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Non-compact symmetries in field theories with indefinite metric
Nuclear Physics B256 (1985) 687-704 © North-Holland Publishing Company
N O N - C O M P A C T S Y M M E T R I E S IN F I E L D T H E O R I E S W I T H INDEFINITE METRIC Yuan K. HA Ph.vsics Department, The Rockefeller University. 1230 York Avenue. New York. N Y 10021, USA
Received 29 October 1984
We motivate the study of non-compact internal symmetries appearing in some renormalizable field theories by the indefinite metric method. The use of the indefinite metric formalism in constructing a positive-definitephysical Hilbert space is illustrated for the non-compact free fields and the non-compact sigma models. We discuss the problems associated with non-compact symmetries occurring in these theories. The notion that positivity of the quantum hamiltonian for a non-compact theory must be based on its classical field theory analogue must be abandoned. For the non-compact O(1, N ) / O ( N ) sigma model we show how to obtain a unitary S-matrix in the physical subspace from a pseudo-unitary S-matrix in the overall Hilbert space, as well as the feature of dynamical mass generation and asymptotic freedom behavior. For the non-compact CP N sigma model we discuss the appearance of a dynamically generated composite gauge boson without external input. The prospect of composite gauge bosons for supergravity theories is assessed. Whenever possible, we compare and discuss the features of compact versus non-compact symmetries in a field theory.
I. Occurrence of non-compact symmetries T h e use of n o n - c o m p a c t symmetry groups in physical theories has gained much interest recently. In solid state p h e n o m e n a , n o n - c o m p a c t sigma models in two d i m e n s i o n s a p p e a r quite i m p o r t a n t l y in the study of the mobility edge of electrons in d i s o r d e r e d systems [1]. The n o n - c o m p a c t symmetry in such cases is essential for the correct b e h a v i o r of the one-loop c o n t r i b u t i o n to the beta function, which would have the o p p o s i t e sign if the symmetry considered is c o m p a c t [2]. Various n o n - c o m p a c t sigma models also appear surprisingly in extended supergravities. These theories offer the hope for an ultimate unified field theory in which all the basic interactions c a n be c o n s t r u c t e d in a single mathematical framework [3]. In particular, the work of C r e m m e r a n d Julia [4] shows that the N = 8 supergravity can be u n i q u e l y reform u l a t e d as a massless theory of a simple N = 1 supergravity in eleven dimensions. W h e n reduced to four d i m e n s i o n s the bosonic sector of the lagrangian admits a n o n - c o m p a c t sigma model with a global n o n - c o m p a c t exceptional group ET( ~7) a n d a local m a x i m a l compact subgroup SU(8). Similar reductions to four d i m e n s i o n s of 687
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the N = 6, 5.4 cases show that the bosonic sectors are all described by non-compact sigma models on the coset spaces SO*(12)/U(6), SU(5.1)/U(5) and SU(1,1)/U(1) respectively. On the other hand, if the eleven-dimensional theory is reduced to three dimensions the resulting sigma model is of the non-compact typc Es~ s~/SO(16): while reduction to five and six dimensions results in the non-compact sigma models on the respective spaces E6t+6~/Usp(8 ) and SO(5,5)/[SO(5)xSO(5)] [5]. More recently, a non-linear sigma model of scalar fields based on the non-compact coset space SU(1, 1)/U(1) has been used to reduce higher-dimensional supergravity theories to lower-dimensional ones with the attractive feature that masslessncss of fermions is not just due to supersymmetry [6]. Thus from many points of view. non-compact symmetries have become more relevant to current problems in particle physics. Earlier attempts in the application of non-compact groups in field theories [7] were perhaps hindered by the non-positive definite nature of the quadratic invariants as well as the lack of finite-dimensional unitary representations of the non-compact groups. For supergravity, one plausible way to handle this situation is to formulate the theory on a coset space G / H so that the non-compact symmetries are realized linearly. In the N = 8 supergravity theory, one would consider the theory to describe new fundamental constituents called preons which would transform according to the representations of SO(8), while the bound states, which are the quarks, ieptons and gauge bosons, would transform linearly according to the representations of SU(8) [8]. A crucial question in this approach is whether these composite gauge fields are dynamical and propagating. In two dimensions, it is known that the compact CP N sigma model possesses a composite U(1) gauge field which becomes truly propagating by acquiring an effective kinetic term via quantum correction [9]. A recent 1/N expansion study of the O(1, N )/O( N ) sigma model [10] and the non-compact CP N sigma model [11] in two dimensions in the negative coupling region has shown that these non-compact models can have dynamical mass generation, a condition necessary for the dynamical generation of a composite gauge boson. Some preliminary studies 112] of a simple non-compact sigma model in four dimensions also point out the possibility of a dynamically generated gauge boson when a mass parameter or a potential giving masses to the fields is introduced into the model. However, for the N = 8 supergravity to be a successful candidate for unification, one has ultimately to demonstrate in the theory that dynamical gauge bosons can appear without introducing any external parameter. The present paper is to motivate the use of the indefinite metric method in some renormalizable field theories with non-compact internal symmetries. The requirement of renormalizability necessarily brings the discussion on the quantum noncompact sigma models to two dimensions, where firm statements and reliable results can be obtained, despite that the classical models can be formulated in any dimensions. Although supergravity is not a renormalizable theory from a perturbative point of view, its tight symmetry structure renders the theory a chance of being
Y.K tta / Non-compact .~Tmmetri¢~
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finite [13] and some of our conclusions in two dimensions on the sigma models may well extend to supergravity in four dimensions. Further, the study of the behavior of quantized field theories with non-compact symmetries is itself of interest regardless of supergravity as the method can be useful in other contexts. We shall illustrate the use of the indefinite metric in the quantization of theories admitting global non-compact symmetries and obtain a positive-definite physical Hilbert space. We discover that the feature of dynamical mass generation with asymptotic freedom behavior is present in a 1/N expansion study of the non-compact sigma models. For the non-compact O(1, N ) / O ( N ) sigma model we show how to obtain a unitary S-matrix in the physical subspace from a pseudo-unitary S-matrix in the overall Hilbert space. For the non-compact CP u sigma model we discuss the appearance of a dynamically generated composite gauge boson without extra input parameter. Whenever possible, we compare and discuss the features of compact versus non-compact symmetries in a field theory. 2. Problems of non-compact theories An apparent difficulty in implementing non-compact symmetries in field theories lies in the non-positive definite nature of the quadratic invariants. For example, the kinetic terms of a non-compact theory would have both positive and negative signs. Thus canonical fields with negative kinetic terms would have negative norm and their associated propagators would develop ghosts. Since physical states must occur with positive norm, the appearance of negative kinetic terms usually means the theory is unphysical. Such argument, although it seems sound, if taken categorically, would render all non-compact symmetries irrelevant to physics. As there are definite systems and theories which require the appearance of non-compact symmetries, one should therefore confront this situation with new insight. In the present approach, we develop a scheme in which the negative norm states induced by non-compact symmetries are eliminated by a superselection rule provided by the indefinite metric to the enlarged Hilbert space to form a new physical subspace. Such a selection rule will forbid transitions between states of positive norm and states of negative norm, while physical observables are to be constructed from operators defined only in the physical subspace. This type of approach has also been used to explain the non-existence of certain decays, e.g. /~--* ey, in the lepton family in a different context [14]. To understand the nature of the problem associated with a non-compact internal symmetry, we consider first a multiplet of free scalar fields = (o,
(o, .....
oM;
.....
(2.1)
transforming according to the defining representation of SO(M. N );
q~,--, ~; = n/q,j.
(2.2)
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The internal components are denoted by i. j = (1 . . . . . M; M + 1. . . . . N). and ~ / a r e elements of the pseudo-orthogonal group SO( M. N). which leave the hermitian form
go =
- 1.. 0
0 ] /~,.
(2.3)
invariant, i.e. /2"rgoI2 = go-
(2.4)
The lagrangian densities for the compact SO(M + N ) case and the non-compact S O ( M , N ) case are respectively ~compact = ! O"l,uO" 0'o.0 OF loq ,'IT- al't"ff
= ½aoo- 0oO - ~ oko. 0~o + ~o,,~,. o,,., - ~ o ~ , . o ~ , .
(2.5)
= - ~ 0{}0. aoa + ½9~.0. a~o + {Oo~. Oo~ - ~Ok,~. afro.
(2.6)
Correspondingly, the hamiltonian densities are '~ompact = ½0,,0" ¢'~ ........
0,,0 + ~Oka" Oka + ½0O4" O0"~+ ~ O~'~" Oa~,
pact = -- ~ 0"100 " 0")0° -- ~OkO" O.L,O + I ao,lT. 00,1T _{._10fflT' Ok'n'.
