Infrared singular fields and nonregular representations of canonical commutation relation algebras

Infrared singular fields and nonregular representations of canonicalcommutation relation algebrasF. Acerbi, G. Morchio, and F. Strocchi Citation: Journal of Mathematical Physics 34, 899 (1993); doi: 10.1063/1.530200 View online: http://dx.doi.org/10.1063/1.530200 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/34/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Representation-theoretic aspects of two-dimensional quantum systems in singular vector potentials:Canonical commutation relations, quantum algebras, and reduction to lattice quantum systems J. Math. Phys. 39, 2476 (1998); 10.1063/1.532631 Hyperfinite-dimensional representations of canonical commutation relation J. Math. Phys. 39, 2682 (1998); 10.1063/1.532413 Operator representations of the real twisted canonical commutation relations J. Math. Phys. 35, 3211 (1994); 10.1063/1.530462 Properties of ``Quadratic'' Canonical Commutation Relation Representations J. Math. Phys. 10, 1661 (1969); 10.1063/1.1665013 Direct-Product Representations of the Canonical Commutation Relations J. Math. Phys. 7, 822 (1966); 10.1063/1.1931213

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Infrared singular fields and nonregular representations of canonical commutation relation algebras

F. Acerbi International School for Advanced Studies, Trieste, Italy

G. Morchio Dipartimento di Fisica dell’UniversitLi and INFN, Piss, Italy

F. Strocchi International School for Advanced Studies, Trieste, Italy and INFN, Trieste, Italy

(Received 28 July 1992; accepted for publication 10 August 1992)

Infrared singular variables which often enter in the formulation of models in quantum field theory, many-body theory, and quantum statistical mechanics are described in terms of nonregular representations of canonical commutation relations algebras. General properties of such representations are discussed. In this way one also obtains insight into the construction and representation of charged field algebras .& as canonical extensions of a neutral (observable) algebra do. The charged state representations of &c are obtained through nonregular vacuum representations of the extended field algebra &‘. The above structures are proven to work in several explicit models.

I. MOTIVATIONS AND RESULTS

Many models of quantum field theory (QFT), many body theory, and quantum statistical mechanics are formulated and sometimes solved in terms of variables or fields which formally satisfy the canonical commutation relations (CCR), but which actually cannot be represented as operators in a Hilbert space because of their bad infrared behavior. As we will discuss in more detail later, typical examples are:

(i) the infinite quantum harmonic lattice in thermal equilibrium and the free Bose gas, both in space dimensions d<2; in the first case the singular variable is the local displacement from the equilibrium positions and in the second case is the local particle density;“’

(ii) the massless quantum electrodynamics in two space-time dimensions (QED, or Schwinger model),3 where the singular field variables are the charged fields;

(iii) the Stiickelberg-Kibble model4 in 1 + 1 and 2+ 1 space-time dimensions, where the singular variable is the analogue of the (phase of the) Higgs field;

(iv) the massless scalar field in two space-time dimensions, which plays an important role as a basic or building block field in the solution of several two-dimensional models;‘-’

(v) the U ( 1) current algebra on the circle, where the singular variables are the charged fields;’

(vi) quite generally statistical models involving unbounded variables with a flat distribution (required by symmetry properties).

The strategies usually adopted to discuss the above models fall essentially in two categories: (A) relax positivity and represent the singular fields as operators in an indefinite metric

space;5*9 (B) use a restricted set of field variables (the ones that can be represented as operators in

a Hilbert space fl) and describe the other degrees of freedom by other methods, e.g., as morphisms of the regular field algebra.6

Both strategies have their own advantages and disadvantages. The aim of this paper is to discuss an alternative and in our opinion useful approach based on the representation of the Weyl exponentials of the singular variables in a positive metric Hilbert space. This will allow

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900 Acerbi, Morchio, and Strocchi: Nonregular representations

us to keep the canonical structure (in Weyl form) also for the singular variables and clarify the relation between infrared singularities and the so arising non regular representation of the CCRs.

Contrary to what one might first be induced to think, the mathematical treatment of nonregular representations of the CCRs can be done in a rather compact way and it yields a convenient and usable framework for discussing explicit models. We will concentrate our attention to the case in which the CCR algebra JS’ has a subalgebra -pPc whose representation is quasi-free; in this case one has:

( 1) nonregular representations of a CCR algebra .c! are determined by the representations of the subalgebra &‘, if the latter is a maximal domain ofregularity (see below); in the various models mentioned above ,pPc can be interpreted as the algebra of observables, the current algebra, etc.;

(2) nonregular representations of a CCR algebra & decompose into sectors corresponding to inequivalent representations of the maximal regular subalgebra do. These structure properties prove very useful for a rigorous mathematical treatment of the infrared singular variables or fields that naturally enter in the Hamiltonian formulation of the above-mentioned models. One also gets insight into the representations of field algebras LZ’ charged under a gaugelike group when the observable algebra do is a CCR algebra. In particular one has the following:

(3) “charged” field algebras JZ’ can be obtained as CCR extensions (called extended CCR algebras) of the algebra of observables JZ!~, briefly “core” subalgebra; furthermore, an extended CCR algebra uniquely determines a “gauge” group G leaving its “core” subalgebra ,pPc pointwise invariant;

