., 31, 494 (1973); A. CASI~ER, J. KoGuT and L. SUSSKIND: Phys. Rev. Lett., 31, 792 (1973); Phys. Rev. D, 10, 732 (1974); K . G . WiLsol<: Phys. Rev. D, 1@, 2445 (1974); Z. F. EZAWA and H. C. TzE: DAMTP 75•5. (5) G. JO~A-LAsINIO: ~VUOVO Cimento, 34, 1790 (1964); R. JACKIW: Phys. Rev. D, 9, 1686 (1974). (6) G. KRAMEtt, H. ]ROLLNIK and B. STEC~: Zeits. Phys., 154, 564 (1959); M. GELr,- 1V[A~<~ and M. L]~v¥: ]~uovo Cimento, 16, 705 (1960); F. Gi3RSEY: N~OVO Cimento, 16, 230 (1960); P. CHA~C~ and F. GtYRs~Y: Phys. Rev., 164, 1752 (1967). (7) R. F. STI~]~ATEI~ and A. S. WIGUTMA~r: POT, Spin and Statistics and All That (New York, N.Y. , ~nd Amsterdam, 1964); R. JOST: The General Theory of Quantized Fields (Providence, R . I . , 1965). (s) R .F . STR~tTER: Proe. Roy. Soc., 287 A, 510 (1965); D. KASTL~R, D. W. ROmNSO~¢ and A. SwIEcA" Comm. Math. Phys., 2, 108 (1966); D. KASTLER" Proceedings o] the 1967 International Con]erence on Particles and Fields (New York, N. Y., 1967), p. 305.

ON THE COVARIANT DESCRIPTION OF SPONTANEOUSLY BROKEN SYMMETRY ETC. 2 8 5

fining relations of A(x), i.e. the field equations and commuta t ion relations, are invar iant under a group ~ of t ransformations a :A(x)->AS(x) . The existence of such a group of automorphisms of the field algebra genera ted by {A(x)} ma y lead ei ther to a quantum-mechanical sy m m et ry (a)), or to a spontaneously broken symmet ry (fl)). These two cases are distinguished by the following prop- erties.

~) There is a quantum-mechanical symmet ry (QMS) if and only if in an irreducible representat ion of A(x) by operators of a Hi lber t space 5/~ the trans- formations a can be implemented by a uni ta ry representat ion U(a) of ~ :

(1.1) A~(x) = U-I(a) A(x) U(a) ,

or of an infinitesimal version with U ( a ) = exp [--iQs]:

(1.2) ~QA(x) = i[Q, A(x)].

The irreducibil i ty assumption implies tha t U(a) and Q are limits of expres- sions formed from A(x). Space-time translations, infinitesimally implemented by the momen tum and Hamil ton operators, are well-known examples of such quantum-mechanical symmetries.

fl) There is a spontaneously broken symmet ry if and only if an irreducible representat ion of A(x) such as U(a) does not exist, i.e. the action of A(x) --> A~(x)

on an irreducible representat ion of the field algebra generates an inequivalent representat ion. We want to ment ion here the simplest example of such a spontaneously broken symmet ry (9). The field equation and the commuta t ion relation of a massless free field

(1.3) []A(x) -~ O, [A(x), A(y)] = iD(x -- y)

are invar iant under the t ransformation

(1.4) A q x ) = A(x) + c , or 8A(x) = 8c.

I t is well known (9), and we discuss it later, tha t for an irreducible represen- ta t ion A(x) there is no uni ta ry operator U(c) (self-adjoint operator Q) which satisfies

U-l(c) A(x) U(c) = A(x) + c

or

(1.5) i[Q, A(x)] = 1.

Therefore, the symmet ry described by eq. (1.4) is spontaneously broken.

(9) G. S. GURALNIK: Phys. Rev. Let&, 13, 295 (1964).

286 R. Joos and E. WEIMAR

I n canon ica l q u a n t u m field theory , a con t inuous a u t o m o r p h i s m g roup

is r e la ted to conse rved cur ren t s j(~)(x), k = 1, ..., n = d i m ~ , 3,j~)(x) = 0. Since

discussions of QMS a nd SSB are for the m o s t p a r t based on the exis tence of

these currents , we w a n t to rev iew shor t ly the wel l -known s i tua t ion for the two

c a s e s •

£ ) A gap in the ene rgy spec t rum be tween the energy of the n o n d e g e n e r a t e

v a c u u m s ta te a n d t h a t of the par t ic le s ta tes implies the exis tence of QMS (~o)

i m p l e m e n t e d b y charge opera tors Q(k); t he m a t r i x e lement of the in tegra l of

the charge dens i ty be tween localized s ta tes ~o and ¢ converges (~o.~):

(1 .6) l im (T , (~o"k'(x)dx 4 ] = ( T , Q~k,¢) . F---~ ¢o \ J /

V

fl') In the e~se of SSB there is no energy gap; this implies the existence of mass-zero particles for manifestly relativistic covariant local theories (<( Gold- stone theorem ~> (8.12)); for gauge theories the appearance of physical Gold- stone particles might be ~bsent because of the Higgs-Kibble mechanism (I~); in irreducible representations, nothing can be said in general on the existence

or meaning of f]o(x) dx.

Conserved currents for which the charge integral cannot be related to the generators Q(k) of a quantum-mechanical symmetry may still be used to describe a spontaneously broken symmetry. Thus current algebra is the classical method to treat spontaneously broken chir~l symmetry (~4). But one should bear in mind that there might be SSB in eases where no currents are related to the automorphism group ~. Spontaneous breaking of the discrete T-symmetry recently proposed by LEE (15) is such an example.

In our analysis of spontaneously broken symmetry, we do not want to

abandon relations of the type of eqs. (1.1) and (1.2)

(].7) A~(x) = U-~(~)A(x) U(~) or ~A(x) =i[Q, A(x)]

(lo) D. KASTLER, D. W. R0m•S0N and A. SWIECA: Comm. Math. Phys., 2, 108 (1966). (11) B. SC~ROER and P. STICHEL: Comm. Math. Phys., 3, 258 (1966). For related problems, see H. REEH: Forts. Phys., 16, 687 (1968); C. A. 0RZALESI: Rev. Mod. Phys. 42, 381 (1970); D. MAISON and H. REEH: NUOVO Cimento, 1 A, 78 (1971). (12) j . GOLDSTONE: XYuovo Cimento, 19, 154 (1961); J. GOLDSTONE, A. SALAM a n d

S. WEINBERG: Phys. Rev., 127, 965 (1962); T. W. B. KIBBLE: Proceedings o] the 1967 International Con]erence on Particles and Fields (New York, N.Y. , 1967), p. 277. (~3) p. H1~6s: Phys. Lett., 12, 132 (1964); G. GURALNIK, C. :R. HA~EN and T. W. B. K~BBLE : Phys. Rev. Lett., 13, 585 (1964) ; T. W. B. KIBBLE : Phys. Rev., 155, 1554 (1967), and ref. (,2). (1~) A review on this subject is contained in S. WEI~EERG: Dynamic and algebraic ~ymmetries, in Lectures on Elementary Particles and Quantum Field Theory, edited by S. DESER, M. GRISARU and H. PENDLETON (Cambridge, Mass., 1970). (15) T. D. LEE: Phys. Rev. D, 8, 1226 (1973).

ON TILE COVA~IANT DESCRIPTION OF SPONTANEOUSLY BROKEN SYI~I1KETRY ETC. 2 ~ 7

because most of the quantum-mechanical symmet ry considerations are based on such equations which define the fields as covariant tensor operators with respect to the group ~. Fields covari~nt with respect to a spontaneously broken symm e t r y group cannot be represented irreducibly by operators in a t t i lber t space W, ra ther we have to consider the direct integral (sum) of the irreducible representat ions genera ted by the action of the group ~. This is evident f rom

the characteristics of SSB. Giving up the irreducibili ty condition has serious consequences. The fields,

i.e. the dynamical variables, no longer generate a complete operator system in W, therefore the relation between observables and operators in W gives rise to questions similar to those in the discussion of superselection rules (1G).

In part icular the relation of the symmet ry generator Q(*) to the local ~( charge densities ~ j~)(x) becomes even more intr icate than in a case of Q]~S. The re- ducibili ty of the fields {A(x)} implies fur ther a degeneracy of the vacuum (~7), which meuns a change in the usual axioms of GFT. In the following we have to deal with all these problems. But the essential point is to show tha t these modifications of the quantum-mechanical scheme do not affect the S-matr ix in terpre ta t ion of quan tum field theory.