(2.7) (2.8)
When appropriate, the following mass terms can be added: (i) compact case - m2(o • 0 + 'IT- 'lr), (ii) non-compact case - m2( -o. 0 + 'rr - 'rr). It can be observed immediately that for the non-compact case, the hamiltonian density in eq. (2.8) for a multiplet of free fields is not positive definite as the a-fields carry negative signs. One might be tempted to conclude that since the classical hamiltonian does not have a lower bound the corresponding quantum theory does not have a vacuum and the theory is therefore not sensible. However, one knows that free fields are non-interacting. Each scalar field in the canonically quantized theory has a ground state energy - the zero point energy - of a positive amount ½h% from each oscillation mode ~ of frequency %. The system of a multiplet of scalar fields should have the positive amount of energy ( M + N)~h% from each mode. Thus there is an apparent contradiction between the naive conclusion drawn from the classical hamiltonian and the result of the quantized hamiltonian. We should emphasize that although the hamiltonian in eq. (2.8) has the correct form in the classical field theory formalism, it is nevertheless useless as a guide for the construction of a viable quantum theory dealing with a non-compact symmetry.
Y.K. Ila / Non-compact.wmmetries
691
The situation is more subtle in an interacting theory. Let us consider the non-linear sigma model which has a very similar appearance to the free field theory. A generalized non-compact sigma model with global SO(M, N ) symmetry is defined on the coset space G / H = SO(M, N ) / [ S O ( M ) x SO(N)]. The lagrangian density in the matrix formalism is given by
~= - ~ Tr[ O~q( x ) O~q( x ) - '] ,
(2.9)
where q ( x ) = g-t(x)gog(X) are those SO(M, N ) group elements g(x) belonging to the coset G / H . They are parametrized by M x N and N x M off-diagonal real matrices X, X t respectively in the Lie algebra as follows [15]: X sinhCX¢~X
coshX~X y q = exp
(2.10)
=
[r
Xtsinh X ¢ ~W
coshCX-~
where the blocks obey the natural relations cosh2 X f ~
- sinh2 X ¢ ~
= 1,w,
c o s h 2 g ~ - ~ - sinhECX*X = 1~..
(2.11)
If the coset space is G / H = SU(M, N ) / [ S U ( M ) x S U ( N ) x U(1)], then the matrices X, X* should be complex and the coefficient in eq. (2.9) should be - ~. The interaction in the model is provided by the constraint
q ( x ) q ( x ) = IM_ N.
(2.12)
It is often convenient to denote X s i n h v ~ *X
Z
(2.13)
in terms of which and the relations of eq. (2.11), the coset elements can also be written as
q
Z*
¢I~, + Z*Z
I
Y.K. Ita / Non-compact.Wmmetrtes
692
The simplest type of these non-compact grassmannian sigma models corresponds to M = 1. In this case, the block Z can be parametrized by a single vector ~r = (~q . . . . . 7ru), which become the independent field variables satisfying the non-compact constraint o 2 = 1 + ,n. ,n. Explicitly, we use the following parametrization: -1 +202 q=
- 2o~r/
2o~',
]
- 1 - 2~r'% "
(2.15)
The lagrangian density for the SO(l, N ) / S O ( N ) sigma model including mass terms can now be written in the form of eq. (2.6), i.e. ~. =
-
½a~,oO~'o
+
~8~,~r.
m2( - 0 2 + ~2),
O~'~r -
(2.16)
with interaction provided by the non-compact constraint - o 2 +'n ' -
N+I
(2.17)
f
,
where the fields are rescaled to include the coupling constant f. Similarly, the hamiltonian density is
~.~= ~_[(O0~)2-(O00)2+(Ok~r)2-(8,0)2+rn2(~r2-02)],
(2.18)
which, again is not positive definite. Depending on the choice of the coupling constant, one can examine the property of the classical hamiltonian by differentiating the constraint in eq. (2.17). For each component t, = O, k, one obtains - oO~,o + 'n- a~,~ = 0 , or 2
o (0,0)
2
2
(2.19)
Thus if f is positive, the condition ,n z < 0 2 holds, and the following inequality is valid for each component: 2
'11.2
2
(8,,0) <~ ~-(anr) < ( 8 ~
)2.
(2.20)
In this case, the classical hamiltonian density is positive definite and is usually considered to be relevant. However, one finds that in the two-dimensional quantum theory there is no dynamical mass generation in the positive coupling region and that the vacuum expectation value of the negative signature scalar field tends to
Y.K. Ita / Non-compact .~Tmmetrws
693
infinity [16]. Thus the positive coupling case is not physically interesting. On the other hand, if the coupling f is negative, the inequality in (2.20) no longer holds and the classical hamiltonian density is unbounded from below. Several important questions can be raised at this point. Is the unbounded classical hamiltonian density in the negative coupling region really an intrinsic property of the non-compact sigma model, or is it only a symptom of the classical field theory formalism? Is the negative coupling region unphysical for the corresponding quantum theory? Does an indefinite classical hamiltonian density preclude the construction of a positive-definite quantum hamiltonian? These are some of the questions we shall answer in the following sections. 3. Use of the indefinite metric formalism for non-compact theories
Indefinite-metric theories have appeared periodically in physics [17]. The Hilbert spaces in these theories contain both positive and negative norm sectors. Notable examples of indefinite-metric theories are the covariant formulation of quantum electrodynamics [18] and the formulation of unitary and gauge-invariant perturbative series in non-abelian theories [19]. In all cases, the physical Hilbert spaces must carry only positive norm. Here we introduce a new use of the indefinite metric formalism in the quantization of theories admitting global non-compact symmetries. As in ordinary theories, theories with an indefinite metric may be canonically quantized. In the case of the non-compact free fields of eq. (2.1), each scalar field has the usual momentum space expansion in the form d3p [ai(p)eip.X+ai(p)*e ~,(x) = f (2~r)32po
,e-x],
(3.1)
where the creation and annihilation operators a~, aj satisfy the canonical commutation rule
[a, ( p ), a J( p')*] = 2 po6/6 ( p - p'),
(3.2)
with 6/ being the Kronecker delta symbol. In the overall Hilbert space of quantization, the multi-particle states are constructed by successive application of the creation operators on the vacuum 10), defined by a,[0) = 0 for all i. Recall that the scalar fields with negative kinetic terms are the o-fields and those with positive kinetic terms the ,n-fields, we have the following classification of the Fock space in terms of (i) single-particle states: Io'> = a~10),
1 ~
1~") = a~10),
M+I<~i<~N;
(3.3)
Y.K. lta / Non-compact ,~vmmetries
694
(ii) two-particle states:
1 <~i,j<~M,
=
]o',n-/> =
a'ota~*[O>,
Irr'Tr)> =
a,,a,, 10>,
~f
)'t
1 < ~ i ~ < M ; M + I <~)<~N,
(3.4)
M+ I <~i,j<~N;
etc. In this way we arrive eventually at any multi-particle state containing a given n u m b e r of o-particles and ,n-particles. These states transform according to the finite-dimensional representations of SO(M, N). The Fock space constructed in this way has an indefinite metric as we associate a negative inner product with the o-fields:
(o'lo'>== - 2 p o ~ ( p - p ' ),
l <~i<~M,
(3.5)
and a positive inner product with the 'n-fields:
(~ril~r'>=(Ola',,(p)a',,(p')*lO)= + 2 p o S ( p - p ' ),
M+I<~i<~N. (3.6)
These sectors can be summarized conveniently by the metric tensor as
(Ola'(p)aS(p')+lO) = 2 p o g ' / 6 ( p - p ' ) ,
for all
i,j.
(3.7)
Let us emphasize that such a Fock space of quantization is not the usual physical space of ordinary theories. To construct the physical Hilbert space from the indefinite-metric Fock space we have to follow the procedure by projecting states which contain only an even n u m b e r of o-fields. These states then have positive norm. Starting from the vacuum, we may apply an appropriate even n u m b e r of creation operators for the n-fields and an arbitrary n u m b e r of creation operators for the ~r-fields to the vacuum. Thus the physical subspace consists of all states given by I '._~___._. o ' . . . o ------.--.---~rr ~... ~r' > = a~,*.. . aoJta~,~+ ... a ~ I0> .
(3.8)
even no. any no. It is evident that states with an odd n u m b e r of o-fields are excluded. One can readily verify that all physical states so defined indeed have a positive-definite norm. For example, the state with two o-fields has the inner product
:
[.;,,,.' ] a 'lo>
= - 2po,~( p - p') = + 4p()p~'$ ( p - p ' ) 8 ( p " - p ' " ) , which is clearly positive definite.
(3.9)
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Y.K. tta / Non-compactsymmetries
Let us next consider the canonically quantized free field hamiltonian H =
f
d3p N (21r)32po p°,=l y"
=fd3p0
at(p)*a,(p)
-ai~(p)ta;(P) + E
,J
)~M+I
i=1
]
(3.10)
On appearance, this hamiltonian is semi-positive as the part pertaining to the o-fields has the negative sign. While the classical free field hamiltonian of eq. (2.8) does not have a lower bound, the quantized hamiltonian in eq. (3.10) nevertheless does. Moreover, this quantized hamiltonian is defined to act on all states of the Fock space of quantization and gives a positive-definite energy. Thus when measured with respect to a single o-particle state, for example, we find a positive-energy eigenvalue H I o ) = - f d3pp0a'~( p
)ta~,(p)lo)
= _ f d3Ppo";(P)' [a;(P),":(P')']10> = + fd3ppo2poS(p -p')a',,(p)*l O) = +EIo).