(4) vacuum representations of extended CCR algebras decompose into disjoint sectors labelled by charges that annihilate the vacuum: more precisely if the subgroup GcC G, leaving the vacuum invariant nontrivial, then the corresponding vacuum representation of the extended CCR algebra d is nonregular (on the other hand if Cc= 1, then the representation is regular and it coincides with the vacuum representation of do). Moreover, nonregular representations of extended CCR algebras allow a full solution of the bosonization problem in 1 + 1 dimensions; in particular,

(5) fermionic degrees of freedom can be obtained in terms of extended bosonic CCR algebras, and

(6) local fermion fields can be proved to exist as ultra strong limits of bosonic (Weyl) operators of an extended CCR algebra. Previous treatments of fermion bosonization were done with reference to specific models and in terms of correlation functions (i.e., with reference to given representations); along this direction the best result seems that of Ref. 10, where local fermion fields are obtained as strong limits on a dense set of vectors in given representations. An algebraic formula that constructs a local fermion field (at a given time) in terms of canonical boson operators seems to be lacking in the literature, apart from the very suggestive Mandelstam formula,“912 whose mathematical meaning is however not clear. The result (6) mentioned above provides such an algebraic fermion bosonization and it is made possible by the use of extended CCR algebras and their nonregular representations.13

Finally, it is worthwhile to remark that the mathematical framework discussed above for the description of infrared singular fields should prove useful in the constructive approach to many body or QFT models which involve severe infrared problems (like Coulomb systems, gauge theories like QCD, etc.). In fact, by introducing an ultraviolet cutoff (where necessary) the models can be formulated in terms of canonical algebras, involving variables associated to the charged fields, which may require nonregular representations. In this sense, the above approach may be regarded as a concrete realization of the Doplicher-Haag-Roberts (DHR) approach.14 The latter is based on the analysis of the representations of the observable algebras LZ’,,~~ and describes their charged state representations in terms of the vacuum representation and of charged morphisms of dEgobs. Here, we exploit the canonical structure of the variables

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Acerbi, Morchio, and Strocchi: Nonregular representations 901

that enter in the definition of the Hamiltonian, of the equation of motion, etc.; the mathematical problems connected with infrared singular variables are resolved by the use of nonregular representations. The construction of the charged morphisms is now directly obtained in terms of the fields that enter in the definition of the model (and/or may be obtained as CCR extensions of the observable algebra), so that the charged state representations of the observable algebra are provided by the vacuum representation of the extended CCR algebra.

The general framework discussed in this paper also provides a useful and relatively simple treatment of quantum fields on a circle, a problem that has recently attracted much interest in connection with string theory and with representations of Kac-Moody and Virasoro algebras.r5 The quantization of such systems leads to nonregular representations of CCR algebras; in this way one has a systematic treatment where infrared problems do not arise and the field algebra is simple.

The framework discussed in Sec. IV naturally leads to the occurrence of superselection rules also for quantum mechanical systems with a finite number of degrees of freedom (i.e., to the breakdown of Von Neumann uniqueness) : The mechanism is that the nonregular factorial representation of the simple Weyl algebra of canonical variables decomposes into inequivalent factorial representations of its regular observable subalgebra. The resulting structure can be regarded as an alternative route to the solution of the problem considered by Landsman16 through a different approach. His starting point is the classification of the representations of a nonsimple Cr algebra, which plays the role of the algebra of observables, whereas our strategy is to construct a simple algebra of canonical variables and to identify its subalgebra of observables.

The use of nonregular representations appears to be useful also for the quantization of systems with constraints, see Refs. 17 and 18.

II. NONREGULAR REPRESENTATIONS OF CCR ALGEBRAS

A CCR Weyl algebra is usually constructed in terms of the exponentials of canonical fields a( f, ), 7r(f2), with F= ( fl,f2) belonging to some real vector space I’ endowed with a symplectic form a(F,G), which encodes the CCR relations:

W(F) =exp i[p(fl) +df2) 1 =exp i@(F),

-it@(F),@(G)1 -dF,G) = (fl,gd - (f2,gl).

Quite generally2 an abstract CCR algebra a( V,a) is the *-algebra generated by abstract elements W(F), with F belonging to a real symplectic space ( V,a), with the product

W(F) W(G) = W(F+G)e-i”(FPG)‘2 (2.1)

and the involution defined by

W(F)*= W( -F). (2.2)

Clearly Eqs. (2.1) and (2.2) imply W(0) =l and W(F)-‘= W(F)*. If (T is nondegenerate, &( V,a) is simple, equivalently every (nonzero) representation is faithful.

For any ( V,o) the abstract CCR algebra &Qp( V,a) admits at least one nondegenerate c” norm (actually that defined by a nonregular representation rm ), and iff u is nondegenerate any other nondegenerate Cc norm coincides with it (uniqueness of the Cc structure).‘9,20 In general, every positive linear functional or state w over JZ!( V,a) is continuous with respect to the minimal c* structure over @’ ( V,a) defined in Ref. 20, and therefore it has a unique continuous extension to the corresponding e closure &( V,a). Through the Gelfand-Naimark-Segal


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(GNS) construction, every state w on ccS( V,a) defines a representation V, of &( ~,o) with cyclic vector Y. and with rJW(F)) unitary operators in the representation space X0. In general a representation rr of &( V,a) is said to be regular if P( W(M) ), il~iR, are strongly continuous in ;1, VFEV. A state o is regular if the corresponding GNS representation rti is regular.

