Chiral symmetry breaking and theta vacuum structure in QCD
aa r X i v : . [ h e p - t h ] J u l Chiral symmetry breaking and θ va uumstru ture in QCDG. Mor hioDipartimento di Fisi a, Università di Pisa,and INFN, Sezione di Pisa, Pisa, ItalyF. Stro hiS uola Normale Superiore, Pisa, Italyand INFN, Sezione di PisaAbstra tThe solution of the axial U (1) problem, the role of the topol-ogy of the gauge group in for ing the breaking of axial symmetryin any irredu ible representation of the observable algebra andthe θ va ua stru ture are revisited in the temporal gauge with at-tention to the mathemati al onsisten y of the derivations. Bothrealizations with strong and weak Gauss law are dis ussed; the ontrol of the general me hanisms and stru tures is obtained onthe basis of the lo alization of the (large) gauge transformationsand the lo al generation of the hiral symmetry. The S hwingermodel in the temporal gauge exa tly reprodu es the general re-sults.Keywords: Chiral symmetry; QCD; gauge group topology; θ va uaPACS: 11.30.Rd; 11.30.Q ; 12.38.-t; 11.40.Ha11 Introdu tionThe solution of the U (1) problem by the dis overy of the θ va uumstru ture and its strong relation with hiral symmetry breaking hasbeen one of the ornerstones in the theoreti al analysis of the standardmodel of elementary parti les. [2℄ [3℄ [4℄The standard (and histori ally the (cid:28)rst) arguments in favor of anon-trivial role of the topology relies on semi lassi al approximations,in terms of boundary onditions of the lassi al (cid:28)eld on(cid:28)gurations andtheir e(cid:27)e t in the (eu lidean) fun tional integral. These results havebeen ex ellently reviewed by Coleman, [5℄ [6℄ who also pointed outthe limitation of su h arguments, sin e smooth (cid:28)eld on(cid:28)gurations, inparti ular those of (cid:28)nite a tion, have zero fun tional measure. The rel-evan e of the lassi(cid:28) ation of the (smooth) (cid:28)eld on(cid:28)gurations in termsof their pure gauge behaviour at spa e in(cid:28)nity, and of the orrespond-ing winding number, is therefore a non-trivial mathemati al problem,espe ially in the in(cid:28)nite volume limit.A de isive step in the dire tion of a mathemati al ontrol of theme hanism of θ va uum stru ture was taken by Ja kiw, who empha-sized the role of the topology of the gauge group, without involvingthe lassi(cid:28) ation of the gauge (cid:28)eld on(cid:28)gurations and thus avoidingthe problem of the semi lassi al approximation of the fun tional inte-gral. [7℄Ja kiw's analysis is done in the temporal gauge and the very pe uliarfeatures of su h a gauge raise some stru tural mathemati al problems,with a non-trivial impa t on the general argument. The aim of thisnote is to present an analysis whi h does not su(cid:27)er of mathemati alin onsisten ies and, in a ertain sense, provides a mathemati al glossaryto Ja kiw's strategy.The paper is organized a ording to the following pattern.As in the abelian ase, the invarian e of the va uum under the gaugetransformations generated by the Gauss operator implies a non-regularrepresentation of the (cid:28)eld algebra (Se tion 2).In Se tion 3, the non-trivial topology of the group G of time indepen-dent gauge transformations is lassi(cid:28)ed in terms of group valued gaugefun tions U ( x ) of ompa t support.This signi(cid:28) antly simpli(cid:28)es the dis ussion with respe t to the on-ventional analysis, [7℄ [5℄ where the gauge fun tions are only requiredto have a limit, for | x | → ∞ , independent of the dire tion (with theneed of requiring for the analysis a faster than /r de ay of the gaugeve tor (cid:28)elds A ai ( x ) ). Furthermore, the exponentials of the topologi- al harge (in bounded regions), at the basis of the standard analysis,are shown to have vanishing matrix elements between ve tors satisfyingthe Gauss law onstraint