(3.11)
The expectation value of the hamiltonian in this state is always positive, despite that the state Io) has a negative inner product, i.e. (°IHI°)
(olo)
= + E(°I°)
(olo)
= + E.
(3.12)
Similarly, for a state chosen from the physical subspace, we certainly find a positive-definite energy eigenvalue, such as the case of the two o-field state:
Hloo) = + Eloo).
(3.13)
It is clear that problems which seem to arise from the classical free field hamiltonian is artificial as the quantized hamiltonian in the indefinite metric formalism does not suffer from a negative energy crisis. This conclusion will also apply to an interacting theory through the use of the reduction formula when the theory is appropriately quantized once we have complete control of the negative signs associated with the indefinite metric.
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Y.K. l t a /
Non-compact svmmetrtex
T o extend the discussion on the free field formalism to an interacting theory with a non-compact symmetry one must necessarily be concerned with the question whether the dynamics still obey the above distinction between positive- and negative-norm sectors of the Hilbert space. In quantum theory, the requirement of probability conservation in the indefinite-metric space implies that the S-matrix is pseudo-unitary (for a review see [20]), i.e. S * ~ S = 71,
(3.14)
where 7/is the metric operator whose eigenvalues are always + 1. If the Hilbert space of the asymptotic states of an interacting theory is constructed out of the positivenorm subspace in an indefinite-metric Fock space, will negative-norm states bc induced by the presence of interaction? Further, is the S-matrix in the physical subspace unitary? These questions will be resolved when we consider the non-compact sigma model in the indefinite-metric formalism. As long as there is a superselection rule consistent with a given symmetry forbidding transitions between positive- and negative-norm states, all the desired properties of a viable physical field theory can be realized.
4. Features of the non-compact sigma model The quantum two-dimensional SO(1, N ) / S O ( N ) sigma model is one of the simplest non-compact theories for the realization of the indefinite-metric formalism in an interacting theory. Here the lagrangian has one o-field and N It-fields. Let us again classify the Fock space of quantization in this model according to the number of particle states as in the prescription of sect. 3. The resulting Hilbert space therefore contains both positive and negative norms, while the physical subspace can be obtained by considering again states which contain only an even number of o-fields, as given in eq. (3.8). In certain two-dimensional models the presence of an infinite number of conservation laws forbids particle production, as a result the S-matrices have very simple analytic structures which can be determined exactly [21]. These conservation laws can also be extended to the non-compact sigma model under discussion. This feature enables us to demonstrate very clearly the significance and the relevance of the indefinite metric formalism for non-compact theories. Due to factorization of N-body S-matrices into products of two-body S-matrices the task of describing N-body scattering reduces to evaluating just two-body scattering. Invoking the general principles of Lorentz invariance, analyticity, crossing symmetry, etc., we can decompose a general two-body S-matrix relating the asymptotic states ]ab)i . = S.h'a]cd)ou,,
a, b, c, d = O, 1 . . . . . N ,
(4.1)
Y.K. Ha / Non-compact ,s3'rnmetries
697
in the indefinite-metric Hilbert space into O(1, N)-invariant combinations as
S, bca( O) = g,bg"aS,( O) + 6,"8haS2(0) + 6h'6~aS3( O),
(4.2)
where 0 is the rapidity parameter and S~, S 2, S 3 are scalar functions to be determined from the compact S N= SO(I + N ) / S O ( N ) sigma model. This S-matrix is pseudo-unitary, since it satisfies the condition of eq. (3.14). To understand the meaning and the features of the above S-matrix we illustrate with the simple case of a single o-field and a single or-field transforming as a doublet under SO(l,1). For two-body scattering in a non-compact theory, the space of scattering states is also an indefinite-metric space as it is a subspace of the multi-particle Fock space. In the present example, there are 4 basis scattering state vectors: I l l ) , 122), 112), 121). Here the label "1" refers to the o-particle and " 2 " the 0r-particle. From eq. (4.1) and eq. (4.2), we obtain explicitly the following relations between the asymptotic states: I l l ) i . = S t ( I l l ) - 122))o., + S / l l l ) o . , + $3111)o.,, 122)~. = S ~ ( - I l l ) + 122)),,~, + $2122)o., + S,122)o.,, 112)i. = S=l 12)o., + $3121)o.,, (4.3)
121)i. = S2121)o~, + S3112)ore.
N o t e that the corresponding relations for the compact S ~ sigma model have all positive signs. We now find that states with positive norm are (11111) = (22122) = + 1 , or
(ooloo)
= (orTrloror) = + 1"
(4.4)
and states with negative norm are (12/12) = (21121) = - 1 , or
= @ o l o r o > = - 1.
(4.5)
One can observe from eq. (4.3) that the S-matrix does not mix states with opposite norms. Relabelling the vectors as [A) = [11),
]B) = ]22),
(4.6)
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Y.K. tla / Non-compact symmetries
we can write, in the physical subspace of the two-body scattering states, the relations
IA)m = ( S~ + Sz + S3)IA>oo,- S, IB),,,, IB>i. = (S~ + S z + S3)IB>° m - SxlA),,u ,.
(4.7)
We now define a new 2 × 2 S-matrix entirely in the positive-norm states subspace by
S' =
S + S 2 + S~ -S~
-Sl ] S 1 + S2 + S 3 '
(4.8)
relating the physical asymptotic states
[
IA>i"]=S'[IA>°ut]
IB>i.]
(4.9)
IB>o.,]
The scalar
functions S(O) occurring here are the same scalar functions appearing in the compact SO(2) sigma model describing massive particles. These functions, by construction, obey the reflection symmetry property
S*(O)=S(-O),
(4.10)
and have the following explicit form [22]:
S~(O) = S 3 ( i ~ - O), S 2(0) = (2/~r)sin(4~r 2/7')sinh(47r0/y')sinh(4~r (i~r - 0 ) / y ' ) U ( 0 ) , S 3(0) = i cot(4~r 2/7')coth(4~rO/v') S 2(0) ;
(4.11)
where
u(o)
=rct ~'iI r (1 +,soi,.(, -V j ~ 8 +,8o ~'
~ <(0)R,(i,,-0)
l'((16~r! + i80)/~,')F(1 + (16~r/+ i80 )/~,') Rt(O) = F((16~r/+ 8~r+ i80)/7')I'(1 + (16~'l- 8~r + i80)fl'y') "
(4.12)
and y' being a real parameter. It can be directly verified using these equations that the reduced S-matrix in the positive norm subspace is always unitary, i.e.
S'*S'= I.
(4.13)
Moreover, when acting on the infinity tower of finite-dimensional representations of
Y.K. Ilu / Non,compact ,sTmrnetries
699
SO(l,1), the multi-particle S-matrix breaks into diagonal blocks associated with each representation. As a consequence of factorization of the N-body S-matrix into products of two-body S-matrices each of the many-body S-matrices is again unitary when restricted to the physical space of asymptotic states. Thus the total S-matrix is also unitary. It is of interest to know whether the original global symmetry is still present or not because of the elimination of negative-norm metric states from the overall Fock space. Evidently, the action of SO(l, 1) no longer has a simple linear realization on the physical subspace. Indeed, the physical subspace may not possess any visible symmetry at all. The role of the SO(l, 1) symmetry is seen here to provide a basis for the construction of the overall Fock space from which the size of the physical subspace can be determined. Thus the non-compact symmetry is essential whether it has a linear realization or not on the physical states. Incidentally, the way SO(1, 1) acts on the overall Fock space is to give a "covariant characteristic" to the physical subspace, i.e. if one chooses instead to use the following basis states: II1> ' = c~lll> + c2122 ) + c3112 > + c4121 >, 122)' -- d a l l l ) + dz122) + d3112) + d4121 ,
(4.14)
with the condition for positive norm 4
Y'. Ic,I z > 0,
4
Y'. Id, I2 > 0,
(4.15)
satisfied, one finds exactly the same reduced unitary S-matrix m the physical Hiibert space. As expected, there should be one and only one physical S-matrix. We next turn to some dynamical aspects of the non-compact sigma model. We shall understand why in the positive coupling region the quantum theory is not interesting and that only in the negative coupling region of eq. (2.17) there is dynamical mass generation. In the quantum theory, the interacting lagrangian Lin t is provided by the non-linear constraint via a lagrangian multiplier h, which can be treated as a field variable. The 1/N expansion method can then be employed for studying the SO(l, N) case [10]. The lagrangian density takes the form
(4.16) where the mass parameter m is now used to regulate the infrared divergences of the massless theory in two dimensions. The value of m can be chosen so that the vacuum expectation value ( h ) = X0 = (0l TXeqe'"'[0) = 0
(4.17)
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Y.K. tla / Non-compact ,sTmmetnes
is satisfied. This condition implies that N+I
- ( o 2) + ('n 2)
f
(4.18)
Since the d y n a m i c s is invariant when the fields are translated by a constant value, we exploit this invariance. Denoting by o o the vacuum expectation value for o and m a k i n g the translation o ---, o ' = o + %, we find the shifted lagrangian to be
if' = £ + ~------~------(oc~ + 2000) +
m2oo,,.
(4.19)
T h e requirement that in the shifted lagrangian o has the zero expectation value further implies the condition m 2% = 0.