Quasi-free states over CCR algebras are slight generalizations of Fock states and, as in the Fock case, are defined by nondegenerate Hilbert quadratic forms q( * ), q(F) = [F,F&, [ * , * I4 a Hilbert product satisfying

I a(F,G) I kq(F)q(G), VF,&V. (2.3)

The corresponding quasi-free state o is then defined by

dWF))=exp(-k(F)); (2.4)

more generally, one can define the shifted quasi-free states by

WO(W(F))=exp(i~(F))exp[ --b(F) 1 (2.5)

with 4(F) a real linear functional over V. In most of the following we will concentrate our attention to generalizations of quasi-free

states of the form (2.4)) but the analysis can be extended immediately to the shifted quasi-free states.

To define nonregular quasi-free states we consider the case in which q(F) is allowed to take also the value + CO, still satisfies (2.3) whenever q(F) and q(G) are finite, and the character- istic properties of a quadratic form (nondegeneracy is not required)

&W =a2q(F), q(F)>O, q(0) =0, (2.6a)

(2.6b)

dF+G) +q(F-G) =24(F) +2q(G), (2.6~)

suitably extended to the case q(F) = CO by means of the following conventions: A2q( F) = 00 if A#O, A2q(F)=0 ifA=O, q(F)+q(G)= CO, etc. In this more general case q will be called a generalized quadratic form (g.q.f.). The relevant point is the following.

Proposition 2.1: Let q( . ) be a g.q.f. over ( V,a), then the linear functional over ,pP( V,a) defined by

q,(WF))=exp[-$q(F>], if q(F) < co

=0 otherwise (2.7)

is positive and therefore it defines a state (generalized quasi-free state). Proofi The proof of this proposition can either be obtained by generalizing the argument of

Ref. 26 which deals with the case q(F) =0 tlFeVO and o nondegenerate, or by showing that wg is a w*-limit of regular states (with a strategy corresponding in models to the removal of an infrared cutoff). In the latter case one decomposes


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V=V,+V,=P+V’+V,,

V,={FEV, q(F) < co), ~={FEV, q(F) =O}; (2.8)

q is then pointwise approximated by finite Hilbert quadratic forms

q(F)= lim lim q,&F), m-m n-m

q,,,(F) = [F,Fl,,,=~ [~,~l”+ [FP,FQ14+n[F,,F,]‘,

where

F=F,+F,, F,=p+F’, F&, F,EV,, F%p,

and [ Jo, [ 1’ are two arbitrary nondegenerate inner products on Ve and V,. Clearly, with fixed m, for any pair F,G in a finite-dimensional subspace E of V, 3nF,G such that qn,m satisfies Eq. (2.3) for n > nF,G, and therefore defines a positive linear functional o,,, on the Weyl algebra over E. Then, for any finite linear combination A of W(Fi), since the space generated by the Fg is finite dimensional, w,(A*A) can be approximated as

lim lim o,,,(A*A) m-m n-m

and it is therefore positive. Remark I: If V&V, i.e., q(F) = ,ZU for some F, then the generalized quasi-free state oq

defined by Eq. (2.7) is nonregular, i.e., w& W(AF)) is not continuous in it ( = 1 for d=O, =0 for L#O). However, wp is regular on the *subalgebra JZ’( V,,a) (regular subalgebra identified by q). Here and in the following, we use the abbreviated notation ( V,,o) , V, C V, instead of (VI44 v,xv,).

Remark 2: Any CCR algebra JZ’( V,a) always admits a nonregular representation defined by q, (0) =0, q,(F) = co, WFEV, F#O. In this way one can always define a ck norm on .@‘( V,a), which is the unique c) norm on &( V,a) if o is nondegenerate.

Every state fl on JZ?( V,o) uniquely identifies a CCR subalgebra dreg= & ( Vreg,~) on which it is regular (the existence of the vector space Vreg follows from the Weyl relations); dIzeg will be called the regular subalgebra (with respect to a).

For the physical applications and also as a step toward the classification of nonregular states it is useful to ask when the knowledge of a regular state o over a CCR subalgebra & ( Vo,a), VoC V, uniquely determines its extension to &( V,a) . To this purpose it is convenient to introduce the following notion: a regular state w defined on d ( Vo,a) has & ( Vo,a) as maximal domain of regularity in .M ( V,o) if there is no regular extension of w to a larger CCR subalgebra & ( Vl,a) 3 JZ’ ( Vo,a) , Vos VI C V. A state o on & ( V,a) is minimally nonregular if its regular subalgebra ,&,,s is its maximal domain of regularity. Similarly, ( Vo,a) is the maximal domain of regularity of a quadratic form q if (q is finite on V. and) there is no finite extension of q (as g.q.f) to a larger space ( V,,CJ), Vo$Vl C V; a generalized quadratic form will be said to be minimally nonregular if, V,= {FE V, q(F) < 00) is its maximal domain of regularity. In the following V\ V. denotes the set of elements of V that are not in Vo.

Proposition 2.2: Let JZ’ ( Vo,a) C .& ( V,a) be CCR algebras and og be a regular quasi-free state over &( V,,o) defined by a quadratic form q on ( Vo,a); then the following are equivalent:

(i) LP’ ( VO,o) is a maximal domain of regularity for wg in J?p ( V,a); (ii) og has no regular and quasi-free extensions to larger CCR subalgebras;

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(iii) for any GczV\ Vo, a(G, . ) is not a bounded linear functional on V. with respect to the scalar product [ , I4 which defines q;

(iv) og admits a unique extension R to &‘( V,a), which is defined by the g.q.f. Q:

Q(F) =dF), ~FEVO, (2.9)

Q(F)=ca, ~IFEV\V~.