and therefore a non-trivial impa t of the largegauge transformations on the stru ture of the physi al states requiresfurther ingredients.In Se tion 4, it is argued that a ru ial role for dis losing the physi alrelevan e of the non-trivial topology of the gauge group is played bythe fermions and the asso iated hiral symmetry.Following Bardeen, [8℄ we show that, ontrary to statements ap-peared in the literature, the presen e of the hiral anomaly does notprevent the hiral symmetry from being a well de(cid:28)ned time indepen-dent group of automorphisms of the (cid:28)eld algebra and of its gauge invari-ant (observable) subalgebra, generated by the operators V R ( λ ) , λ ∈ R ,formally the exponentials of the harge density J ( f R α R ) , J µ denotingthe onserved gauge dependent axial urrent .The solution of the U (1) problem (i.e. the absen e of massless Gold-stone bosons asso iated to hiral symmetry breaking) is provided by thefailure of a ru ial hypotheses of the Goldstone theorem, namely theimpossibility of writing the symmetry breaking Ward identities, sin ethe va uum expe tations < J ( f R α R ) ¯ ψ ψ > do not exist, as a on-sequen e of the non-regularity of the lo al implementers of the hiralsymmetry, V R ( λ ) .The interplay between the topology of the gauge group and the hi-ral transformations whi h for es hiral symmetry breaking (in any ir-redu ible, or even fa torial, representation of the observable algebra)and gives rise to a non-trivial va uum stru ture is learly displayed un-der general assumptions. By assuming that the lo alized large gaugetransformations may be implemented by unitary operators whi h are(cid:16)fun tions(cid:17) of (cid:28)elds with the same lo alization, we show that the non-trivial topology of the gauge group re(cid:29)e ts in a non-trivial enter of thealgebra of observables, whi h is not left pointwise invariant under the hiral transformations (Se tion 5).The standard labeling of the irredu ible (or even fa torial) repre-sentations of the observable algebra by an angle θ ∈ [0 , π ) , ( θ se tors) isobtained by analyzing the redu ible representation of the (cid:28)eld algebrade(cid:28)ned by a hiral invariant va uum and by a spe tral de ompositionover the enter of the observable algebra.In Se tion 6, we onsider a realization of the temporal gauge inwhi h only a weak Gauss invarian e of the va uum is required, as dis- ussed in the abelian ase. [9℄ As a onsequen e of the la k of Gaussinvarian e of the va uum, the onserved axial urrent J µ an be rep-resented by a well de(cid:28)ned (cid:28)eld operator and Bardeen analysis dire tlyapplies.Under the same assumption of implementation of the large gaugetransformations by lo al operators, we show that, if the hiral symmetryis unitarily implemented, the va uum de(cid:28)nes a redu ible representation π of the observable algebra; hiral symmetry is broken in ea h irre-du ibile omponent. Independently of the possibility of extending the θ va ua to non-positive weakly gauge invariant fun tionals on the (cid:28)eld al-gebra, the hiral urrent J µ does not exist in the physi al representationspa e of the observable algebra, in parti ular in a θ se tor.It is worthwhile to stress that, ontrary to statements appearedin the literature, the derivation of the physi al onsequen es of thetopology of G ru ially relies on the presen e of fermions and their hiral transformations. The essential point, whi h distinguishes theabelian ase from the non-abelian one, beyond the existen e of the hiralanomaly, is the existen e of a enter of the lo al observable algebras,following from the topology of G , whi h is not pointwise invariant underthe hiral transformations.All the general results are exa tly reprodu ed by the S hwingermodel in the temporal gauge, analyzed in Se tion 7, both in the posi-tive non-regular