(4.20)
T h u s there are two distinct possibilities to be considered according to eq. (4.20): (i) If o 0 =~ 0, then m 2 must vanish identically. This is the case corresponding to no d y n a m i c a l mass generation [16]. Eq. (4.18) shows that, for the shifted field o and to lowest order in the large-N limit,
od
N+ 1 +(N+ f
1)f
d2k
i
(4.21)
(2~r) 2 k 2 - t~/2 + ie
in which the integral, with an appropriate ultraviolet cutoff, is logarithmically divergent when m = 0. Thus if the coupling f is chosen to be positive, then the value o 0---, ~ , which is unsuitable for mass generation, although it implies symmetry restoration. (ii) If o 0 = 0, then m 2 can be non-vanishing and finite. In this case, the evaluation of the integral in eq. (4.21) after a Wick rotation k o ~ ik o gives the following relation: 1
d2k~
(27r)2 f k~ +m 2
1
(4.22)
f'
where k 2 = k02+ k~. Since the integral is positive definite, the coupling must necessarily assume negative values in order that eq. (4.22) be valid, in which case we obtain the typical asymptotic freedom behavior between the bare coupling of the n o n - c o m p a c t sigma model and the mass of the o-particle, i.e. f ~ ( A ) = c o n s t x In(AZ/m2), where A is a Pauli-Villars cutoff. This is the case corresponding to mass generation. T h e mass m introduced earlier as a free parameter is now related non-perturbatively to the coupling f and is the dynamically generated mass.
Y.K. Ita / Non-compact .~ymmetrtes
701
We conclude from the above analysis that there is a clear possibility of dynamical mass generation in the non-compact sigma model and that this result is the same as the behavior in the corresponding compact sigma model case [9]. Note that although a negative coupling implies an unbounded hamiltonian from below in the classical theory, this fact does not preclude the construction of a positive-definite hamiltonian in the quantum theory. A similar example in which this situation can be compared with is the ?~4 theory with negative coupling. Again, a careful analysis [23] in that model shows that problem with positivity of the classical theory does not imply the same feature in the quantum theory. Our conclusion of mass generation in the non-compact sigma model in the negative coupling region is entirely meaningful and physical as the result has been further supported by the S-matrix previously discussed describing the scattering of massive particles.
5. Dynamical gauge bosons and supergravity The phenomenon of dynamical mass generation is most useful when the sigma model possesses a gauge degree of freedom. While the SO(1, N ) / S O ( N ) sigma model considered so far does not have a gauge structure, the non-compact CP N model defined on the coset G / H = SU(I, N ) / [ S U ( N ) × U(1)] does accommodate a U(1) gauge field. The formulation of the non-compact CP u model [11] is entirely similar to the pseudo-orthogonal case as outlined in sect. 2. The coset elements are now parametrized by a single complex vector z, = (z 1. . . . . z~,) obeying the non-compact constraint 20z 0 = 1 + £,z,. Analogous to eq. (2.15), we have the coset elements in the form
22oZ ,
- 1 + 22oZ o q=
- 1~2,z,
- 2Zo,~;
] ]"
(5.1)
The lagrangian in terms of canonical fields can be written in the gauge-invariant form
= D~,z,, D~'z ",
a = O, 1 ..... N.
(5.2)
with the help of a composite U(1) gauge field A~, = ff,,O~,z". The covariant derivative is defined by D~, = 0, + iA~,, while the coupling f is introduced via the constraint ,~,,z" = -,~oZo + ,~,z,
N+I f
(5.3)
The analysis of the quantum dynamics in this gauge model in the 1 / N expansion proceeds quite similarly to that of the SO(1, N ) / S O ( N ) case and has been presented in ref. [11]. The result shows that there are also two distinct possibilities depending
702
KK. lla / Non-compactsymmetrie.g
on the choice of the coupling in eq. (5.3). For positive coupling f, there is no dynamical mass generation for the z 0 field and the composite gauge field A~,, although present in the classical theory, fails to appear in the quantum theory. This is because the pole of the gauge field propagator is absent in the massless limit. For negative coupling f, however, there can be dynamical mass generation together with asymptotic freedom behavior, much like the SO(l, N ) / S O ( N ) case. In addition, there is a genuine gauge field in the quantum theory since its proper two-point function, given by
F~,~( p ) = const x
( &,~
P~,P~ ) £1 dct ( 1 - 2ot)2p 2 pa(l - a)p 2- m 2 "
(5.4)
shows the propagator develops a pole at p2 = 0. This pole vanishes at m = 0, thus explaining why dynamical generation of gauge boson in this model depends crucially on mass generation. Sigma models with a U ( N ) gauge structure [24] are physically more interesting. For example, in order to accommodate the gauge symmetry S U ( N ) x U(1), one should naturally consider a sigma model on the coset space G / H = SU(M, N ) / [ S U ( M ) x S U ( N ) x U ( 1 ) ] . In this case, the analysis of the dynamics are more complicated but qualitatively one should expect the same conclusion as that of the non-compact CP ~ model, which is a special case of the generalized non-compact grassmannian models. For small values of M, N, however, we note that the noncompact grassmannian is a submanifold of E71 + 7~/SU(8), the manifold on which the scalar fields in the N = 8 supergravity theory reside, if SU(8) ~ [SU(M) x SU( N ) x U(I)]. Thus it is very likely that gauge bosons in the theory of strong and electroweak interactions are composite particles whose structure may be revealed ultimately at very high energies. In the field theoretical framework, it still remains to be shown that composite gauge objects can appear unambiguously in a nonrenormalizable theory.
6. Conclusion
We have extended the use of the indefinite metric formalism to the study of global non-compact internal symmetries in a class of renormalizable field theories and the results indicate that field theories with non-compact symmetries are just as viable as theories constructed with compact symmetries, as long as one can construct all physical quantities on a positive-definite subspace projected from an overall indefinite-metric Hilbert space. The notion that the quantum hamiltonian for a non-compact theory must be based on its classical field theory analogue must be abandoned. In the non-compact sigma model, although a negative coupling implies an unbounded hamiltonian from below in the classical theory, this fact does not preclude
Y.K. lla / Non-compact.wmmetries
703
the construction of a positive-definite hamiltonian in the quantum theory. Our results regarding the non-compact sigma model is entirely meaningful and physical and offers the desired feature of dynamical mass generation with asymptotic freedom behavior, a condition necessary for the generation of composite gauge bosons in models which can accommodate a gauge structure. The extensions of these results to theories based on the non-compact manifolds is obviously interesting, especially viewed in the context of supergravity theories in various dimensions. It becomes clear that the role of the non-compact symmetry is to provide a basis for the selection of the physical Hilbert space. Thus the size of the physical subspace depends on the size of the indefinite-metric Fock space, which itself is determined by the form of the non-compact symmetry. Different non-compact groups, of course, will result in different physical spaces and accommodate different physical phenomena. We advocate that there should be no prejudice against the use of non-compact groups in physical theories, as there are many instances in which non-compact symmetries have become indispensable in modern particle theories. I thank Professor John Lowenstein, Dr. M. Giinaydin, and especially Dr. M. Gomes for valuable discussions. This research was supported by the Department of Energy under contract number DE-AC02-81 ER40033B.000.