In particular, if oq is pure, so also is a. Proofi (i) j (ii) is obvious. (ii) j (iii). If for some GE V\ V,, a( G, * ) is a bounded functional on V,, then o( G,F) =O

if q(F) =0 and

Q(G)= sup “r;;;)12< ~. FE v,

dF)#o

Then q can be extended as a finite q.f. to V,+Span[Gj by

[G,F],=O ~‘FEV~, q(G) =Q(G>.

In this way one obtains a quasi-free state which extends wq to JZ’ ( V. + Span[Gj) contrary to ii). (iii) j (iv). Let fi be an extension of wq to ccS( V,a), then, given GEV\ V,, by (iii) there

exists a sequence {I;,) in V. such that

Then

lim a(G,F,) =a#O, lim q(F,) =O. “-Cl7 ?I-.co

lim a(W(F,)W(G)W(--F,))=n(w(G))e’” n-m

and, on the other hand,

(2.10)

IlmW(F,t) - 1)~,l12=2-2~(~(F,))=2(1 -wq( HV’,)) + 0, n-m

which implies s-lim vn(W(Fn))Yn=Yn and

lim fl(W(F,) W(G) W( -Fn))=fi(W(G)). (2.11) n+cu

Equations (2.11) and (2.12) are compatible (a#O) only if fi (G) = 0, i.e., R coincides with the extension defined by Eq. (2.9). Clearly, if wq is pure, so is R by uniqueness.

(iv) j (i). A regular extension of wq to a larger subalgebra would imply an extension a’ to JZ’( V,a) different from a (since the latter does not extend the regularity domain), contrary to (iv).

With the help of Proposition 2.2, we can analyze the quasi-free representations of a CCR algebra ZZ’( V,a) defined by minimally nonregular states wq in terms of the representations of the corresponding regular subalgebra. In fact, let &( Vo,a) be the regular subalgebra with respect to wq and r. its representation given by wo--wq 1 dc v,,O,. For every fixed FE V the automorphism PF

W(G) --+ W(G)exp ia(F,G), GEVo, (2.12)


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is unitarily implementable in r. iff q(F) < CO. In fact, if q(F) < 00, FEV~ and pF is implemented in rro by ?V( F); conversely, if PF is implementable in ?ro, then oF is strongly continuous (with respect to the strong topology of ro) and therefore o( F,G,) must converge whenever G, converge strongly with respect to the topology defined by q. Hence, a(F, * ) is bounded and by Proposition 2.2 FEV,,. In general, the state aF=tic * oF defines a representation VF of &/( Vo,a) that is irreducible iff r. is so and it is factorial iff n-O is so. Two such representations PF and ?T~ are equivalent iff F - G cVO, i.e., q(F - G) < 00. Clearly, for each FEV, rF is contained in rrU,(sP( V,a)) (and %FCSY’@), and if FEV~, xF=x@ In conclusion we have the following.

Proposition 2.3: The representation rU of a CCR algebra JS’( V,a) given by a minimally nonregular generalized quasi-free state ti: decomposes into the direct sum of inequivalent representations ‘a-F of the regular subalgebra .&‘( Vo,o), labeled by the equivalence classes FE V/ V,:

ztiq= $ dli”F, vu,= $ VF. FE V/ V, FE V/ V,

(2.13)

As we will see later in the models below, the SYF have the meaning of the charged disjoint sectors of the “observable” algebra L%‘( Vo,a).

To characterize the generalized quasi-free states that are pure, it is convenient to introduce the following.

Definition 2.4: A generalized quadratic form q is said to be minimal on a symplectic space ( V,a) (with o not necessarily nondegenerate) if there does not exist a g.q.f. q’#q on ( V,a) such that

q’(F) <q(F) VFEV. (2.14)

Clearly a minimal g.q.f. always exists (by Zorn’s lemma). Furthermore, it follows from the definition that q is minimal on ( V,a), iff (i) its restriction q 1 vq to V,={FEV,q(F) < CO} is minimal on ( Vpa), and (ii) this is its maximal domain of regularity in ( V,o).

Then we have the following. Proposition 2.5: Given a symplectic space ( V,a), with o possibly degenerate, a generalized

quasi-free state wq on s!( V,a) is pure iff q is minimal on ( V,o). Prooj ( 1) q minimal implies wq pure. By Proposition 2.2 it is enough to show that the restriction of wq to its maximal domain of

regularity LS’( V,+T) is pure. To this purpose we note that the restrictions q and 5 to Vq have the same kernel. In fact, by Eq. (2.3), Ker q CKer i? and a( *, . ) =[ * ,D*], with 1 D( 4; on the other hand, if FEKer 5 and @(F)#O, then FEKer ) DI and

qmin(G) = [G, 1 DI G],G~G)

would be smaller than g, contrary to the minimality of @ Hence rwq( IV(F)) = I tlF E Ker q, and n,q(&‘( V,,a)) = nU4(&‘( VJKer q,a)) which is irreducible since o is not degenerate on V,/Ker q and 1 D 1 =I, again by the minimality of q, and the standard argument applies.2’

(2) oq pure implies q minimal. In fact, if q is nonminimal, one explicitly exhibits a decomposition of We It is convenient

to distinguish two cases depending on whether ( V,,a) is the maximal domain of regularity of q or not. In the first case, q nonminimal implies q[ v4 nonminimal, and therefore wg can be decomposed essentially as in the standard case:21


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oqWW))= da e--(r20,( W( F)),

=o, VF6 Vq,

where HE vq is a normalized vector with spectral support relative to I D I below 1 - E (which exists since q) v q nonminimal implies I D I < I> and

qdF) =q(F) -4H,Fl;. In the second case, there exists a q’ that extends ql vp’ it is finite on I-‘;={ Vq + AG, A.