Seeing all these complications, why do we still believe tha t it is worth- while to s tudy fields which ,~re covarian~ with respect to a spontaneously broken symmet ry group? We uli'eady mentioned tha t such fields allow the applica-

t ion of the group-theoretical techniques generally used in physics. With the help of these methods the group-theoretical s t ructure of the consequences of SSB becomes more clear. Fur thermore , this s t ructure shows often a new feature, called (( dynamical rear rangement of symmet ry )) by U~EZAWA (~8), which requires flexible group-theoretical methods for its investigation. We shall i l lustrate this notion for the example of chiral S U2 X S U2 symmet ry realized by nonlinear t ransformations of the pion field r:(x). In this example the in- finitesimal isospin ~nd chiral t rnnsformations have the form (19)

(1.8) { 6 ~ ( x ) = ~x3¢o~,

(16) G. C. WICK, A. S. WIGnTMAN and E. P. WINNER: Phys. l~ev., 88, 101 (1952); S. DOrLICHER, R. HAAG and J. E. ROBERTS: Comm. Math. Phys., 13, 1 (1969); 15, 173 (1969); 23, 199 (1971); 35, 49 (1974). (1~) H. ARAKI: Prog. Theor. Phys., 32, 844 (1964); H. J. BOtChERS: Comm. Math. Phys., l , 49 (1965). (18) H. UMEZAWA." in Renormalization and Invariance in Quantum Field Theory, edited by E. R. CAIANIELLO (New York, N.Y., 1974), p. 275; H. MATSAMOTO, N. J. PAPA- STAMATIOU and H. UMEZAWA: Phys. Lett., 46 B, 93 (1973); Nucl. Phys., 68 B, 236 (1974); 82 B, 45 (1974). (19) S. WEINBERG: Phys. Rev., 166, 1568 (1968).

288 H. JOOS and ]g. WEIMAI~

with /7(n 2) = (~2)-~(1--1~] e tgl~l) in the exponential model. They satisfy the commutat ion relations

(1.9) { [&,, &~] = - e ~ &, ,

[~ , , d~,~] = _ _ e ~ &,.

In the LSZ limit (20) these t ransformat ion laws change to those of the free asymptot ic fields for massless particles:

(1.10) ~ , . ( x ) = ~e. X 8¢O~, d~o,(X) = 8 % .

In contrast to eq. (1.9), these infinitesimal t ransformations have the commu- ta t ion relations

(1.11) [~v,, ~v~] = -- e~k~dv,, [dr,, da,] = - - e,k~&, [~a,, ~,4,] = 0 ,

which are the commuta t ion relations of a group isomorphic to the three-di- mensional Eucl idean group ISO3. This (( contract ion )) (21) of the automorphism

group SU2 × SU2 of the interact ing fields ~(x) to tha t of the massless free pion fields go~(x), induced by the LSZ limit, is called (( dynamical rear rangement of symmet ry ~). The only systematic approach to questions raised by this new phenomenon, e.g. in a chiral invar iant per turba t ion theory (22), are based on funct ional methods. We hope tha t the use of covariant representat ions of fields, together with the group-theoret ical technique of contract ion and deforma- t ion (23) of representations, m a y give a s tar t ing point for an operator approach to these problems. All these more mathemat ica l questions are closely related to pract ical physical problems. Thus the incorporat ion of spontaneously broken chiral symmet ry in the phenomenology of e lementary particles based on the quark model is not ye t performed satisfactorily. Bu t it is ev iden t f rom the a t t empts of this problem up to now (24) tha t a systematic s tudy of SSB in

GFT, part icular ly with respect to the bound-s ta te problem, would be ve ry help- ful for an unders tanding of this question of (( current and const i tuent quarks ,~.

(20) H. L]~HMANI% K. SY~AI~zII~ and W. ZIMMER~AI~lg: 2VUOVO Cimento, 1, 425 (1955); 6, 1 (1957). (~!) Contraction: E. I1~61~i3 and E. P. WInNeR: .proc. Acad. Sci., 39, 510 (1953); E. W]~I?aAR: Nuovo Cimento, 15 B, 245 (1973). (22) Chiral invariant perturbation theory: J. tIoN~.RXA~tl- and K. MEETZ: .Phys. Rev. D, 3, 1996 (1971); L. D. FADDEEV and A. A. SLAVSIOV: Soy. Phys. (TMP), 8, 297 (1971); J. HON]~RKAMP: Nucl. Phys., 36 B, 130 (1972). (2a) R. H]~RMA~I~I: Comm. Math. Phys., 2, 251 (1966); 3, 53, 75 (1966); E. W]~IMAI~: £Vuovo Cimento, 15 B, 245, 257, 272 (1973). (2a) M. B6HSt, H. Joos and M. KRAMMER: CERN preprint TII. 1949 (1974); R. CARLITZ, D. HECKATHORN, J. KAUR and W. K. TunG: Chicago University EFI 74-32.

ON THE COVAI%IANT D E S C R I P T I O N OF SPONTANEOUSLY BROKEN SYMMETRY ETC. 289

After this ra ther lengthy sketch of questions raised b y the prob lem of spon-

taneous s y m m e t r y breaking in general field theory, the content of this pape r appears re la t ively modest , and it represents only a beginning. I n the following

section we describe complete ly the covar ian t fields of the noninterac t ing model defined b y eqs. (1.3)-(1.5). This is r ecommended b y simplicity, bu t also by the

impor tance of such fields in scat ter ing theory. Besides the clarification of

the conceptual questions, the precise description of massless part icles in a theory

with degenerate v a c u u m forms an impor t an t pa r t of this section. I n addit ion,

we discuss the proper t ies of the (~ charge ~> re la ted to the current of this spon-

taneous ly b roken pion gauge symmet ry . I n the other section we discuss the in terac t ing fields and the scat ter ing theory in the ease of such a pion gauge

invariance. The main result will be to show how certain zeros of the scat ter ing ampl i tude originate in the spontaneous broken s y m m e t r y . These correspond to the well-known Adler zeros (25) in ehiral s y m m e t r y . Since we assume the same

s y m m e t r y for the in terac t ing and asympto t i c fields, we do not ye t t r ea t dy-

namica l r ea r r angemen t of s y m m e t r y in this paper . Bu t we consider our s tudy

main ly as a guide-line to the consideration of more complex examples.

This work is ve ry much influenced b y the recent discussion of ehiral sym-

m e t r y b y IJEIIlgANN (2e), in which he based m a n y of his a rguments directly on

the scat ter ing theories in GFT. As for the methodica l aspects we consider our work as a cont inuat ion of an invest igat ion done by ];(RAMER and PALI~ER (27).

However , there are two impor t an t points in which we do not follow these authors.

We show tha t i t is unnecessary to describe covar ian t fields in nonseparable t t i lbe r t spaces (2s), and we shall s t ay in the convent ional one. Fur thermore , our discussion is s t r ict ly based on the conceptional f r amework of GFT. This

leads to an i m p r o v e m e n t of the ma thema t i ca l and conceptional f r amework for the t r e a t m e n t of q u a n t u m fields covar iant with respect to a spontaneously broken s y m m e t r y .

2. - Free fields w i th covar iant scalar g a uge s y m m e t r y .

2"1. Definitions. - We shall invest igate with appropr ia te ma thema t i ca l rigour the free massless q u a n t u m field A(x) which is eovar ian t with respect to

scalar gauge t ransformat ions . This field-theoretical model is defined b y

eqs. (1.3)-(1.5) which get f rom the (( smeared )) field operators A(]) = f A ( x ) ] ( x ) d x

(25) S. L. ADLER: Phys. Rev., 137, B 1022 (1965); 139, B 1638 (1965). (2e) H. LEttMANN: in Recent Developments in Mathematical Physics, edited by P. URBAN (Wien and New York, N.Y. , 1973), p. 139. (27) G. KRAMER and W. F. PALMER: Phys. Rev., 132, 1492 (1969). (2s) j . VON ]~EUMANN: Compositio Math., 6, 1 (1938); L. VA~ HOVE: Physica, 18, 145 (1952). In avoiding the nonseparable Hilbert space, we follow the approach of R. HAAG: NUOVO Cimento, 25, 287 (1962).

290

the following form:

(2.1)

(2.:[')

(2.1")

H. ZOOS and E. "WEIMAR

A(D ]) = 0 ,

[A(/), A(g)] = i(], g),

u-~(c) Af t ) U(c) = Af t ) + c~/(x) dx = A( / ) + G(,,~:)~/(0) ; d

] ~}, are infinitely differentiable, fas t decreasing tes t functions (29):

(2.2)

(2.3)

](x) = (2=)-*f/(p) exp [ - ipx] dp,

1 (2.4) D(x) = - - ~ ~(Xo)~(x~).