References [1] A. Houghton, A. Jevicki, R.D. Kenway and A.M.M. Pruisken, Phys. Rev. Lett. 45 (1980) 394: A.M.M. Pruisken and L. Schaeffer, Phys. Rev. Lett. 46 (1981) 490; Nucl. Phys. B200 (1982) 20 [2] E. Brezin, S. Hikami and J. Zinn-Justin, Nucl. Phys. B165 (1980) 528 [3] J. Wess and J. Bagger. Supersymmetry and supergravity (Princeton University Press, 1983) [4] E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141 [5] B. Julia, in Superspace and supergravity, ed. S. Hawking and M. Ro~ek (Cambridge University Press, 1981) [6] M. (.;ell-Mann and B. Zwiebach, Phys. kett. 141B (1984) 333 [71 Y. Dothan, M. (}eli-Mann and Y. Ne'eman, Phys. Lett. 17 (1965) 148: Y. Chow, Non-compact groups in particle physics (Benjamin. Reading, Mass., 1966) [8] J. Ellis, M.K. (}aillard and B. Zumino, Phys. l,ett. 94B (1980) 343 [9] A. D'Adda, P. Di Vecchia and M. Liischer, Nucl. Phys. B146 (1978) 63; B152 (1979) 125: E. Witten, Nucl. Phys. B149 (1979) 285 [10] M. Gomes and Y.K. Ha, Phys. Lett. 145B (1984) 235 [11] M. Gomes and Y.K. Ha, The non-compact CP" Model, R U 8 4 / B / 9 3 [12] N. Ohta, Phys. Lett. 134B (1984) 75: J.W. Van Holten, Phys. Lett. 135B (1984) 427: Nucl. Phys. B242 (1984) 307 [13] M. Gell-Mann. From renormalizability to calculability?, Shelter Island, 1983; F. Gtirsey, Quaternionic and octonionic structures in physics, YTP 84-08 [14] J.P. Crawford and A.O. Barut, Phys. Rcv. D27 (1983) 2493 [15] R. Gilmore, Lie groups, Lie algebras, and some of their applications (John Wiley & Sons, New York, 1974); S. tlelgason, Differential geometry, Lie Groups, and symmetric spaces (Academic Press, New York. 1978)
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Y.K. Ila / Non-compact.~Tmmetrtes
[16] A.C. l)avis, A.J. Macfarlane and J.W. Van Holten, Phys. l.ctt. 125B (1983) 151: Nucl. Phys. B232 (1984) 473 [17] P.A.M. Dirac, Commun. Dublin Inst. Adv. Stud. A. No. 1 (1943): W. Pauli, Rev. Mod. Phys. 15 (1943) 175; T.D. Lee, Phys. Rev. 95 (1954) 1329; W. lleisenberg, Rev. Mod. Phys. 29 (1957) 269: EC.G. Sudarshan, Phys. Rev. 123 (1961) 2183: T.D. Lee and G.C. Wick, Nucl. Phys. B9 (1969) 209 [18] S.N. (iupta, Proc. Phys. Soc. A63 (1950) 681: K. Bleuler, Helv. Phys. Acta 23 (1950) 567 [19] (i. "t Hooft. Nucl. Phys. B33 (1971) 173; B35 (1971) 167 [20] K. Nagy, Nuovo Cim. Suppl. 17 (1960) 92 [21] M. Ltischer and K. Pohlmeyer, Nucl. Phys. B137 (1978) 46: T.L. Curtright and C.K. Zachos, Phys. Re,,'. D21 (1980) 411:D24 (1981) 2661: R. Shankar and E. Witten, Phys. Rev. D17 (1978) 2134: AB. Zamolodchikov and AI.B. Zamolodchikov, Ann. of Phys. 120 (1979) 253 [22] A.13. Zamolodchikov and AI.B. Zamolodchikov, Nucl. Phys. B133 (1978) 525: R. Rajaraman, Solitons and instantons (North-Holland, Amsterdam, 1982) [23] R. Brandt, Phys. Rev. D14 (1976) 3381 [24] A.M. Perelomov, Physica 4D (1981) 1; M. Gomes and YK. Ha, Phys. Lett. 137B (1984) 197
N O N - C O M P A C T S Y M M E T R I E S IN F I E L D T H E O R I E S W I T H INDEFINITE METRIC Yuan K. HA Ph.vsics Department, The Rockefeller University. 1230 York Avenue. New York. N Y 10021, USA
Received 29 October 1984
We motivate the study of non-compact internal symmetries appearing in some renormalizable field theories by the indefinite metric method. The use of the indefinite metric formalism in constructing a positive-definitephysical Hilbert space is illustrated for the non-compact free fields and the non-compact sigma models. We discuss the problems associated with non-compact symmetries occurring in these theories. The notion that positivity of the quantum hamiltonian for a non-compact theory must be based on its classical field theory analogue must be abandoned. For the non-compact O(1, N ) / O ( N ) sigma model we show how to obtain a unitary S-matrix in the physical subspace from a pseudo-unitary S-matrix in the overall Hilbert space, as well as the feature of dynamical mass generation and asymptotic freedom behavior. For the non-compact CP N sigma model we discuss the appearance of a dynamically generated composite gauge boson without external input. The prospect of composite gauge bosons for supergravity theories is assessed. Whenever possible, we compare and discuss the features of compact versus non-compact symmetries in a field theory.
I. Occurrence of non-compact symmetries T h e use of n o n - c o m p a c t symmetry groups in physical theories has gained much interest recently. In solid state p h e n o m e n a , n o n - c o m p a c t sigma models in two d i m e n s i o n s a p p e a r quite i m p o r t a n t l y in the study of the mobility edge of electrons in d i s o r d e r e d systems [1]. The n o n - c o m p a c t symmetry in such cases is essential for the correct b e h a v i o r of the one-loop c o n t r i b u t i o n to the beta function, which would have the o p p o s i t e sign if the symmetry considered is c o m p a c t [2]. Various n o n - c o m p a c t sigma models also appear surprisingly in extended supergravities. These theories offer the hope for an ultimate unified field theory in which all the basic interactions c a n be c o n s t r u c t e d in a single mathematical framework [3]. In particular, the work of C r e m m e r a n d Julia [4] shows that the N = 8 supergravity can be u n i q u e l y reform u l a t e d as a massless theory of a simple N = 1 supergravity in eleven dimensions. W h e n reduced to four d i m e n s i o n s the bosonic sector of the lagrangian admits a n o n - c o m p a c t sigma model with a global n o n - c o m p a c t exceptional group ET( ~7) a n d a local m a x i m a l compact subgroup SU(8). Similar reductions to four d i m e n s i o n s of 687
688
K K. Ila / Non-compact ,~Tn~metrtes
the N = 6, 5.4 cases show that the bosonic sectors are all described by non-compact sigma models on the coset spaces SO*(12)/U(6), SU(5.1)/U(5) and SU(1,1)/U(1) respectively. On the other hand, if the eleven-dimensional theory is reduced to three dimensions the resulting sigma model is of the non-compact typc Es~ s~/SO(16): while reduction to five and six dimensions results in the non-compact sigma models on the respective spaces E6t+6~/Usp(8 ) and SO(5,5)/[SO(5)xSO(5)] [5]. More recently, a non-linear sigma model of scalar fields based on the non-compact coset space SU(1, 1)/U(1) has been used to reduce higher-dimensional supergravity theories to lower-dimensional ones with the attractive feature that masslessncss of fermions is not just due to supersymmetry [6]. Thus from many points of view. non-compact symmetries have become more relevant to current problems in particle physics. Earlier attempts in the application of non-compact groups in field theories [7] were perhaps hindered by the non-positive definite nature of the quadratic invariants as well as the lack of finite-dimensional unitary representations of the non-compact groups. For supergravity, one plausible way to handle this situation is to formulate the theory on a coset space G / H so that the non-compact symmetries are realized linearly. In the N = 8 supergravity theory, one would consider the theory to describe new fundamental constituents called preons which would transform according to the representations of SO(8), while the bound states, which are the quarks, ieptons and gauge bosons, would transform linearly according to the representations of SU(8) [8]. A crucial question in this approach is whether these composite gauge fields are dynamical and propagating. In two dimensions, it is known that the compact CP N sigma model possesses a composite U(1) gauge field which becomes truly propagating by acquiring an effective kinetic term via quantum correction [9]. A recent 1/N expansion study of the O(1, N )/O( N ) sigma model [10] and the non-compact CP N sigma model [11] in two dimensions in the negative coupling region has shown that these non-compact models can have dynamical mass generation, a condition necessary for the dynamical generation of a composite gauge boson. Some preliminary studies 112] of a simple non-compact sigma model in four dimensions also point out the possibility of a dynamically generated gauge boson when a mass parameter or a potential giving masses to the fields is introduced into the model. However, for the N = 8 supergravity to be a successful candidate for unification, one has ultimately to demonstrate in the theory that dynamical gauge bosons can appear without introducing any external parameter. The present paper is to motivate the use of the indefinite metric method in some renormalizable field theories with non-compact internal symmetries. The requirement of renormalizability necessarily brings the discussion on the quantum noncompact sigma models to two dimensions, where firm statements and reliable results can be obtained, despite that the classical models can be formulated in any dimensions. Although supergravity is not a renormalizable theory from a perturbative point of view, its tight symmetry structure renders the theory a chance of being
Y.K tta / Non-compact .~Tmmetri¢~
689
finite [13] and some of our conclusions in two dimensions on the sigma models may well extend to supergravity in four dimensions. Further, the study of the behavior of quantized field theories with non-compact symmetries is itself of interest regardless of supergravity as the method can be useful in other contexts. We shall illustrate the use of the indefinite metric in the quantization of theories admitting global non-compact symmetries and obtain a positive-definite physical Hilbert space. We discover that the feature of dynamical mass generation with asymptotic freedom behavior is present in a 1/N expansion study of the non-compact sigma models. For the non-compact O(1, N ) / O ( N ) sigma model we show how to obtain a unitary S-matrix in the physical subspace from a pseudo-unitary S-matrix in the overall Hilbert space. For the non-compact CP u sigma model we discuss the appearance of a dynamically generated composite gauge boson without extra input parameter. Whenever possible, we compare and discuss the features of compact versus non-compact symmetries in a field theory. 2. Problems of non-compact theories An apparent difficulty in implementing non-compact symmetries in field theories lies in the non-positive definite nature of the quadratic invariants. For example, the kinetic terms of a non-compact theory would have both positive and negative signs. Thus canonical fields with negative kinetic terms would have negative norm and their associated propagators would develop ghosts. Since physical states must occur with positive norm, the appearance of negative kinetic terms usually means the theory is unphysical. Such argument, although it seems sound, if taken categorically, would render all non-compact symmetries irrelevant to physics. As there are definite systems and theories which require the appearance of non-compact symmetries, one should therefore confront this situation with new insight. In the present approach, we develop a scheme in which the negative norm states induced by non-compact symmetries are eliminated by a superselection rule provided by the indefinite metric to the enlarged Hilbert space to form a new physical subspace. Such a selection rule will forbid transitions between states of positive norm and states of negative norm, while physical observables are to be constructed from operators defined only in the physical subspace. This type of approach has also been used to explain the non-existence of certain decays, e.g. /~--* ey, in the lepton family in a different context [14]. To understand the nature of the problem associated with a non-compact internal symmetry, we consider first a multiplet of free scalar fields = (o,
(o, .....
oM;
.....