E R} for a suitable fixed G E V\ VP and it is infinite otherwise. Now, by decomponing V = Vi + V;, each FE V has a unique decomposition F=F,+AFG

+F”, FqeVP ilF EW, and F”ctV$ and one may define the states

w~,(W(F))=exp(idF)oqt(W(F)), tI&V, a&.

Hence one has

1 L wq=w*- lim -

s L&2= -L a:, da

and this allows a decomposition of oq into states defined by a suitable splitting of the above integral, e.g.,

o,=w*-- lim - n-m 2iT jjn R”“G da

so that oq=~(o++o-). Remark: Proposition 2.5 generalizes the standard result2* on the characterization of pure

quasi-free states in two respects: (i) o can be degenerate on V; (ii) the quasi-free states can be nonregular.

III. EXAMPLES

A. The harmonic oscillator in the limit of zero frequency or of infinite temperature

The model is defined by the Hamiltonian

H= (2m)-‘(p2+m2~2x2)

and the Weyl algebra is generated by

v= R2, (T(u,v)=up2-ulu2.

The equilibrium states at inverse temperature p= l/T are quasi-free states defined by

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~~ww)=exp( -; [ ~+mwuijcoth q) . The zero temperature (/!I- 00 ) state in the limit w-+0 (free particle limit) is a generalized

quasi-free state defined by

aa+YO,u,))= 1, %(w(u,,o))=o, ut#O, (3.1)

corresponding to q(O,u,) =0, q(u,,O) = 03, ut#O. Furthermore V4= {u&, ut =0} is the maximal domain of regularity for a,-, (if not, by Proposition 2.2, there would be a regular and quasi-free extension and this would contradict the nondegeneracy of a).

For nonzero temperature states the w -0 limit also defines a generalized quasi-free state given by

(3.2)

fidW(O,uz))=exp ( 1

-5 uz .

It is worthwhile to remark that, in an equilibrium state at nonzero temperature, the Maxwell distribution of the velocity of a free quantum particle arises only at the infinite volume limit and it requires a uniform distribution of the position so that the use of nonregular representations is necessary. For fixed o and p-0 the state R, converges on the Weyl algebra to the nonregular state a,( W( F)) = 0, VFfO.

6. Quantum particle on a circle

The canonical variables q and p have the meaning of an angular variable 9, and the conjugated angular momentum L= L,. The CCR algebra SZ’ is generated by the Weyl operators, heuristically given by

W( n,O) =exp inq, r&E, W( 0,~) = exp iv L, vdR,

i.e., LZY = .LY’ ( V,a) with V=Z x R, a( n,v;m,w) =nw-mv. It is worthwhile to remark that here V is not a (real) vector space as a consequence of the periodicity condition required by the circle. The Hamiltonian is H= L2/21 and the ground states are quasi-free states of the form

fl(W(n,v))=O if n#O

=l if n=O. (3.3)

It is the restriction to the subalgebra &(ZX R) of the nonregular quasi-free representation (3.1) of &(WXR).

The occurence of nonregular representations displayed by the above simple examples is actually a general fact whenever one deals with states that are homogeneous with respect to one field variable. In the functional integral language this occurs when the functional measure is homogeneous along one direction in field configuration space.

It should not be a surprise that such phenomenon already occurs at the level of quantum mechanical systems with a finite number of degrees of freedom. The point is that Von Neu- mann theorem on the unitary equivalence of representations of the CCR algebras is evaded by the lack of continuity of the Weyl group, a feature that is sometimes required by the physics of the problem.


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C. Electrons in a periodic potential

Another example of nonregular representation of the CCR algebra is provided by the ground state of an electron in a periodic potential V(x) = V(x+a). By Bloch’s theorem the energy levels are described by the Bloch wave functions (Bloch eIectrons)22

$4(x) =eikxv3x), v;(x) =v;(x+u), kE[0,27r], n&i, (3.4)

s o= ~v[G(x>~2dx=l,

which are also (generalized) eigenfunctions of the discrete translations U(a). It is not difficult to see23 that each Bloch wave function t/k gives rise to a nonregular representation of the CCR algebra &( V,a) defined in Sec. III A. In particular, the ground state (n=O, k=O) is given by

~~(W(u,,u2))=0, if u,#27rn/u, n&, (3.5)

4 wif ,*2)] =e~77w/~ s,” &X)V:(X+U2)e2”~ dx.

Clearly, whereas @i(x) is a non-normalizable wave function, the state a: is normalized as a state on .&( V,a), i.e., &,(I) = 1. The Weyl operators W( ~t,0), which generates the boosts on &‘( V,a), are nonregularly represented by @; any other state fi; defined by the Bloch wave functions (3.4) belongs to the GNS representation space given the state 9,. The nonregularity of the representation is forced by the invariance of @ under the noncompact discrete group of lattice translations.