Our a im is to describe the representa t ions of {A(])} by operators in a separable Hi lber t space ~f, res t r ic ted b y the postula tes of GFT (7):

I) A(]) is defined on a dense set ~ c W with A ( / ) g c ~ , and with (¢, A ( / ) ~ ) = (A(/*)¢, T) for ¢, ~ e ~ ;

I I ) there is a un i ta ry representa t ion in ~%f of the Poinear6 group U(A, a) with an ene rgy -momen tum spec t rum in the forward cone V+: P~,P'>~O, PolO, and there is at least one vacuum sta te

U(A, a) Do = Do, #20 ~ ~;

III) the fields t r ans form covar ian t ly with respect to U(A, a):

U-I(A, a)A(x) U(A, a) = A(Ax + a).

Because we t r ea t fields covar iant with respect to a spontaneously b roken sym-

met ry , we would run into a contradict ion, if we would assume in I I ) the unique-

ness of the vacuum. On the contrary , we have to expect an infinite-dimensional

subspace ~fo of vectors ~ i , each invar ian t under U(A, a). Let us first clarify

this aspect of the problem.

(29) For the distribution theoretic notions if GFT, see R. JOST: The General Theory o] Quantized Fields (Providence, R . I . , 1965); K. HEPP: in Particle Symmetries and Axiomatic Field Theory, edited by M. C~R~TI]~N and S. DnsE~ (New York, N. Y., 1965), p. 135.

ON THE COVARIANT D E S C R I P T I O N OF SPO~TA:N-:EOUSLY B R O K E N SYMMETX~Y :ETC. 291

2"2. The vacuum subspace. - In order to avoid complications in our dis-

cussion, which originate in the consideration of physically implausible situations,

we make the addit ional assumption (~o)

(2.5) [U(c), U(A, a)] = o.

This has the immediate consequence that with ~o, from (2.1) II) , also -Qo =

= U(c) f2o is a Poincar6-invariant state : U(A, a) U(c).(-2o = U(c) U(A, a)-('2o = = U(c).(-20. Since the expectation values of the eovariant field A(x) are dif-

ferent in these states, ( ~ , A ( x ) Q o ) = (z'-2o, A ( x ) ~ o ) + c, there is a multi-

dimensional subspaee ~ o of Poincar6-invariant states, the (( vacuum subspaee )).

I t contains a subspace ~ o generated by U(c) from f2o as a cyclic vector

(2.6) g(c) sufficiently decreasing.

I t is sufficient for our purpose to consider cases where ~fo = ~ 0 , and we as-

sume this always, unless otherwise stated. Because of

IIA(/) ~ll ~ : f f d ~ dc ' a(c') a*(c).

• ( u ( c - c') no, (A~q) + (2~)~ c'/*(o))(a(1) + (2~)~ c'](0)) ~o) <

< I~ol =. lIAr(/)A(/)nol[ + 2(2=)~1Go e j (o ) l . IIA(/)no]i + (2=)~leo G=[-[](0) I ~,

Gi =fg(c) ci de, i = O, 1, 2,

A(]) can be natural ly defined on a dense set in 5~fo. Therefore we assume gen-

erally for the domain ~ of A(]) tha t U ( c ) ~ c ~ . We want to construct an orthonormal basis in J F o ( ~ o ) related to A(x).

For this we have to consider certain Ra means of A(x):

(2.7) ,f C = lira dx A(x , xo) F--~R~ V

v

This operator is defined by the following procedure (~1): instead of the mean

value of A(x), we first consider tha t of a smeared field As(x ) = U(-- x)A(]) U(x). The averaging is performed with help of a family of functions Zv(X), called

(30) A slightly more general discussion might be based on the work of H. ARAKI: P~'og. Theov. Phys., 32, 844 (1964). (31) D. KASTrnR and D. W. ROnINSON: Comm. Math. Phys., 3, 151 (1966).

2 9 2 ~ . JOOS a n d ~ . W~IMA~

an M-filter (31), with the properties

i) Xv(x)/> 0 ,

ii) f Zv(X) dx = 1 ,

iii) l i ~ f dx[zv(x) - Zv(X + Y)I = 0 for all y ~ R3.

An example is given by the characterist ic funct ion VZ~,(x ) of spheres with volume V, which may be smoothed if necessary. Then C is defined on vacua ~2 e ~ n 2 as the limit

(2.S) l i m f d x z ~ ( x ) A / x ) ~ = l i m f d ~ z ~ ( ~ ) U ( - - x ) A ( l ) ~ =

= E(o) A(/) Q = (2~)~/(0) Vt2 ;

/iJ(0) projection on 9~o. As a consequence of locMity, expressed by eq. (1.3) or (2.1'), we have~ for

/ , ( x ) " ' ' =l , (x ) l , ( xo) , ]~(xo) with compact support

[I [A(/,), A,(~)] ¢1] -~ 0 for I ~ l - ~ ,

stronger than any power in Ix[ -~, and therefore

(2.9) li fd ' Zv(x)[A1(x), A(]~)] ¢ = 0 for ¢ e 9 .

Wi th the help of this proper ty , it is easy to show tha t C can be defined as a strong limit on states ¢ ---- A(]~) ... A(],~) Q, ~ ~ ~ n ~'~. We get

(2.10) s-lim.ldx Zv(X) A , (x ) A(]I) ... A (/,~) [2 =

= (2=?/ (0) ~A(/I) . . . A( I~)~ = (2~)U(0)A(f l ) ... Af t . ) e l2 .

I t follows from the wave equat ion (2.1) tha t compactness in x, of the support

of the ]~ means no restrict ion on the states ¢. Equat ions (2.8) and (2.10) give a precise meaning to the (( mean ~ in (2.7) and define C on a dense set of ~f.

The following properties of C are contMned in this derivat ion:

(2.11)

i) [C,A(/)] = 0

ii) [C, U(A, a)] = 0

iii) U-I(c) CU(c) = C + c

iv) 0 9 n ~zo c ~fc

f rom eq. (2.9),

f rom eq. (2.10),

f rom eqs. (2.1"), (2.7),

f rom eq. (2.8).

ON THE COVARIANT DESCRIPTION OF SPONTANEOUSLY BROKEN SYMMETRY ETC. 2~3

Equat ion iii) states t ha t the generator Q of U(c)= exp [ - i Q c ] and C satisfy canonical commutat ion relations

(2.12) [Q, O] = 1 .

Since Q and C leave Wo invariant , s tandard procedures of quan tum mechanics

allow the introduct ion of an (improper) complete basis of 5~fo ( -- 9fro), in which C is diagonal:

Vie> = c[e>, <elc'> = ~ ( c - c') ,

(2.13) +® ~9 ~ ~ 0 : ~9 =IG(c)le} dc

- - c o

if and only if

(2.14)

-~-co

I1~11 = =fde [ G( e ) I ~ < ~ , - - ¢ o

The basis Ic} transforms under U(c) like

(2.15) U ( e ' ) l c ) = Ic ÷ c ' } ,

if the relative phases of the vacua are chosen appropriately. I t is this basis which allows a ve ry simple discussion of the covariant free fields.

2"3. Vacuum expectation values. - The s t ructure of the covariant field A(x) can be exhibi ted with help of their vacuum expectat ion values (VEV):

(2.16) =ffdcde' O:(c)O2(e') <clA(x,)... (xo) Je'>,

Q~ ---- f de G~(c)]c} e Y f o .

Since in the basis {]e}} the central element C is diagonal, it follows f rom eq. (2.11), i), i.e. [C, A(x)] = 0, tha t

(2 .17) <c[A(x~) ... A(xn)[c'> = W : ( x l , ..., x~) ~ (e - - c') .

These generalized (( Wightman functions ~) W:(xl, ..., xn) have to be considered as tempered distributions in x~ according to GFT (2~), and a]so with respect to their c-dependence, because of the possible restrictions of the domain of

A(ll) . . .A(]~). We mention the distr ibution-theoretic aspects of our formal manipulations only when this seems to be necessary.