realization with a Gauss invariant va uum and in theinde(cid:28)nite regular (quasi free) realization, with the weak form of theGauss law onstraint.2 Temporal gauge in QCD and Gauss lawFor the dis ussion of the non-perturbative aspe ts of QCD, in parti ularthe θ va uum stru ture, its relation with the topology of the gauge (cid:28)eld on(cid:28)gurations and its role in hiral symmetry breaking, the temporalgauge has proved to be parti ularly onvenient.However, as noted before for the abelian (QED) ase, [9℄ the on(cid:29)i tbetween the Gauss law onstraint and anoni al quantization raisesproblems of mathemati al onsisten y, whi h, as we shall see, a(cid:27)e tthe derivation of the general stru tures leading to signi(cid:28) ant physi alimpli ations.For simpli ity, we start by onsidering the ase with only ve tor(cid:28)elds (no fermion or s alar (cid:28)eld being present). Then, at the lassi allevel the QCD Lagrangean density redu es to the Yang-Mills form L = − X a F aµ ν F µ ν a = X a ( E a − B a ) , (2.1)where in the temporal gauge, de(cid:28)ned by A a = 0 , E a = − ˙ A a , B a = ∇ × A a − gf abc A b × A c , (2.2)( a is a olor index and f abc are the stru ture onstants of the Lie algebraof the olor gauge group G ).The orresponding equations of motion, obtained by variations withrespe t to A a , are ∂ t E a = ∇ × B a + gf abc A b × B c ≡ ( D × B ) a , (2.3)whi h imply ∂ t G a = 0 , G a ≡ ∇ · E a + gf abc A b · E c ≡ ( D · E ) a . (2.4)The operators G a are alled the Gauss law operators.In the standard quantum version of the temporal gauge it is assumedthat the (cid:28)elds A a and their powers an be de(cid:28)ned and quantization isgiven by the anoni al ommutation relations.In parti ular one has the following ommutation relations − i [ D · E a ( x , t ) , A b ( y , t ) ] = δ ab ∇ δ ( x − y ) + gf abc A c ( x , t ) δ ( x − y ) . (2.5) − i [ D · E a ( x , t ) , E b ( y , t ) ] = gf abc E c ( x , t ) δ ( x − y ) . They state that the Gauss operators generate the in(cid:28)nitesimal timeindependent gauge transformations, δ Λ , with Λ a ( x ) ∈ S ( R ) the -number gauge fun tion δ Λ A a ( x ) = ∇ Λ( x ) + gf abc A b ( x ) Λ c ( x )) . (2.6)Sin e the variables A a are missing in the Lagrangean, one annot ex-ploit the stationarity of the a tion with respe t to them and thereforeone does not get the Gauss law G a = 0 . A tually, the Gauss law isin ompatible with eq. (2.5) and therefore with anoni al quantizationand more ru ially with the Gauss operator being the generator of thetime independent gauge transformations, eq. (2.6).A proposed solution of this on(cid:29)i t, widely adopted in the litera-ture and in textbook dis ussions of the temporal gauge, is to requirethe Gauss law onstraint as an operator equation on the (subspa e of)physi al states and, in parti ular, on the va uum state. However, su ha solution is not mathemati ally onsistent. In fa t, the va uum expe -tation of eq. (2.6) gives zero on the left hand side and non-zero on theright hand side.It has been proposed [7℄ [10℄ to ope with this paradox by admittingthat the va uum ve tor is not normalizable. In our opinion, su h asolution is not a eptable, be ause it does not yield a representation ofa (cid:28)eld algebra ontaining both gauge dependent and gauge independent(cid:28)elds.A mathemati ally a eptable solution for the Gauss law onstraintis to adopt a Weyl quantization and admit non-regular representations.As a preliminary step in this dire tion, we re all that from a math-emati al point of view the quantum (cid:28)elds are operator valued distri-butions and a smearing with test fun tions (typi ally in(cid:28)nitely di(cid:27)er-entiable and of ompa t support) is needed for obtaining well de(cid:28)nedHilbert spa e operators. Therefore, we onsider the (cid:28)eld algebra gen-erated by the polynomials of