(2.1)
transforming according to the defining representation of SO(M. N );
q~,--, ~; = n/q,j.
(2.2)
690
Y.K. tta / Non-compact,wmmetrtes
The internal components are denoted by i. j = (1 . . . . . M; M + 1. . . . . N). and ~ / a r e elements of the pseudo-orthogonal group SO( M. N). which leave the hermitian form
go =
- 1.. 0
0 ] /~,.
(2.3)
invariant, i.e. /2"rgoI2 = go-
(2.4)
The lagrangian densities for the compact SO(M + N ) case and the non-compact S O ( M , N ) case are respectively ~compact = ! O"l,uO" 0'o.0 OF loq ,'IT- al't"ff
= ½aoo- 0oO - ~ oko. 0~o + ~o,,~,. o,,., - ~ o ~ , . o ~ , .
(2.5)
= - ~ 0{}0. aoa + ½9~.0. a~o + {Oo~. Oo~ - ~Ok,~. afro.
(2.6)
Correspondingly, the hamiltonian densities are '~ompact = ½0,,0" ¢'~ ........
0,,0 + ~Oka" Oka + ½0O4" O0"~+ ~ O~'~" Oa~,
pact = -- ~ 0"100 " 0")0° -- ~OkO" O.L,O + I ao,lT. 00,1T _{._10fflT' Ok'n'.
(2.7) (2.8)
When appropriate, the following mass terms can be added: (i) compact case - m2(o • 0 + 'IT- 'lr), (ii) non-compact case - m2( -o. 0 + 'rr - 'rr). It can be observed immediately that for the non-compact case, the hamiltonian density in eq. (2.8) for a multiplet of free fields is not positive definite as the a-fields carry negative signs. One might be tempted to conclude that since the classical hamiltonian does not have a lower bound the corresponding quantum theory does not have a vacuum and the theory is therefore not sensible. However, one knows that free fields are non-interacting. Each scalar field in the canonically quantized theory has a ground state energy - the zero point energy - of a positive amount ½h% from each oscillation mode ~ of frequency %. The system of a multiplet of scalar fields should have the positive amount of energy ( M + N)~h% from each mode. Thus there is an apparent contradiction between the naive conclusion drawn from the classical hamiltonian and the result of the quantized hamiltonian. We should emphasize that although the hamiltonian in eq. (2.8) has the correct form in the classical field theory formalism, it is nevertheless useless as a guide for the construction of a viable quantum theory dealing with a non-compact symmetry.
Y.K. Ila / Non-compact.wmmetries
691
The situation is more subtle in an interacting theory. Let us consider the non-linear sigma model which has a very similar appearance to the free field theory. A generalized non-compact sigma model with global SO(M, N ) symmetry is defined on the coset space G / H = SO(M, N ) / [ S O ( M ) x SO(N)]. The lagrangian density in the matrix formalism is given by
~= - ~ Tr[ O~q( x ) O~q( x ) - '] ,
(2.9)
where q ( x ) = g-t(x)gog(X) are those SO(M, N ) group elements g(x) belonging to the coset G / H . They are parametrized by M x N and N x M off-diagonal real matrices X, X t respectively in the Lie algebra as follows [15]: X sinhCX¢~X
coshX~X y q = exp
(2.10)
=
[r
Xtsinh X ¢ ~W
coshCX-~
where the blocks obey the natural relations cosh2 X f ~
- sinh2 X ¢ ~
= 1,w,
c o s h 2 g ~ - ~ - sinhECX*X = 1~..
(2.11)
If the coset space is G / H = SU(M, N ) / [ S U ( M ) x S U ( N ) x U(1)], then the matrices X, X* should be complex and the coefficient in eq. (2.9) should be - ~. The interaction in the model is provided by the constraint
q ( x ) q ( x ) = IM_ N.
(2.12)
It is often convenient to denote X s i n h v ~ *X
Z
(2.13)
in terms of which and the relations of eq. (2.11), the coset elements can also be written as
q
Z*
¢I~, + Z*Z
I
Y.K. Ita / Non-compact.Wmmetrtes
692
The simplest type of these non-compact grassmannian sigma models corresponds to M = 1. In this case, the block Z can be parametrized by a single vector ~r = (~q . . . . . 7ru), which become the independent field variables satisfying the non-compact constraint o 2 = 1 + ,n. ,n. Explicitly, we use the following parametrization: -1 +202 q=
- 2o~r/
2o~',
]
- 1 - 2~r'% "
(2.15)
The lagrangian density for the SO(l, N ) / S O ( N ) sigma model including mass terms can now be written in the form of eq. (2.6), i.e. ~. =
-
½a~,oO~'o
+
~8~,~r.
m2( - 0 2 + ~2),
O~'~r -
(2.16)
with interaction provided by the non-compact constraint - o 2 +'n ' -
N+I
(2.17)
f
,
where the fields are rescaled to include the coupling constant f. Similarly, the hamiltonian density is
~.~= ~_[(O0~)2-(O00)2+(Ok~r)2-(8,0)2+rn2(~r2-02)],
(2.18)
which, again is not positive definite. Depending on the choice of the coupling constant, one can examine the property of the classical hamiltonian by differentiating the constraint in eq. (2.17). For each component t, = O, k, one obtains - oO~,o + 'n- a~,~ = 0 , or 2
o (0,0)
2
2
(2.19)
Thus if f is positive, the condition ,n z < 0 2 holds, and the following inequality is valid for each component: 2
'11.2
2
(8,,0) <~ ~-(anr) < ( 8 ~
)2.
(2.20)
In this case, the classical hamiltonian density is positive definite and is usually considered to be relevant. However, one finds that in the two-dimensional quantum theory there is no dynamical mass generation in the positive coupling region and that the vacuum expectation value of the negative signature scalar field tends to
Y.K. Ita / Non-compact .~Tmmetrws
693
infinity [16]. Thus the positive coupling case is not physically interesting. On the other hand, if the coupling f is negative, the inequality in (2.20) no longer holds and the classical hamiltonian density is unbounded from below. Several important questions can be raised at this point. Is the unbounded classical hamiltonian density in the negative coupling region really an intrinsic property of the non-compact sigma model, or is it only a symptom of the classical field theory formalism? Is the negative coupling region unphysical for the corresponding quantum theory? Does an indefinite classical hamiltonian density preclude the construction of a positive-definite quantum hamiltonian? These are some of the questions we shall answer in the following sections. 3. Use of the indefinite metric formalism for non-compact theories
Indefinite-metric theories have appeared periodically in physics [17]. The Hilbert spaces in these theories contain both positive and negative norm sectors. Notable examples of indefinite-metric theories are the covariant formulation of quantum electrodynamics [18] and the formulation of unitary and gauge-invariant perturbative series in non-abelian theories [19]. In all cases, the physical Hilbert spaces must carry only positive norm. Here we introduce a new use of the indefinite metric formalism in the quantization of theories admitting global non-compact symmetries. As in ordinary theories, theories with an indefinite metric may be canonically quantized. In the case of the non-compact free fields of eq. (2.1), each scalar field has the usual momentum space expansion in the form d3p [ai(p)eip.X+ai(p)*e ~,(x) = f (2~r)32po
,e-x],
(3.1)
where the creation and annihilation operators a~, aj satisfy the canonical commutation rule
[a, ( p ), a J( p')*] = 2 po6/6 ( p - p'),
(3.2)
with 6/ being the Kronecker delta symbol. In the overall Hilbert space of quantization, the multi-particle states are constructed by successive application of the creation operators on the vacuum 10), defined by a,[0) = 0 for all i. Recall that the scalar fields with negative kinetic terms are the o-fields and those with positive kinetic terms the ,n-fields, we have the following classification of the Fock space in terms of (i) single-particle states: Io'> = a~10),
1 ~
1~") = a~10),
M+I<~i<~N;
(3.3)
Y.K. lta / Non-compact ,~vmmetries
694
(ii) two-particle states:
1 <~i,j<~M,
=
]o',n-/> =
a'ota~*[O>,
Irr'Tr)> =
a,,a,, 10>,
~f
)'t
1 < ~ i ~ < M ; M + I <~)<~N,
(3.4)
M+ I <~i,j<~N;
etc. In this way we arrive eventually at any multi-particle state containing a given n u m b e r of o-particles and ,n-particles. These states transform according to the finite-dimensional representations of SO(M, N). The Fock space constructed in this way has an indefinite metric as we associate a negative inner product with the o-fields:
(o'lo'>=
l <~i<~M,
(3.5)
and a positive inner product with the 'n-fields:
(~ril~r'>=(Ola',,(p)a',,(p')*lO)= + 2 p o S ( p - p ' ),
M+I<~i<~N. (3.6)
These sectors can be summarized conveniently by the metric tensor as
(Ola'(p)aS(p')+lO) = 2 p o g ' / 6 ( p - p ' ) ,
for all
i,j.