294 H. Joos and ~. wn~rAl~

lqow the usual procedure gives the symmet ry properties of the Wightman functions related to the gauge group

(2.18) Wc~.~(x~, ..., x=) 5(v-- e') ~- <c]U(-- d) A(xx) ... A(x~) U(d)lc'> ~-

----- < e I ( A ( x l ) ÷ t/) . . . (A(x~) ÷ d)lc'> =

,{° - - W e ( X 1 ~ * . . , X r _ 1 ~ X r _ b l ~ . . . , - ~ ( c - c ) Wo(x~, ..., x~) + ~ ~-~ x~) +

r

d~ ~-~ I + Wo (x l , . . . , x~_~, x~+~, . . . , x~_~, x~+~, . . . , x~) ÷ . . . ÷ d ~ . ,I

This formula expresses the W~ as functions of c with the help of the Wightman functions for a given value c-= Co, e.g. co =-0. We repeat this expression in

distribution form, compact ly wri t ten:

(2.19) w:(ll, ..., l.) = ~ (2~) ~'~ ~" Z L.(o)... L.(o) w;-"(i, ..., I 0 . ~ 0 p a r t i t i o n m

The indices of the arguments of W0 must have the natura l order. The Wight- man functions of lowest order are

(2.2o) [ w~(x) = c,

W ~ ( x l , x~) = W~o(xl - x~) + c~.

In our case of a free massless field

w ~ ( x ) satisfies the wave equation [] W2o(x) = 0;

is Lorentz invar iant ;

has a Fourier t ransform with support on {p ip2= O, p o > 0 ) ;

which is l im (1 /V) ;W2o(x)dx= c21c=o= O, accord- has a R~ mean~ ing to (2.8) ; ~o=O

W~(x) - - W~(-- x) ---- iD(x), according to eq. (2.1').

Therefore, we have (33)

(2.21) 1 ( d~p exp [ - i ( Ip lx0 -p .x ) ] W2o(x) ~- D+(x) -~ (27~)--- ~ J 2]Pl

which is the conventional two-point funct ion of a massless free field. Indeed,

(a~) K. BAU~AN~ and W. SCHmDT: NUOVO Cimento, 4, 860 (1956).

ON T H E COVA--~IANT D : E S C R I P T I O N O F S P O N T A N E O U S L Y B I ~ O K E N SY)SM:ETRY :ETC. 2 ~

if we insert in (2.19) the usual n-point function of a massless free field

W~"(xl, ..., x~,) = ~ Dt(xi~-- xiJ D*(xi .... - - xi~) , partitions in pairs

( 2 . 2 2 ) (il,i2) natural order

2n÷1 W0 (x~, ..., x~) = 0 ,

we get the W i g h t m a n functions of a covar iant free field. I t is one of the re- mMning problems to show tha t this describes the only solution to our problem.

2"4. The appropriate Hilbert space o / the solutions o /V~A(x) ~ O. - The well-

known t r e a t m e n t of free fields (m3) is based on the construct ion of a Fock

space f rom a Hi lber t space of solutions of the linear field equations. I t is crucial

for the solution of our p rob lem to use in this approach the r ight Hi lber t space

of the solutions of [ ] A ( x ) = 0. This follows f rom the discussion of vectors

T E ~ o Q ~ 1 of the type

(2.23) =ffdxdcF(x, c) A(x)[c> +fdcO(c)lc>

The n o r m ll li is de te rmined by the one- and two-point functions eqs. (2.20) and (2.21), and it is expressed s imply with the help of the Fourier t r ans form

P(p , c), eq. (2.2):

The first t e r m is the convent ional expression, the second one, which introduces

a discrete pa r t of the measure in m o m e n t u m space a t p ~ 0, is a consequence 2

of the c2-term of Wc(x~, x~), eq. (2.20), and is typica l for a covar iant two-point function. We see f rom this expression for the norm tha t the T ~ o (~ 5/fl are de te rmined by G(c) and the restr ict ion of l~(p, c) to the forward m o m e n t u m cone ~ + ~ - { p l p ~ = O, po>~O}. Test functions F(p , c) with suppor t y outside ~ + :y n ~ + = 0 represent pure vacuum states T ~ f d c G(c)[c} E ~fo. On the other hand, s tates T c ~ , orthogonM on the v a c u u m space ~ o , have the physical in te rpre ta t ion of single-particle s ta tes of mass zero. According to the inner p roduc t in t roduced b y eq. (2.24), t hey mus t satisfy the following conditions:

(2.25) ~ 5/z~: G~(c) = O, P~(O, c) = O .

I n order to make the s t ructure of ~ o ( ~ ~f l clearer, we introduce on the m o m e n t u m cone ~--~ ( p I p ~ = O} the following measure :

(2.26) d#(p) ---- d/z+(p) ~- d/zo(p) ÷ d/z_(p) =

= (O(po) ~(p~) + ~'(p) + 0(-- Pc) ~(P~)} alp,

(a3) S. DOrLZCH]~n: Comm. Math. Phys., 3, 228 (1966).

296 m JOOS and ~. W E I ~

and cons t ruc t in t he usual m a n n e r (34) the H i lbe r t space of func t ions on which are square in tegrable wi th respec t to this measu re :

(2.27) 95(p)~-97 ~)" II~l[~=fV(p)p~(p~)dp+ #(0)1~<oo - - / z • *

This ~(2~ is na tu r a l l y decomposed in to o r thogona l subspaces

,u - - , u + " , ~ - - , a a

with

95 ~ Ae(~2 ) : 95 = ae0, eo = 1 for p = 0, 0 for p ¢ 0 ,

(2.28) 95~Lf(,~): (95, e o ) = 0 , suppor t 95c(p]p~:O, po>O},

95 e ~(~' : (¢, eo) = 0 , suppor t 95 c {p Ip ~ = o, po < o}.

The funct ions on ~3 are Fom' ier t r ans fo rms of solut ions of [ ] ~v(x)= 0. I f we pa r am e t r i z e p ~ ~ b y p and t he ene rgy sign, t he Four i e r t r a n s f o r m a t i o n of

95 ~ Ae(~ ~ has the fo rm

J~IP

• (95-(p)exp [i(Jplxo-p'x)] ~- 95+(p)exp [ - i ( ] p i x o - p . x ) ) + ~ ( o ) ,

and the invers ion

] 95(0) : l i m ~ f~v(x, Xo)dx,

r v j

95-~(p) = ± i(2~)- xp [:k ip-x] oo(~(x, Xo)-95(0)) dx

(2.30')

(2.30")

with

p=(leL,,),

~ (x) [~ ) ~o z = ~(x) ~o z - [ , ~ ~(x) z(x) .

As a consequence of [ ] ~v(x) : 0, ~0±(p) is i n d e p e n d e n t of Xo. The inner p r o d u c t

r e la ted to t he n o r m eq. (2.26) reads in conf igura t ion space

(2.31) (~v, Z) = 4- if(q~*(x, xo) -- 95(0)) Oo(g(x, Xo) -- i (0)) dx + 95*(0)~(0),

(3~) p. R. HALLOS: Introduction to Hilbert Space and the Theory o] Spectral Multiplicity (New York, N.Y. , 1951).

ON TILE COVARIANT D E S C R I P T I O N OiF SPONTANEOUSLY B R O K E N SYMMETRY ETC. 2 ~ 7

This t t f lbert space of solutions of the wave equation, in which the constant solution is normalizablc, allows (( second quantization ,~ of [] A(x) ----- O. The (~ c-components ,) of single-particle states T=- ffdxacF(x, c)A(x)]c> with /~(0, c) ---- 0 (eq. (2.25)) can be natural ly identified with elements of -~(~'+~, whereas the elements of Lf~. are related to the vacuum components. The Fourier de- composition of A(x), according to (2.30), leads to the introduction of creation and annihilation operators, which allows a complete characterization of the free eovariant operator fields. We shall formulate this in the following subsection.

2"5. The main theorems o] /tee fields. - Let A(x) be a free, massless field covariant under scalar gauge transformations, as defined by eq. (2.1), which is represented by operators in a separable Hilbert space 3ff in such a way tha t the general postulates of GFT (subsect 2"1, I)-III)) are satisfied and tha t the system of operators la , generated by A(]) and U(e) -= exp [-- iQc], is irreducible. Then we consider the Fourier t ransformation of A(x) as defined for solutions of the wave equation in eqs. (2.29) and (2.30):

(2.32) A(x) = (2g)-tfddp _~(p) exp [--ipx] =

= [-- i( JplXo--p " -)] + a*,p)exp [ i ( J p l X o - p ' - ] ) + Ao,

o r

(2.33')

(2.33")

Ao = lirm ~ d x A ( x , Xo), F

a(p) = i(2 )-Jfdx exp [ipx] 50) -- _40),

a*(p) = -- i(2z)-J f dx exp [-- ipx] O~(A(x, xo) -- Ao) .

l~rom eq. (2.1) the commutat ion relations

(2.34')

[a(p), ai(p')] = 21p I ~ ( p - - p ' ) ,

[a(p), a(p')] = [a*(p), a*(p')] = O,

iEAo, Q] = - 1 ,

(2.34") [a(p), Q] = [at(p), Q] = [a(p), Ao] = [a+(p), Ao] ---- 0

follow by formal calculations. Since A(x), or a(p), a+(p), Ao, respectively, are operator-valued distributions, we have to explain the meaning of the expres- sions eq. (2.33). We discussed already the precise definition of eq. (2.33') in

20 - I1 Nuovo Cimento A .