D. The massless scalar field in 1 +l dimensions

The Weyl algebra can be thought as formally generated by the exponentials of the canonical “time zero” fields p,(ft ), 7r(f2), f,, f2~9’red(R) ~9’ and the canonical commutation relations are described by the standard symplectic form o (see beginning of Sec. II) over the (real) vector space V= (Y'xY). Since (T is nondegenerate, ( V,a) identifies a unique CCR F-algebra LZ’( V,a). The time evolution is described by a one-parameter group of * automorphisms af of (;9( V,o) and it is well known’ that for infrared reasons there does not exist a regular representation of ,pP( V,o) induced by a space-time translation invariant ground state. Such representation exists if one considers the subalgebra &( V~,(T), Vc=ay X Y, &Y =cw, f =a&, ~993, and it is given by the regular quasi-free state K&, identified by the following quadratic form on ( Vc,o)

q(F)= dp~s~~P~~,~P~w~P~-‘+~2;(P~f2~P~0~P~1, o(p)= IPI. s

(3.6)

It is not difficult to show24 that ( Vc,o) is the maximal domain of regularity of q in (V,a) and therefore, by Proposition 2.2, q has a unique extension to ( V,a) given by

QW =4(F), VFEVfj,

Q(F)=co, VFEV\Vl). (3.7)

Correspondingly, sic has a unique extension to LZ’( V,o> which defines a nonregular representation. Since Rc is pure so is its extension by uniqueness.


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Similarly, equilibrium (KMS) states over JX!‘( Vc,a) are given by the regular quasi-free states

(3.8)

where Qrs is related to Q by the standard factor coth(w(p)fl/2), p= l/T, which yields the fullfillment of the Kubo-Martin-Schwinger (KMS) condition.

Furthermore fiB has an extension to &pp( V,o) given by

nf;‘(W(F))=I(ZB(W(F)), VFEV,, (3.9)

i-gW(F))=O, VFEV\Vo.

The state fiFt (i) is invariant under space translations, (ii) is primary, and (iii) satisfies the KMS condition; it is actually the unique extension of Qp with the properties (i) and (ii). In fact, any extension sli with such properties must satisfy the cluster property

lim fq(,i[o(f)-W,)l) = (nl;(,iW)) 12, a-m

where f is real and f a(x) z f (x-u); since f-f a E a,$“, the left-hand side can be computed on ag, and the limit a-+ UJ vanishes. Hence “b = a;‘.

In conclusion, ground state or equilibrium state representations of the CCR algebra of the massless scalar field in 1 + 1 dimensions exist compatibly with positivity but they are nonregular.

E. Free Bose gas

The CCR algebra is defined in the following way. For each parallelepiped AC I@, d= space dimensions, one defines VA = L2 (A) and the canonical symplectic form (T, o( f ,g) = Im ( f ,g) , with ( *, + ) the L2 scalar product. Then G!( V,,,a) denotes the CCR algebra relative to A and JZ! ( V,a) = U ,,Jx’( V,,,a). The finite volume Hamiltonian H,, is given by the self-adjoint extension of the Laplacian on L2(A) corresponding to Dirichlet boundary conditions on aA. In the thermodynamical limit, the space and time translationally invariant states are the quasi-free gauge invariant states2

&Wf ))=expt -d (f,(I+zemPH) (I-eMPH)-‘f )] (3.10)

for z=e@< 1, p= the chemical potential, fl the inverse temperature, and H the Laplacian in L2( I@) (single phase region); whereas for z= 1 and d<2,2

&JW(f ))=O, if f(O)#O (3.11)

&Y1(W(f ))=e-llf112’4 exp[-2(:?,)d SddplS(p)/3e-8p2(l-e-~~)-‘], if f(O)=O.

One then obtains nonregular representations, which are not locally Fock and the physical meaning of this mathematical phenomenon is that for z= 1 and d<2 the local particle density p(&z= 1) zpp,(p) is infinite.


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F. Quantum harmonic lattice

The model describes quantum crystal lattices in the harmonic approximation.’ The lattice points are identified by the vector R, the vector q(R) denotes the displacement of the oscillator in R and p(R) its conjugate momentum. The CCR algebra is generated by the Weyl operators heuristically defined by

W(agt2) =exp i c [a,(R) -q(R) +%(R) *p(R)] R

(3.12)

with al(R), a2(R) sufficiently decreasing for large R, e.g.,

;a,W2< 001 &2W2< ~0, (3.13) R

and the symplectic form u is

a(al,a2;&/32> = c [q(R) *L%(R) -dR) *PI(R) 1. R

The finite volume Hamiltonian is taken to be that of coupled harmonic oscillators. In the thermodynamical limit A -+ HP’, d= space dimensions, the Gibbs canonical equilibrium state gives rise to the generalized quasi-free state at temperature T= l//3:

q&V(a~,a2))=exp -4 ( 1 s & i, [ ia;~~\~f’12+mo,(k) I&(k)

*e,(k) 12]cothPq]

with .s,( k) the polarization vectors that define the normal modes and o,(k) the normal fre- quencies. Since for small I k I, w,(k) - I k 1, the 5, terms behave like km2, so that in dimensions d= 1,2 the above integral diverges if

El(O)= C q(R)#O R

and correspondingly wp(M’(a,,a2))=0. Thus one has a nonregular representation. The physical meaning is that in this case the variable q(R) does not exist; the point is that all the variables q(R) have a uniform distribution in the infinite volume limit and the mean of q(R) does not exist (crystals do not exist in d= 1, 2).