(3.7)
Let us emphasize that such a Fock space of quantization is not the usual physical space of ordinary theories. To construct the physical Hilbert space from the indefinite-metric Fock space we have to follow the procedure by projecting states which contain only an even n u m b e r of o-fields. These states then have positive norm. Starting from the vacuum, we may apply an appropriate even n u m b e r of creation operators for the n-fields and an arbitrary n u m b e r of creation operators for the ~r-fields to the vacuum. Thus the physical subspace consists of all states given by I '._~___._. o ' . . . o ------.--.---~rr ~... ~r' > = a~,*.. . aoJta~,~+ ... a ~ I0> .
(3.8)
even no. any no. It is evident that states with an odd n u m b e r of o-fields are excluded. One can readily verify that all physical states so defined indeed have a positive-definite norm. For example, the state with two o-fields has the inner product
:
[.;,,,.' ] a 'lo>
= - 2po,~( p - p')
(3.9)
695
Y.K. tta / Non-compactsymmetries
Let us next consider the canonically quantized free field hamiltonian H =
f
d3p N (21r)32po p°,=l y"
=fd3p0
at(p)*a,(p)
-ai~(p)ta;(P) + E
,J
)~M+I
i=1
]
(3.10)
On appearance, this hamiltonian is semi-positive as the part pertaining to the o-fields has the negative sign. While the classical free field hamiltonian of eq. (2.8) does not have a lower bound, the quantized hamiltonian in eq. (3.10) nevertheless does. Moreover, this quantized hamiltonian is defined to act on all states of the Fock space of quantization and gives a positive-definite energy. Thus when measured with respect to a single o-particle state, for example, we find a positive-energy eigenvalue H I o ) = - f d3pp0a'~( p
)ta~,(p)lo)
= _ f d3Ppo";(P)' [a;(P),":(P')']10> = + fd3ppo2poS(p -p')a',,(p)*l O) = +EIo).
(3.11)
The expectation value of the hamiltonian in this state is always positive, despite that the state Io) has a negative inner product, i.e. (°IHI°)
(olo)
= + E(°I°)
(olo)
= + E.
(3.12)
Similarly, for a state chosen from the physical subspace, we certainly find a positive-definite energy eigenvalue, such as the case of the two o-field state:
Hloo) = + Eloo).
(3.13)
It is clear that problems which seem to arise from the classical free field hamiltonian is artificial as the quantized hamiltonian in the indefinite metric formalism does not suffer from a negative energy crisis. This conclusion will also apply to an interacting theory through the use of the reduction formula when the theory is appropriately quantized once we have complete control of the negative signs associated with the indefinite metric.
696
Y.K. l t a /
Non-compact svmmetrtex
T o extend the discussion on the free field formalism to an interacting theory with a non-compact symmetry one must necessarily be concerned with the question whether the dynamics still obey the above distinction between positive- and negative-norm sectors of the Hilbert space. In quantum theory, the requirement of probability conservation in the indefinite-metric space implies that the S-matrix is pseudo-unitary (for a review see [20]), i.e. S * ~ S = 71,
(3.14)
where 7/is the metric operator whose eigenvalues are always + 1. If the Hilbert space of the asymptotic states of an interacting theory is constructed out of the positivenorm subspace in an indefinite-metric Fock space, will negative-norm states bc induced by the presence of interaction? Further, is the S-matrix in the physical subspace unitary? These questions will be resolved when we consider the non-compact sigma model in the indefinite-metric formalism. As long as there is a superselection rule consistent with a given symmetry forbidding transitions between positive- and negative-norm states, all the desired properties of a viable physical field theory can be realized.
4. Features of the non-compact sigma model The quantum two-dimensional SO(1, N ) / S O ( N ) sigma model is one of the simplest non-compact theories for the realization of the indefinite-metric formalism in an interacting theory. Here the lagrangian has one o-field and N It-fields. Let us again classify the Fock space of quantization in this model according to the number of particle states as in the prescription of sect. 3. The resulting Hilbert space therefore contains both positive and negative norms, while the physical subspace can be obtained by considering again states which contain only an even number of o-fields, as given in eq. (3.8). In certain two-dimensional models the presence of an infinite number of conservation laws forbids particle production, as a result the S-matrices have very simple analytic structures which can be determined exactly [21]. These conservation laws can also be extended to the non-compact sigma model under discussion. This feature enables us to demonstrate very clearly the significance and the relevance of the indefinite metric formalism for non-compact theories. Due to factorization of N-body S-matrices into products of two-body S-matrices the task of describing N-body scattering reduces to evaluating just two-body scattering. Invoking the general principles of Lorentz invariance, analyticity, crossing symmetry, etc., we can decompose a general two-body S-matrix relating the asymptotic states ]ab)i . = S.h'a]cd)ou,,
a, b, c, d = O, 1 . . . . . N ,
(4.1)
Y.K. Ha / Non-compact ,s3'rnmetries
697
in the indefinite-metric Hilbert space into O(1, N)-invariant combinations as
S, bca( O) = g,bg"aS,( O) + 6,"8haS2(0) + 6h'6~aS3( O),
(4.2)
where 0 is the rapidity parameter and S~, S 2, S 3 are scalar functions to be determined from the compact S N= SO(I + N ) / S O ( N ) sigma model. This S-matrix is pseudo-unitary, since it satisfies the condition of eq. (3.14). To understand the meaning and the features of the above S-matrix we illustrate with the simple case of a single o-field and a single or-field transforming as a doublet under SO(l,1). For two-body scattering in a non-compact theory, the space of scattering states is also an indefinite-metric space as it is a subspace of the multi-particle Fock space. In the present example, there are 4 basis scattering state vectors: I l l ) , 122), 112), 121). Here the label "1" refers to the o-particle and " 2 " the 0r-particle. From eq. (4.1) and eq. (4.2), we obtain explicitly the following relations between the asymptotic states: I l l ) i . = S t ( I l l ) - 122))o., + S / l l l ) o . , + $3111)o.,, 122)~. = S ~ ( - I l l ) + 122)),,~, + $2122)o., + S,122)o.,, 112)i. = S=l 12)o., + $3121)o.,, (4.3)
121)i. = S2121)o~, + S3112)ore.
N o t e that the corresponding relations for the compact S ~ sigma model have all positive signs. We now find that states with positive norm are (11111) = (22122) = + 1 , or
(ooloo)
= (orTrloror) = + 1"
(4.4)
and states with negative norm are (12/12) = (21121) = - 1 , or
(4.5)
One can observe from eq. (4.3) that the S-matrix does not mix states with opposite norms. Relabelling the vectors as [A) = [11),
]B) = ]22),
(4.6)
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Y.K. tla / Non-compact symmetries
we can write, in the physical subspace of the two-body scattering states, the relations
IA)m = ( S~ + Sz + S3)IA>oo,- S, IB),,,, IB>i. = (S~ + S z + S3)IB>° m - SxlA),,u ,.
(4.7)
We now define a new 2 × 2 S-matrix entirely in the positive-norm states subspace by
S' =
S + S 2 + S~ -S~
-Sl ] S 1 + S2 + S 3 '
(4.8)
relating the physical asymptotic states
[
IA>i"]=S'[IA>°ut]
IB>i.]
(4.9)
IB>o.,]
The scalar
functions S(O) occurring here are the same scalar functions appearing in the compact SO(2) sigma model describing massive particles. These functions, by construction, obey the reflection symmetry property
S*(O)=S(-O),
(4.10)
and have the following explicit form [22]:
S~(O) = S 3 ( i ~ - O), S 2(0) = (2/~r)sin(4~r 2/7')sinh(47r0/y')sinh(4~r (i~r - 0 ) / y ' ) U ( 0 ) , S 3(0) = i cot(4~r 2/7')coth(4~rO/v') S 2(0) ;
(4.11)
where
u(o)
=rct ~'iI r (1 +,soi,.(, -V j ~ 8 +,8o ~'
~ <(0)R,(i,,-0)
l'((16~r! + i80)/~,')F(1 + (16~r/+ i80 )/~,') Rt(O) = F((16~r/+ 8~r+ i80)/7')I'(1 + (16~'l- 8~r + i80)fl'y') "
(4.12)
and y' being a real parameter. It can be directly verified using these equations that the reduced S-matrix in the positive norm subspace is always unitary, i.e.
S'*S'= I.