298 H . J O O S a n d E . W E I M A R

"X

subsect. 2"2 (eqs. (2.7), (2.8) and (2.10)). Since the assumption 9~o= 5~fo (eq. (2.6)) follows from the irreducibility of Ia, we conclude

(2.35) A0 = C,

where C is the unbounded operator defined by eqs. (2.8), (2.10) with the prop- erties (2.11), i), iii)) corresponding to eq. (2.34"). On the other hand, the Fourier transformation, eqs. (2.29) or (2.33"), defines the creation and anni- hilation operators a(p), a+(p) with respect to the improper momentum basis in 5¢(~2, ~_) , respectively. Following the usual procedure we introduce in ~e~

complete orthonormal system {f~(p)) of sufficiently smooth functions

(2.36) ~L(p)f:(p') = 2]pl~(p-p'), G¢

satisfying the condition

(2.37) (i~, Co) = f~(o) = o

characteristic for ] e ~(~) - - ~ t + "

as equivalent to

(2.38)

Equations (2.34') therefore should be considered

a(,~) -- f z~pla~,)f~(p) , a+(/~) = f dp a+(p)f.(p )

[a( /~) , a~(/~)] = ~. .

Tow we sum up our main results in the form of theorems.

Theorem 1. A free massless relativistic field A(x) covariant under scalar gauge transformations, satisfying our general assumptions, is equivalent to the uniquely determined irreducible representations of the canonical commutation relations eq. (2.34) with the vacuum subspace ~o :

(2.39) a(])~fo ---- 0, Qg~o c~¢~o, G~o c 5¢Fo.

_Prop]. An irreducible representation of the commutation relations eq. (2.34) which in addition satisfy the condition eq. (2.39) is up to equivalence uniquely defined: in a C-representation of 5/fo according to eq. (2.13), we have a(p)[c> = O. Thus, for each v the a(p), at(p) generate a unique Fock repre- sentation (7) with vacuum ]c>.

OR- TILE COVARIANT DESCRIPTION OF SPONTANEOUSLY BROKER- SYMMETRY ETC. 2 ~

On the other hand, given a covariant field A(x) satisfying our assumptions, we define C and Q according to (2.7) and U(c)~--exp [--iQC] and a(p), at(p) according to (2.33") or (2.38) which satisfy the commuta t ion relations eq. (2.34). We already have shown CWo c o%fo in eq. (2.11), iv) and QWo c Yfo in eq. (2.15). The remaining condition a(])Wo = 0 follows from t]a(],) f2 II ~ =- (2z)~f ]F(c) l ~ c~ de. ]]o(0)12=0beeause of eq. (2.37) calculated with the help of eq. (2.24), g2= =-- fF(c) ]c}dc. This theorem settles also the problem of the Wightman func- t ions of covariant free fields. The direct calculation, following the usual lines (7), leads to the s ta tement :

Corollary. The Wightman functions of the covariant free fields are uniquely given by eqs. (2.19) and (2.22).

The second theorem summarizes the general s t ructure of the fields {A(])} defined in subsect. 2"1. I t is also a direct consequence of theorem i and the definitions given above.

Theorem 2. The irreducible representat ion of the algebra generated by {A(x)~ U(c)} restr ic ted to the fields A(]) allows a factor decomposition with respect to the centre {C}:

9ff = f , ~ c de, C,ZZ~ = ~ c ,

where J¢% is the Fock space over ~ + . The action of A(]) in 9fr ° looks like

(2.40) A(/) = a([+) + a*(L) + (2~)~ c/(0),

]± being the restriction of the tes t functions to

The gauge group acts t ransi t ively on the family of Hilber t spaces

(2.40') exp [-- iQc']~f ~ = jfc+,'.

We want to elaborate a little bit fur ther on the properties and the physical in terpre ta t ion of the vectors of the t t i lber t space iF. First we remark that , in

spite of the reducible representat ion of the field, there are cyclic vectors~ in part icular cyclic vacua. Because of eq. (2.40'), the cyclic vacua have the form fG(e)[e} de with G(c)V-0 and decreasing faster than ,~ny power of e.

The covariant description of SSB implies some modifications of the particle in terpreta t ion of the quantized field obeying the linear field equation. As already ment ioned in subsect. 2"4, we have to define the one-particle states as normalizable eigenstatcs of the mass operator orthogonal on ~ o . The one-

3 0 0 H. J o o s a n d ~ . W-EIMwa

particle states have mass zero and look like

a'(lo)lZ> = c> ,

=fG(c)lc>dce o, lxe .v'" P + ,

with the improper basis being normalized as

<p, e l f , c'> = 21pl ~ ( p - - p ' ) ~(c-- c').

The degeneracy of the vacuum carries over to the one-particle states. I t is re la ted to a kind of superselection rule (le) because all observables genera ted from A ( ] ) commute with C and therefore do not mix the subspaces ~ ° . Fur- thermore, the S-matr ix commutes with Q too, which implies t h a t the descrip- t ion of the scat ter ing processes of these particles is identical in each factor space 5if c. This will be shown in sect. 3.

Wi th the help of the creation operators at(]x) we can construct as usual a complete or thonormal basis in Yf:

(2.4~) I]¢~,, " " , ]x,,; c> = N(ot i )aq( fx , ) . . . aq(fxn)]c> =

ra. , so.(.,,)i.,,. ;o> " " 3 2 1 p , I '

n = O~ 1, 2, . . . ,

and with

(2.42)

<Ix,, . . . , Ix,; el/x,, . . . , Ix;n; c'> = an,n a ( c - c ' ) N ' ( ~ , ) ~ ax,x'/, ... x,,x, ,

<p,, ..., p . ; ~ IE, --., P',; ~'> = - - 3 ! - ~nm ~(e-- C')21p11 ... 21p=l ~ ~ (P l - -P~ , ) . . . ~ ( p n - - p , ) ,

yg

N(al, ..., a n ) = n~! ... n,! for ni equal indices a~.

In spite of the degeneracy in c, these states can be in terpre ted as n part icle states~ since the degeneracy plays the same r61e as discussed above. F ro m the

considerations of the multipart icle states, it is evident tha t the infinite degeneracy

of the one-particle states does not describe multiplets of mass-zero part icle states, because the composition of multiplets would require the tensor product also with respect to the label c of the mult iplet members.

2"6. W i c k order ing . H a a g ' s e x p a n s i o n . - Part icle properties and react ion ampli tudes are described by mat r ix elements of the scattering operator, cur- rents, etc., in the many-part ic le basis corresponding to eq. (2.41). The well-

ON TH]~ COVARIANT D]~SCRIPTION OF SPONTANEOUSLY B R O K E N 8Y~MF, TRY ~,TC. ~ 0 1

known techniques like Wick ordering (~) and IIaag's expansion (~ )~deve loped

for the evaluat ion of such mat r ix e lements - -need some slight modifications

in our formalism. In the definition of Wick ordering we have to take into consideration the

t e rm propor t ional to C in

with

(2.43)

.~(p) = X~_(p) + (2~) ~ t,(p) c = .~(p) t(p~) + (2z)~ t,(p) c

A(p) =- O(po) a(p) + 0(-- p°) at( - p)

according to eq. (2.32). Therefore, we define

(2.44) :A(]~) A ( / , ) : = ~ (2~)"~L.(o) L,(0) .A±(],,+,)...zI~(L,)

l~artitlons of

with the usual convention of Wick ordering of the A±(]) on the r ight-hand side• ~ o w we consider Haag 's expansion of (< observables >>7 i.e. linear operators

Z in ~ f with [L, C] =- 0

(2.45)

As usual this infinite sum is mean t us weak limit of mat r ix elements with re- spect to the mult ipart icle states. Whereas under conventional conditions only a finite number of terms contr ibute to mat r ix elements between states with a finite number of purticles~ in our ease the situation is more involved. Because of the C-terms hidden in :~(p~). . . ~ (p~) : , tv~ with arb i t rary n con- t r ibute to such ma t r ix elements. Equat ion (2.44) leads to

(2.46')

with

(2.46")

iPI , . . . ,Pm; c]LIP,~+I, . . . ,P , ; c ' } = ~ o ~ " ... q~...dq, G~(q~, . . . ,q,) .