IV.EXTENDEDCCRALGEBRASANDCHARGEDFlELDS

The previous examples show that in many cases some of the canonical variables that enter in the definition of the Hamiltonian are infrared singular and only their Weyl exponentials can be represented as operators in a Hilbert space (nonregular representation). The important role played by maximal regular subalgebras, which in fact determine the representation of the whole algebra and its decomposition into disjoint sectors, suggests that we look at the models in the perspective according to which such regular subalgebra has a distinguished status independent of the state. This is also suggested by gauge theory models where one has the distinguished algebra of observables, which is pointwise invariant under the gauge group and the larger field algebra which includes charged fields that may be infrared singular. From this point of view the


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question that we want to address is the following: Given an algebra of observables dot,, does it determine its extension to a field algebra 9 which includes the fields that intertwine between the vacuum sector and the other sectors and therefore define the gauge group? We will answer the question in the framework of CCR algebras discussed in the previous sections and show the connection between unbroken gauge group and nonregular representations.

Definition 4. I: Let &a=&( Ve,oc) be a CCR algebra and ( V,a) a symplectic space such that V>Vcandal yoX v, = 00; then the CCR algebra & ( V,a) is called a canonical extension of JP’ ( Vo,ao). More generally one can consider the case in which V and V. do not have the structure of vector spaces but only that of Abelian groups.

One may then show that a canonical extension & of -cPo identifies a gauge group 3 and can be interpreted as a charged jield algebra. To this purpose we consider the algebraic dual ( V/V,) ’ of the quotient space V/V,; it is isomorphic to the space Y, of real linear functionals on I’ which are zero on Vo:

~~“{fpEV;,,,,rp(F) =o, VFEVO}. (4.1)

In the case in which V has the structure of a vector space V, plus an additive group, Y. is defined as the set of real functionals which are additive on V and linear on V, and vanish on V. modulo 2~.

Each *To defines an automorphism a, of &( V,a) by the equation

(4.2)

The gauge group 9 identified by the structure ( V,p,> 3 ( Vo,oo) and therefore also denoted by Y v,rb is then defined as the group of automorphisms aP, KEY, Clearly ,pP, is pointwise invariant under 9. If V is a vector space, 9 is isomorphic to some W”, whereas, if WV0 has the structure of a finitely generated Abelian group, then 9 is U ( 1) X * * * X U ( 1) .

By definition, the gauge group acts nontrivially on the Weyl operators W(F), &V. (charged fields). They define charged automorphisms of &( Vo,ao> by

rdW(G))=exp(ia(F,G))W(G), kfGeVo

[as we will see below W(F) acts as an intertwiner between inequivalent representations of d”( Vo,~o>l.

We now discuss the properties of representations of extended CCR algebras that are related to the gauge group.

We consider a CCR algebra &‘( Vo,oo) and a canonical extension of it, JZ!( V,a); for convenience in the following &( Vo,ao) will be referred to as the core subalgebra of ccS( V,a).

Let fi4 be a generalized quasi-free state on & ( V,a), whose restriction o to d ( Vo,oo) is a regular state. As in Sec. II we denote by Vg the subspace of V on which the generalized quadratic form 4, which defines R, is finite:

V,={&V, q(F) < CO}. (4.3)

Proposition 4.2: The subgroup Yfi of the gauge group 9, which leaves the state f12, invariant is characterized by a&9o iff

Q)Eq=CqEvreal, qo-1 =o, V~EVJ. (4.4)

Furthermore Yn is implemented, in the representation n-n defined by Q, by strongly continuous unitary operators. Their generators define the gauge charges that leave s2 invariant.


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If Yn is nontrivial, then s1, defines a nonregular representation of the field algebra d( VP).

Proq? It follows from the above definitions that VJ V. and therefore Y4c Yo. Clearly if F.Y~ aq is the identity automorphism on ccS( Vqa); moreover, VFEV\ Vq

(4.5)

Then

a&= R, .

Conversely, if @.V,r, there is at least one FEV, such that q,( F)#O and

flq(a,(W(F))=ei~cF)~q(W(F))#f2q(W(F)).

Hence a&#$ Furthermore, ‘d-Yp let a$ be the one-parameter group of automorphisms defined by

a$W(F))=e”qcF)W(F), YFEV, k/A&. (4.6)

Since (a$)*R,=R, , a$ is implemented by a unitary operator U,(n) in 7rn and

Il(U,(~)--H)~~(W(F)>~nll=Iexp[i~~(F)l--ll -0, 1-O

i.e., strong continuity holds. The existence of gauge charges as generators of the one-parameter groups u,(n), p EZ;-, then follows from Stone’s theorem.

Finally, if Yo is nontrivial there is at least one p E.Y~ that is nontrivial and therefore at least one FEY\ Vq

Additional information is obtained when the core subalgebra is the maximal domain of regularity of R+ in the interpretation of -c9, as the observable algebra and of fi2, as the ground state this property can be viewed as a sort of completeness of do.

Proposition 4.3: If the core subalgebra do is a maximal domain of regularity of 02, then (i) Yn=Y (unbroken gauge group), (ii) the representation rn defined by R, decomposes into the direct sum of inequivalent

representations of,the core subalgebra do, labelled by the gauge charges

&“n= 8 XF, (4.7) FE V/ V4

(iii) the charged fields W(F), FEV\ Vq act as intertwiners between the inequivalent representations of the “observable” algebra de.