(4.13)
Moreover, when acting on the infinity tower of finite-dimensional representations of
Y.K. Ilu / Non,compact ,sTmrnetries
699
SO(l,1), the multi-particle S-matrix breaks into diagonal blocks associated with each representation. As a consequence of factorization of the N-body S-matrix into products of two-body S-matrices each of the many-body S-matrices is again unitary when restricted to the physical space of asymptotic states. Thus the total S-matrix is also unitary. It is of interest to know whether the original global symmetry is still present or not because of the elimination of negative-norm metric states from the overall Fock space. Evidently, the action of SO(l, 1) no longer has a simple linear realization on the physical subspace. Indeed, the physical subspace may not possess any visible symmetry at all. The role of the SO(l, 1) symmetry is seen here to provide a basis for the construction of the overall Fock space from which the size of the physical subspace can be determined. Thus the non-compact symmetry is essential whether it has a linear realization or not on the physical states. Incidentally, the way SO(1, 1) acts on the overall Fock space is to give a "covariant characteristic" to the physical subspace, i.e. if one chooses instead to use the following basis states: II1> ' = c~lll> + c2122 ) + c3112 > + c4121 >, 122)' -- d a l l l ) + dz122) + d3112) + d4121 ,
(4.14)
with the condition for positive norm 4
Y'. Ic,I z > 0,
4
Y'. Id, I2 > 0,
(4.15)
satisfied, one finds exactly the same reduced unitary S-matrix m the physical Hiibert space. As expected, there should be one and only one physical S-matrix. We next turn to some dynamical aspects of the non-compact sigma model. We shall understand why in the positive coupling region the quantum theory is not interesting and that only in the negative coupling region of eq. (2.17) there is dynamical mass generation. In the quantum theory, the interacting lagrangian Lin t is provided by the non-linear constraint via a lagrangian multiplier h, which can be treated as a field variable. The 1/N expansion method can then be employed for studying the SO(l, N) case [10]. The lagrangian density takes the form
(4.16) where the mass parameter m is now used to regulate the infrared divergences of the massless theory in two dimensions. The value of m can be chosen so that the vacuum expectation value ( h ) = X0 = (0l TXeqe'"'[0) = 0
(4.17)
700
Y.K. tla / Non-compact ,sTmmetnes
is satisfied. This condition implies that N+I
- ( o 2) + ('n 2)
f
(4.18)
Since the d y n a m i c s is invariant when the fields are translated by a constant value, we exploit this invariance. Denoting by o o the vacuum expectation value for o and m a k i n g the translation o ---, o ' = o + %, we find the shifted lagrangian to be
if' = £ + ~------~------(oc~ + 2000) +
m2oo,,.
(4.19)
T h e requirement that in the shifted lagrangian o has the zero expectation value further implies the condition m 2% = 0.
(4.20)
T h u s there are two distinct possibilities to be considered according to eq. (4.20): (i) If o 0 =~ 0, then m 2 must vanish identically. This is the case corresponding to no d y n a m i c a l mass generation [16]. Eq. (4.18) shows that, for the shifted field o and to lowest order in the large-N limit,
od
N+ 1 +(N+ f
1)f
d2k
i
(4.21)
(2~r) 2 k 2 - t~/2 + ie
in which the integral, with an appropriate ultraviolet cutoff, is logarithmically divergent when m = 0. Thus if the coupling f is chosen to be positive, then the value o 0---, ~ , which is unsuitable for mass generation, although it implies symmetry restoration. (ii) If o 0 = 0, then m 2 can be non-vanishing and finite. In this case, the evaluation of the integral in eq. (4.21) after a Wick rotation k o ~ ik o gives the following relation: 1
d2k~
(27r)2 f k~ +m 2
1
(4.22)
f'
where k 2 = k02+ k~. Since the integral is positive definite, the coupling must necessarily assume negative values in order that eq. (4.22) be valid, in which case we obtain the typical asymptotic freedom behavior between the bare coupling of the n o n - c o m p a c t sigma model and the mass of the o-particle, i.e. f ~ ( A ) = c o n s t x In(AZ/m2), where A is a Pauli-Villars cutoff. This is the case corresponding to mass generation. T h e mass m introduced earlier as a free parameter is now related non-perturbatively to the coupling f and is the dynamically generated mass.
Y.K. Ita / Non-compact .~ymmetrtes
701
We conclude from the above analysis that there is a clear possibility of dynamical mass generation in the non-compact sigma model and that this result is the same as the behavior in the corresponding compact sigma model case [9]. Note that although a negative coupling implies an unbounded hamiltonian from below in the classical theory, this fact does not preclude the construction of a positive-definite hamiltonian in the quantum theory. A similar example in which this situation can be compared with is the ?~4 theory with negative coupling. Again, a careful analysis [23] in that model shows that problem with positivity of the classical theory does not imply the same feature in the quantum theory. Our conclusion of mass generation in the non-compact sigma model in the negative coupling region is entirely meaningful and physical as the result has been further supported by the S-matrix previously discussed describing the scattering of massive particles.
5. Dynamical gauge bosons and supergravity The phenomenon of dynamical mass generation is most useful when the sigma model possesses a gauge degree of freedom. While the SO(1, N ) / S O ( N ) sigma model considered so far does not have a gauge structure, the non-compact CP N model defined on the coset G / H = SU(I, N ) / [ S U ( N ) × U(1)] does accommodate a U(1) gauge field. The formulation of the non-compact CP u model [11] is entirely similar to the pseudo-orthogonal case as outlined in sect. 2. The coset elements are now parametrized by a single complex vector z, = (z 1. . . . . z~,) obeying the non-compact constraint 20z 0 = 1 + £,z,. Analogous to eq. (2.15), we have the coset elements in the form
22oZ ,
- 1 + 22oZ o q=
- 1~2,z,
- 2Zo,~;
] ]"
(5.1)
The lagrangian in terms of canonical fields can be written in the gauge-invariant form
= D~,z,, D~'z ",
a = O, 1 ..... N.
(5.2)
with the help of a composite U(1) gauge field A~, = ff,,O~,z". The covariant derivative is defined by D~, = 0, + iA~,, while the coupling f is introduced via the constraint ,~,,z" = -,~oZo + ,~,z,
N+I f
(5.3)
The analysis of the quantum dynamics in this gauge model in the 1 / N expansion proceeds quite similarly to that of the SO(1, N ) / S O ( N ) case and has been presented in ref. [11]. The result shows that there are also two distinct possibilities depending
702
KK. lla / Non-compactsymmetrie.g
on the choice of the coupling in eq. (5.3). For positive coupling f, there is no dynamical mass generation for the z 0 field and the composite gauge field A~,, although present in the classical theory, fails to appear in the quantum theory. This is because the pole of the gauge field propagator is absent in the massless limit. For negative coupling f, however, there can be dynamical mass generation together with asymptotic freedom behavior, much like the SO(l, N ) / S O ( N ) case. In addition, there is a genuine gauge field in the quantum theory since its proper two-point function, given by
F~,~( p ) = const x
( &,~
P~,P~ ) £1 dct ( 1 - 2ot)2p 2 pa(l - a)p 2- m 2 "
(5.4)
shows the propagator develops a pole at p2 = 0. This pole vanishes at m = 0, thus explaining why dynamical generation of gauge boson in this model depends crucially on mass generation. Sigma models with a U ( N ) gauge structure [24] are physically more interesting. For example, in order to accommodate the gauge symmetry S U ( N ) x U(1), one should naturally consider a sigma model on the coset space G / H = SU(M, N ) / [ S U ( M ) x S U ( N ) x U ( 1 ) ] . In this case, the analysis of the dynamics are more complicated but qualitatively one should expect the same conclusion as that of the non-compact CP ~ model, which is a special case of the generalized non-compact grassmannian models. For small values of M, N, however, we note that the noncompact grassmannian is a submanifold of E71 + 7~/SU(8), the manifold on which the scalar fields in the N = 8 supergravity theory reside, if SU(8) ~ [SU(M) x SU( N ) x U(I)]. Thus it is very likely that gauge bosons in the theory of strong and electroweak interactions are composite particles whose structure may be revealed ultimately at very high energies. In the field theoretical framework, it still remains to be shown that composite gauge objects can appear unambiguously in a nonrenormalizable theory.
6. Conclusion
We have extended the use of the indefinite metric formalism to the study of global non-compact internal symmetries in a class of renormalizable field theories and the results indicate that field theories with non-compact symmetries are just as viable as theories constructed with compact symmetries, as long as one can construct all physical quantities on a positive-definite subspace projected from an overall indefinite-metric Hilbert space. The notion that the quantum hamiltonian for a non-compact theory must be based on its classical field theory analogue must be abandoned. In the non-compact sigma model, although a negative coupling implies an unbounded hamiltonian from below in the classical theory, this fact does not preclude
Y.K. lla / Non-compact.wmmetries
703
the construction of a positive-definite hamiltonian in the quantum theory. Our results regarding the non-compact sigma model is entirely meaningful and physical and offers the desired feature of dynamical mass generation with asymptotic freedom behavior, a condition necessary for the generation of composite gauge bosons in models which can accommodate a gauge structure. The extensions of these results to theories based on the non-compact manifolds is obviously interesting, especially viewed in the context of supergravity theories in various dimensions. It becomes clear that the role of the non-compact symmetry is to provide a basis for the selection of the physical Hilbert space. Thus the size of the physical subspace depends on the size of the indefinite-metric Fock space, which itself is determined by the form of the non-compact symmetry. Different non-compact groups, of course, will result in different physical spaces and accommodate different physical phenomena. We advocate that there should be no prejudice against the use of non-compact groups in physical theories, as there are many instances in which non-compact symmetries have become indispensable in modern particle theories. I thank Professor John Lowenstein, Dr. M. Giinaydin, and especially Dr. M. Gomes for valuable discussions. This research was supported by the Department of Energy under contract number DE-AC02-81 ER40033B.000.
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