• (P~, . . . , Pro; el :~±(q~) .. . Ax(q, )" ]P~+~, . . . , P , ; e ' )

G,(ql, . . . , q,) = ~ = ~. (2~)3~/~E,+~(ql, ... , qt, 0 . . . 0) c ~ .

The convergence of these sums is crucial for the existence of Haag 's expansion.

In the applications the convergence follows from the t ransformat ion properties of Z with respect to the gauge group.

(35) G. C. WICK: .Phys. Rev., 80, 268 (1950). (~6) R. HAAG: Dan. Math..Fys. Medd., 29, 13 (1955).

302 E. aoos and E. WEIMAR

I t is known tha t the mutr ix elements of ]5 determine the coefficients G.(q~, ..., q~) of the conventional Hang expansion (2.46') uniquely for ' q~ ~ 0~ q~ =/= 0 (as usual up to points of measure zero). We have to show tha t also the coefficients ~.(p~, ..., p . ) of the generalized Hung expansion (2.45) are uniquely de termined for points p~----0, including the special points with nonvanishing measure p~----0. Bu t this follows from the uniqueness of G, because the F ' s are coefficients of a power series expansion of G in c according to eq. (2.46"). Since we used an improper vacuum basis, the derivat ion is somewhat formal. A more rigorous derivat ion can be given.

Now we consider gauge-invariant observables 35, i.e. those with [35, Q] -~ 0. Their expansion coefficients have the characterist ic p roper ty

(2.47) ~,,(Pl, ..., p~---- O, ..., p . ) = 0 , p~j = O,

as a consequence of the uniqueness of Haag 's expansion and

[ L , Q ] = ( 2 ~ ) t ~ o ~ . T ... ~ . . . dp .F .+~(p~ , . . . , p . ,O) :~ (p~) . . .~ (p . ) " = 0 .

This implies for the infinite sum in eq. (2.46") tha t only the first t e rm is dif- ferent f rom zero. Henc% for invariant operators there is no convergence prob- lem. These zeros o f / v have physical consequences when the F . have analyt ic propert ies which often follow from locality.

I t is worth-while to ment ion that , for p~----0, p~ =~ 0, the well-known ex- pression for the coefficients of Haag's expansion remains valid:

(2.48) F.(p~, ..., p.) = e(p~)...e(p.)<~2l[... [35, .~ ( -p~)] , . . . . ~ ( - p . ) ] ] ~ >

with A defined in eq. (2.43). This follows from the calculation of the mat r ix elements of L (37) in the multipart icle basis ~nd the fact t ha t mat r ix elements

of invar iant operators are independent of c:

<p,; cl lp~; c'~ <p,; c l ~ ( - d)35U(d)lp'~; c'} <p~; c + dL351p~; c '+ d~

Covariant operators with the t ransformat ion proper ty [L, Q] = i m ay be t rea ted along the same line by considering the invar iant operator L - - A ( x ) .

2"7. Relation between the generator Q o] the gauge symmetry and the conserved current. - In the case of a quantum-mechanical symmetry , the space integral

of the (( charge density ,) jo(X) of a conserved current ja(x), (~ja(x)= O, con-

(37) V. GLASS]~IL H. LE~MANX and W. ZI~MER~A~N: N'UOVO Cimento, 6, 1122 (1957).

O:N" THE COVAI~IANT DESCRIPTION OF SPONTANEOUSLY BROKE~ 8"IrMMETRY ETC. ~ 0 ~

verges weakly to the generator of the symmetry , i.e. the (( global charge ~ Q (11) :

(2.49) lim [dxjo(X, t) --~ Q. v--R, J

F

The scalar gauge symmet ry is related by the :Noether theorem to the conserved current

(2.50) jr(x) = ~ A ( x ) .

In the following we want to derive a relation similar to eq. (2.49) between the

charge integral of this current and the generator Q of the scalar gauge group as defined for our covariant free massless field. Since for the usual irreducible fields such a symmet ry generator Q does not exist (as), the impossibility to give a meaning to the charge integral is often considered as characterist ic for a spon- taneously broken symmetry . For this reason the discussion of the charge in- tegral is i l luminating for the comparison of our t r ea tmen t of SSB based on covar iant fields to the conventional one based on irreducible fields.

In order to give the expression eq. (2.49) a precise meaning we follow the usual procedure (11) and introduce functions gr(x) ---- v~(Ixl) ~(xo) with the prop- erties

(2.51a) v~(lx[) = 1 for IxI<r, v ~ ( [ x l ) = 0 for Ix[>~r~-d;

(2.51b) v~(y) = ~ for y>~O;

(2.51e) f~(Xo) dxo ---- 1 , ~(Xo) > 0 ;

(2.51d) support (v(xo)) :compact , ~(xo) : ~ •

Then we consider as a definition of the charge integral the limit of the smeared field operator Q~, r --> ~ :

(2.52) Qr =rio(x) g,(x) dx = A ( - - ~o g~) .

Because of (2.51c) and 5,j~(x) ---- O, the t ime averaging in eq. (2.51) is for- mally irrelevant.

We continue with the known remark (lO.11) on the existence of l im [Q~, A(/)]

for ] E 7. In our special example of spontaneously broken symmet ry we get f rom eqs. (2.1') and (2.3)

(2.53) [Qr, A(/)] = - i(Oogr, ]) ~,~> - i (2z)If(0) .

(38) •. F. STIr]EATER; Proc. Roy. Soe., 287A, 510 (1965); E. F ~ R I and L. E. PICASSO: t)hys. Rev. JSett., 16, 408 (1966).

304 H. JOOS and ~. W~IM~R

This agrees with [Q, A(])] according to eqs. (1.5) or (2.1"), therefore we have

(2.54) l~m [Q~, A(])] = [Q, A ( ] ) ] .

Now we consider the limit of matrix elements of Q~ between states

d ~ = A(]~)... A(] , ) .('2, f2 =Jdc.F(c)le> e .~o, .E(e) e F ,

and compare it with matrix dements of Q. From eqs. (2.53)~ (2.54) it follows directly that

(2.55) lim <121~*Q,d ' l~>- <QI~+Q~'I~> : I*-~qO

= lira <Dlz¢ + zd'@ID> -- <DI M+ zC'QID > . r - - ~ m

Therefore we have to calculate only matrix elements <D]~Q]f2>, and l i ra <~l~¢Q~]~>. These follows immediately from eq. (2.15) and (2.17):

(2.56) 1<oI (t:) ... (J:)olo> = - - i f de W:<1:, ...,

and from eqs. (2.19) and (2.22), together with o®limffdxd~l*(x)D+(x--x') • • ~ o g , ( x ' ) = ½ (2~)t](0):

(2.57) lira l<~lA(/*) ... A ( I * ) Q , IQ> = l im <QIA(/*) . . . A(I*)A(-- aog,)l~2> = $,--> ~ ¢ - - ~ cO

= - w o i / , , . . . , (c)F(e). 2

For real /~(e), /~*(e):/~(e), the right-hand sides of eqs. (2.56) and (2.57) are equal. This allows us to state the following

Theorem 3. I~et Q be a cyclic vacuum vector: Q = f d e 2 ' ( e ) l e > , _ F ( e ) e r ,

/~(e) # 0, with the conjugation property /~*(e)=/~(e) relative to the phase convention of the vacuum basis exp [-- iQd] le> = [e + d> and let ~¢ c g f be the dense set generated from ~ by the polynomials pol [A(],)] ~2 of the quasi- local field operators A(/), ] e F . Then the charge integral converges weakly on ~ . towards the symmetry generator Q:

~ m (¢', Q~¢) = (¢', Q¢), ¢, ¢ ' e 2 , .

O ~ TtIE COVARIANT DESCI~IPTIO/q OF SPONTANEOUSLY BI:tOKElq SY~t[M]~TRY :ETC. 3 0 ~

3. - R e m a r k s o n in terac t ing fields and scat ter ing theory .

3"1. Interacting /ields with covariant scalar gauge symmetry. - ~ o w we s tudy the covuriant description of spontaneously broken symmet ry for fields with a nontr ivial S-matrix. For tha t we follow the general line of GFT. We assume the existence of a field operator A(x) satisfying the postulates I ) - I I I ) of subsect. 2"1. Ins tead of the specific commuta t ion relations (2.1') of free

fields we postulate

IV) locali ty: [A(x), A(x')] = 0 for (x- - x ' )~< 0.