ProoJ By hypothesis Vo= V, i.e., 7-q= To and therefore Yo= 9. The second part of the Proposition follows from Proposition 2.3. Finally, one can show that under the condition of Proposition 4.3 (the regular subalgebra

is maximal) the symmetries of the vacuum sector automatically extend to the charged sectors. Proposition 4.4: Given & = &( V,a) and its subalgebra do= J ( Vo,ao), let wq be a gen-

eralized quasi-free state on do, and ,pP, a maximal domain of regularity of wq in &‘. Let p be an automorphism of JZ’ such that

(i> PdoCdo, (ii) p*wq=wq Then, the unique extension R of wq to d satisfies


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ProoJ By Proposition 2.2 wq has a unique extension R to &; since /3*n is an extension of p*w4=w4 it must coincide with Q.

A number of realizations of the general structures of this section will be presented else- where;” we only discuss here an application to a quantum field model which is relevant (see, e.g., Refs. 29 and 30) for understanding the Higgs and confinement mechanisms in different space-time dimensions.

Buchholz and Fredenhagen have argued that nonregular representations naturally arise in the construction of charged states and explicitly implement their idea in the case of the algebra of the free electromagnetic field.26 For the use of nonregular representations in the canonical formulation of the Chern-Simons theories see Refs. 27 and 28.

The Sttickelberg-Kibble (SK) model is defined as an approximation of the Abelian Higgs- Kibble model, by freezing the modulus of the Higgs field x= Ix I exp iq to Ix I = 1.2g,30 The observable algebra is generated by the current jp=apq+eA, and by the electromagnetic field Fp,+ In the Coulomb gauge, such a field algebra can be realized in terms of a CCR algebra

jo=r, ji=de, F,=J&, Ao=A-‘n; F,=aJj-a/Ii, (4.8)

with Q, and rr canonical fields. In more than 1 + 1 dimensions one has a nontrivial transverse algebra generated by FV but it decouples from the other fields and it will not be considered for simplicity. Hence, the observable algebra at t=O can be identified with the CCR algebra:

where A-‘Y is the space of C” functions, f, bounded by polynomials, with A~EY, A the Laplace operator; dh-‘Y denotes the space of (partial) derivatives of functions in A-‘5“.

The dynamics is defined by the following infrared cutoff Hamiltonians:

d”x d’y dx)n(y) u,(x-Y), (4.9)

where V,(x) =Jr(x) V(x) with AV(x) = --6(x), and fL(x) =f(x/L), fog, f(x) = 1 for 1x1 < 1, f(x) =0 for 1x1 > 1 +E.

The removal of the infrared cutoff can be done by taking strong limits, in a class of representations, of the (infrared cutoff) time evolution of .!7Zot,,; one obtains a one-parameter group a’, PER, of automorphisms of dabs, as in Ref. 24.

The charged field algebra &’ of the SK model can be obtained as a canonical extension of .d ObS, with Vo=i3YxdA-‘S C V=.YxXAvlS, i.e., &=&(YxdA-‘S), with the natural extension of o.

The gauge group is isomorphic to R and is defined by p-+97+/2, aA-‘n-+dA-‘n; it is a U ( 1) group on the “compact fields” exp iq(f ), sfd’x = m E Z, which are the original variables of the SK model.

The ground state R. on &,,bs is given by the quadratic form (3.6) with o(p) replaced by p2/w,(p), w,(p) E ,/pw. The state R, has a unique extension Q from dabs to JZ’ (since all FE V\ V. define unbounded functionals on V,).

The representation 7rn defined by fi on &’ is nonregular in 1 + 1 and 2 + 1 dimensions and regular in 3 + 1. In 1 + 1 and 2 + 1 dimensions the gauge group is therefore unbroken in Rio, and correspondingly the Hilbert space &“n decomposes into irreducible inequivalent representations of -pP,& labeled by the gauge charge:

xi2 = @ &~a ;

in 3 + 1 dimensions the gauge group is spontaneously broken and there are no charged sectors.


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The charged sectors in 1 + 1 and 2+ 1 dimensions are not stable under the time evolution, and the energy cannot be defined, except on the vacuum sector, i.e., conjinement takes place as spontaneous breaking of the time translations on the observable algebra, in the charged sectors. Under the time evolution each charged sector is mapped into a one-parameter family of inequivalent representations and this can be viewed as the result of a time evolution of a larger field algebra .GP~= .@‘(YXA-‘9) (which gives rise to the larger gauge group WXR), and the (unique) extension of s1 to .Pxt defines a Hilbert space in which the time translations are implementable by unitary operators which are not strongly continuous, except on the vacuum sector. Charged states are therefore excluded in different space dimensions by two different mechanisms, one (confinement) involving infinite energies, the other (Higgs) resulting from the fact that charged fields applied to the vacuum give rise to states in the vacuum sector (for related results see Refs. 25 and 29).

Note added in prooJ After this paper had been accepted we received a preprint by W. Thirring and H. Namhofer “Covariant QED without indefinite metric” in which, among other things, they present a discussion of the Stiickelberg-Kibble model in 1 + 1 dimensions very similar to ours.

The content of this note was part of a longer paper SISSA 39/92/FM (March 1992).

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Infrared singular fields and nonregular representations of canonical commutation relation algebras

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