We assume no special field equation, bu t the field algebra generated by {A(])} should allow the same spontaneously broken symmet ry group as the free fields do. Therefore in our covariant description we postulate

(3.1) U-,(c)A(/) ~(c) = Aft) + (2~)nf(0),

[U(c), U(A, a)] = 0.

I f we bear in mind the possibility of dynamical rear rangement of symmetry , this means an essential simplification. )Tevertheless, a lready this simple model reveals some features typical for S-matr ix theories with SSB. The scalar gauge symmet ry may be related to a local conserved current jr(x):

(3.1') U(c)j,(x) U-'(c) = j~(x) and ~ j , ( x ) = O .

The charge integral formally generates the gauge symmetry . The precise de- finition of this charge integral raises questions similar to those already discussed in subseet. 2"7 for the free field.

Since the interact ing and the free fields are subject to the same postulates of GFT, some general features of A(]) can be derived as for the free fields in

sect. 2.

i) There is an infinite-dimensional vacuum subspace Jfo.

ii) The unbounded operator

(3.2) C = lim x A ( x , Xo) v--~-R= "V

Ip

can be defined according to the procedure following eq. (2.7), and it has the propert ies (2.11). Whereas in the case of the free field we could make use of the wave equation in order to show tha t the states ¢ = A(]I).. . A ( / . ) Q are

306 ~. zoos and E. w ~ i ~ m

dense in 5/z, even if all f~(Xo) have compact support with a common bound, we have to assume this p roper ty for interact ing fields ((( Zeitsehicht axiom >> (39)).

iii) There is an improper basis in ~fo related to C:

(3.3) C, ic > = cic> , U(e')lc> = [c d- v'> ,

as in eqs. (2.13)-(2.15). Related to this basis in Wo there is a factor decomposit ion of the I t f lbert space ~W:

(3.4) ~=fdc~o, ~t(S~)... A(/.)I~> e ,.~f'o.

iv) F rom the covariance of the fields it follows the symmet ry p roper ty of the Wightman functions W~(x4, ..., x~), defined according to eq. (2.17) as

(3.5) (X71, A(xO ... A(x,) ~ ) = f dcV:(c) G(c) WT(x~, ..., x,) ,

and symmetr ic according to eqs. (2.18) and (2.19)

(3.6) w:(s~, ..., s.) = ~ (2~) ~'~ ~" Z L.(0) ... L(0) ~"-",+o , , , , ..., s , . ) . • =0 partitions

i l . . . i n

v) The clustering proper ty (40)

[ <~, A U(1, x)BQ> ~ <~IAI~> <~[BI~> , (3.7) Ixl~°

A = A(]~) ... A ( / , ) , B : A(]~) ... A( f~) ,

which plays an impor tan t r61e in the Haag-Ruelle scattering theory (~1), is

not valid for our fields, because covar iant fields are necessarily reducible and

it is well known (~v.3~) tha t irreducibili ty and unique clustering are equivalent

conditions. However, we have weak clustering for the factors, and therefore

(3.8) l im W~+'~(x~ + a, x , + a, Yl, , y~) = W~(x,, ..., x~). W~(y~, y,~) l a l_+~ • . . } . . . . . . ~ •

(3,q) R. HAAG and B. SCHRO]~: Journ. Math. Phys., 3, 248 (1962). (40) R. HAAC~: Phys. Rev., 112, 669 (1958); H. ARAKI, K. It]~el > and D. RVELLE: Helv. Phys. Acta, 35, 164 (1962). (41) R. HAAG: Phys. Rev., 112, 669 (1958); D. RVELLE: Helv. Phys. Acta, 35, 147 (1962); R. J0ST: The General Theory o] Quantized Eields (Providence, R.I . , 1965).

ON T HE COVAI~IANT D E S C R I P T I O N OF S P O N T A N E O U S L T B R O K E N S Y M M E T R Y :ETC. S D ~

This p roper ty suggests the introduct ion of t runca ted Wightman functions W z ,

defined as usual b y

( 3 . 9 )

with

(3.1o)

!" 1 w2(~ , , ..., x ) = Z w ; , , ( x , , , , , . . . , x~,,,,) ... w;.~(x~.,~, . . . , x , , , , , ) par~lttonl

{ w~ = 1 , w~ = w~o(x0 = ~,

2 C2 W~o(x~, x~) + . W~(x~, x~) =

Inside distinct Wz the order of the space-time points x~ has to be the same as in W on the lef t -hand side of eq. (3.9). Of course, in our case the sum on the r ight-hand side contains factors corresponding to the nonvanishing one-point function, e.g.

(3.]]) $ W ~ c ( X l , X 2 W~(x~, x~, x~) = ~ , x~) +

+ c(WU~,, ~) + w,°(x,~ , x~) + ~ . ( ~ , ~)) + c..

In our ve ry simple model the t runca ted functions Wr are independent of v for n va 1; the ent ire c-dependence comes from the one-point function. There- fore, the decomposition (3.5) looks similar to eq. (3.6). Now we prove this.

Theorem 4. W~(x~ , . . . , x~) is independent of c for n # 1.

Proo]. We assume the val idi ty of the s ta tement for m = 1, 2, ..., n - - 1 . Differentiation of W~" according to (3.9) leads to

x . ) = 8 (3.12/ ~ W~(x~, . . . , - ~ W ~ , ( x , , . . . , x . ) +

rs ~ + ~ Z w2(xj,,~,..., x,,,,,).., w~o(~,,,~ ... x,,,,,). k~l Dartition$

On the other hand, differentiation of (3.6) yields

x.) ~ "-~ x.) - - " z W c ( x l ~ . ~ X k _ l ~ X k + l ~ . . ~ • (3.13) ~c W , ( x l , . . . , .. .

According to the definition (3.9) this sum is equal to the second te rm on the right- hand side of eq. (3.13), so tha t (~/~c)W: = 0.

The unique Wightman functions of the free fields are simply characterized by the t runca ted functions

(3.14')

(3.14")

2 W~,(x) = c , W~°(x,, xO = iD~(x~ -- x~) ,

n W,°(x~ , . . . , xn) = 0 for n > 2 .

3 0 8 H. J 0 0 S a n d E. W ~ I ~ A R

3"2. The invariant S-matrix. - The physical in terpre ta t ion of a general field theory is based on a theorem which ensures the existence of a un i ta ry S-matrix. One of the essential assumptions of the Haag-Ruelle scattering theory is the existence of one-particle states with masses m ~ 0, isolated in the mass spec- t rum. This assumption contradicts the well-known result t h a t in a field theory with SSB there exist s tates with masses arbi t rar i ly close to zero (~). There- fore we should be prepared to encounter difficulties in the formulat ion of the scattering theory. On the other hand, infra-red divergences do not appear in per turbat ion theory calculations in models showing this type of SSB. I t is beyond the scope of this paper to clarify possible relations between SSB and the existence of the S-matrix. We restr ict ourselves to the discussion of scat ter ing theory based on the assumption of the LSZ asymptot ic condition.

Because of our general postulates on A(x) , in part icular t ha t the sponta- neously broken gauge symmet ry is related to a conserved current and tha t A(/ ) are operators in a Hflbert space with positive norm, the assumptions of

the Goldstone theorem are satisfied (~2). This theorem ensures the existence of massless part icle states [P, c} with

(3.~5) i F

<vljAx)lp , v'> = ,-~_,~p. exp [-- ipx] 6(v-- v').

We assume tha t the fields A(x) are interpolating, renormalized, fields of these Goldstone particles, i.e.

(3.16) <elA(x) IP, c'> = (2~)-~ exp [-- ipx] ~(c-- c').

Now we introduce the local operator (~6)

B~(x) = j~(x) + F~.A(x), (3.17)

which satisfies

(3.18) <clB .(x)Ip, ¢> = o .

As a consequence of current conservation and definition (3.17), we calculate

(3.19')

(3.19")

DA(x) = F - 1 . 8.B~(x),

D/ . (x ) = & ( & B . - 8.B.) = i.(x).

Now we can formulate the asymptot ic condition. We assume that eqs. (3.19)

can be solved by

(3.2o') A(x) = A , o ( x ) - f a t D a ( x - ~) F -19. B~,(~) ,

(3.20") j~,(x) = -- F~,A,~(x) - - f d~ D ~ ( x - ~) i,(~) ,

O N T H E C O V A I ~ I A N T D E S C R I P T I O N O F S P O N T A N E O U S L Y B R O K E N SYMM~TI:~Y E T C . ~ 0 ~

with free fields A~=(x) satisfying

[] A,.(x) = 0 , [A~o(x), A, . (x ' ) ] = il)(x-- x').

Similar solutions exist with the advanced Green's function D~ and Aout(x), as follows from C P T . As usual, the integral in (3.20) is unders tood as a weak limit.

Similarly to the case of a quantum-mechanical symmetry , we will show tha t the covariance properties of the interact ing field A(x) with respect to the sponta- neously broken gauge symmet ry imply covariance of the asymptot ic fields Ao,(x) (ex = in, out), and invariance of the S-matrix.

First we remark tha t

(3.21) U-~(c) A.x U(c) = A°.(x) + c ,

which follows from applying the uni ta ry operator U(c) on (3.20') and using

the invariance of B(x) according to eqs. (3.1') and (3.17). In the same way we get

(3.22) [C, A¢~(x)] = 0.

For reasons of simplicity we restrictie our model fur ther by assuming asymp- tot ic completeness in the following:

(3.23)

where 2[ex denotes the algebra generated by the fields Ae:(/). Under these circum- stances we have

(3.24) C -- lin~, -~ ox(x) dx ,

V

corresponding to the definition eq. (3.2). Now we define the S-operator in the usual way as an uni ta ry equivalence

t ransformat ion between the free fields A~n and Aou,:

(3.25') A~n(x ) = SAou~(X ) S -~

with the phase condition

(3.25") Sic> = Ic>.

This definition is unique and leads to the impor tan t properties

(3.26') [C, S] = 0 ,

(3.26") [Q, S] = 0, or U(c) SU-I(e) = S.

310 H. JOOS and v.. WEIMAR

The first p roper ty [C, S] : 0 follows ra ther directly f rom the definition eq. (3.25) : since A~,(x) and Aou,(x ) ~re irreducibly represented in the factor spaces 9~ (~), inequivalent for different c, S must leave the factor spaces invar iant ; this is

equivalent to eq. (3.26'). This result allows us to postulate the phase condition of the S-matr ix (3.25"), which is necessary for the proof of eq. (3.26"). For this we consider

[Q, A~o] = [Q, SA°~ S-~],

which implies

0 : SAo.t[Q, S -~] + [Q, S] Ao, tS - ~ = A~,S[Q, ~-1] _~ [Q, HI ~-IA~.

Because of

we get

o = [Q, $8-~] = S[Q, ~-~] + [Q, S] S -~,

[A,o, S[Q, S-~]] = 0.

Therefore S[Q, S -~] has to be an element of the centre:

k ~ [ Q , k~ - 1 ] : 0~" ~ .

The vacuum expectat ion values (t)]S[Q, S-1][Y2 '} (~, ~9 'e~o) vanish because of eq. (3.25"). Therefore we have ~----0, which proves [Q, S ] - - 0 . Since S t ransforms the irreducible algebra generated by {Al~, Q} into tha t generated b y (Ao~t~ Q}, it is uniquely determined up to a phase which is fixed by eq. (3.25#).

According to eq. (2.42) we introduce n particle bases related to A~ or Ao.t, respectively, which describe states with n incoming particles lP~, ..., P~, c} ' '

t t or m outgoing particles IPl, "." ~ P~, c} °'t. As usual the S-matr ix elements describe the physical transit ion ampli tudes

t t t \ I n (~I¢ (3.27) °at(p1 , . . . ,p, ,; t ip: , ...,p,~; C'~'n=ln(pl, ...,p~lS(C)]p~, . . . , ~ 2 ~ -- C').

The two eqs. (3.26) contain the main s ta tements on the s t ructure of the

S-matr ix in a covariant field theory with a spontaneously broken symmetry . The first one means tha t the S-matr ix leaves invar iant the irreducible factors 5~f (~ of the field algebra. I t follows directly f rom the second one t h a t the S-matr ix elements are identical in different factors, i.e. the matr ix elements (3.27) are independent of c. Therefore, in principle, the consideration of the S-matr ix could have been restr icted from the beginning to a single factor. This means t ha t the modification of quan tum mechanics by giving up irreducibil i ty of

the field algebra is i rrelevant for the physical interpretat ion. However , the

O:N TILE COVARIA:NT DESCRIPTIO:N OF SPONTAlqEOUSLY B R O K E N SYMMETRY ETC. 311

spontaneously broken symmet ry has consequences for the S-matr ix which become clear by embedding the S-operator in a covariant field theory. Under

the conditions of eqs. (3.26), the coefficients in Haag 's expansion of the S-matr ix

(3.28) S = ~ .. p ~ . . . d p ~ S ~ ( p ~ , . . . , p ~ ) : ~ ( p ~ ) . . . A ' ~ ) :

vanish whenever one of the moment~ is zero

(3.29) S n ( p l , . . .~ P i ~-- O~ . . . , p~) = O .

This we have proved in subsect. 2"6. I t is the main advantage of the covar iant description of SSB tha t such zeros of the connected par t of the S-matrix, which are similar to the Adler zeros (~5.4~) of spontaneously broken SU~×SU~ in current algebra, appear as self-consistency conditions on the S-matr ix with the properties (3.26), independent of any specific dynamical model.

I t would be possible to derive from the asymptot ic condition the reduct ion formulae for the S-matr ix (20.37). This would allow us to express the coefficients S,(p~, ..., p , ) by re tarded or t ime-ordered functions which would combine the zeros (3.29) with the analyt ic properties of u local field theory.

3"3. Concluding remarks. - Our discussion of a scalar gauge sy m m et ry exhibits the characteristic problems related to spontaneously broken symmet ry : the s t ructure of the degenerate vacuum, the covariance properties of the vacuum

expectat ion values, the connectiou between conserved currents and symmet ry generators, the representat ion of the free fields and the form of gang ' s ex- pansion, the asymptot ic condition and the consistency properties of the S-matrix. Of course~ now it would be desirable to s tudy this general s t ructure in special dynamical models. The simplest model with scalar gauge symmet ry is the gradient coupling model (27) defined by the Lagrangian

L = ½ (~A) 2 -- ~f(-- iy , ~' + m -~ i17~,75 ~F'A) y~

formally invar iant under the t ransformat ion A ( x ) - - > A ( x ) + c. In so far as

this model contains a coupling to the ma t t e r field yJ we would have to generalize slightly our approach. I t is more serious tha t this model is not renormalizable

and also ra ther unphysical. Ex tended to the coupling of isoveetor pions and nucleons it would become more realistic. This would imply a generalization

of the scalar gauge group to tha t of ISO3, already ment ioned in sect. 1, which is possible wi thout serious complications. But , of course, the really challenging

(a2) K. NISHIJII~A: -/YUOVO Cimento, 11, 698 (1959); J. IIAMILTO~¢: 2Vucl. Phys., 1 B, 449 (1967); M. MARTIniS: NUOVO Cimento, 56A, 935 (1968).

312 H. J00S and ~,. w ~ I ~

problem would be tha t of the t r ea tment of SU~×SU~, because it is believed

to be an approximate symmet ry of Nature. I n addit ion this would ~llow ~ close

comparison of our method with the familiar techniques of current a lgeb ra - -

in part icular with respect to the <~ existence ,~ of the charge integral in a co-

var iant field theory. Besides the appearance of dynamical rearrangement of

symmet ry (~s) a first look made us believe tha t a covariant description is also

possible in this case and leads to the typical problems of SSB discussed here.

We shall leave such model considerations in the f ramework of SU~ ×SU~ for

discussions in the future.

Studying these problems, we learned much from discussions with our col-

leagues at Berlin, Hamburg and at (3ERN. Particularly, we have to t hank

D. BUCHHOLZ~ H. HAAG~ H. LEHMANN, L. STREIT for their s t imulat ing remarks.

One of us (E.W.) would like to t hank the Theoretical S tudy Division of

CERI~ for hospital i ty during a short visit.

• RIASSUNTO (*)

Si esaminano i campi ridueibili A(x) con vuoto degenere the eonsentono la trasformazione eli simmetria unitaria U-l(e)A(x) U(e) = A(a~) q- c. Si deserivono lo propriet~ naatema- tiche dell~ integrale di eariea, correlate alia eorrente conservata di questa simmetria spontaneamente infranta. Inoltre si diseuto la struttura della teoria della matrice S in tale teoria dei eampi generalizzata come guida per il trattamento di esempi pi5 complessi di simmetrie spontaneamente inffan~e.

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