the smeared (cid:28)elds A ia ( f ) , f ∈ S ( R ) ; weshall assume that by a suitable point splitting pro edure one an on-sider as (cid:28)eld variables the powers of A a ( x ) and its derivatives, like e.g.the Gauss operator G a ( g ) , the (cid:16)magneti (cid:17) (cid:28)eld B ia ( g ) , g ∈ S ( R ) , et .We then take the polynomials of su h (cid:28)eld variables as a lo al(Bor hers) (cid:28)eld algebra F , transforming ovariantly under the spa etime translations α y , y ∈ R , α y ( A ia ( f )) = A ia ( f y ) , f y ( x ) = f ( x − y ) . In order to simplify the bookkeeping of the indi es, it is onvenientto introdu e the following notations: T a denote the hermitean repre-sentation matri es of the Lie algebra of the gauge group, normalized sothat Tr T a T b = δ a b , the (cid:28)eld A i ( x ) = P a A ia T a are Lie algebra valueddistributions on Lie algebra valued test fun tions f i ( x ) = P a f ia ( x ) T a , f ia ∈ S ( R ) , A ( f ) ≡ Z d x Tr ( A ( x ) f ( x )) = Z d x X i, a A ia ( x ) f ia ( x ) . Unless otherwise stated, in the following the sum over repeated spa eand gauge indi es will be understood.With this notation the time independent gauge transformations α U are labeled by gauge group valued unitary C ∞ fun tions U ( x ) , whi hmay be taken to di(cid:27)er from the identity only on a ompa t set K U , and α U ( A ( f )) = A ( U f U − ) + U ∂ U − ( f ) , (2.7) U ∂ U − ( f ) ≡ Z d x Tr ( X i U ( x ) ∂ i U − ( x ) f i ( x )) . The (spa e-time) lo alization of U is given by the ylinder C U ≡K U × R , so that α U ( A ( f )) = A ( f ) if supp f ∩ C U = ∅ .We denote by U λ , λ ∈ R , the gauge fun tions orresponding toone-parameter subgroups of the gauge group. They an be written inthe form U λ ( x ) = e iλ g ( x ) with g = P a g a ( x ) T a a Lie algebra valuedfun tion, in(cid:28)nitely di(cid:27)erentiable and of ompa t support ( g a ∈ D ( R ) ).All gauge transformations of ompa t support in a neighborhood of theidentity, in the C ∞ topology, are of this form; they generate the Gausssubgroup G of the gauge group G .For dis ussing the Weyl quantization one has to onsider the ex-ponential (cid:28)eld algebra F W generated by the unitary operators W ( f ) , f i = P a f ia ( x ) T a , f ia ∈ S ( R ) , formally the exponentials e i A ( f ) , and byunitary operators V ( U λ ) , representing G , formally the exponentials ofthe Gauss operators, V ( U λ ) = e i λ G ( g ) , G ( g ) = X a G a ( g a ) , g a ∈ D ( R ) , transforming ovariantly under spa e translations. Their time indepen-den e formally follows if the dynami s α t is generated by lo al gaugeinvariant Hamiltonians H R , so that ( d/dt ) α t ( V ( U λ )) = i lim R →∞ [ H R , V ( U λ ) ] = 0 . (2.8)The observable (cid:28)eld subalgebra of F W is hara terized by its point-wise invarian e under gauge transformations.A representation of F W also de(cid:28)nes a representation of F only if itis regular, i.e. if (the representatives of) the (cid:28)eld exponentials W ( λ f ) , λ ∈ R de(cid:28)ne weakly ontinuous one-parameter groups.A state ω on the exponential (cid:28)eld algebra F W , in parti ular a va -uum state, is said to satisfy the Gauss law in exponential form, if ω ( V ( U λ )) = 1 , equivalently if its representative ve tor Ψ ω in the GNSHilbert spa e H ω (de(cid:28)ned by the expe tations of F W on ω ) satis(cid:28)es V ( U λ ) Ψ ω = Ψ ω , ∀ U λ . (2.9)Brie(cid:29)y, a ve tor state Ψ satisfying eq. (2.9) is said to be Gauss invari-ant. An operator in H is Gauss invariant if it ommutes with all the V ( U λ ) .In the following we shall onsider the realization of the temporalgauge de(cid:28)ned by a va uum state ω satisfying the Gauss law. As inother interesting quantum me hani al models, like the ele tron in aperiodi potential (Blo h ele tron), the quantum parti le on a ir le,the Quantum Hall ele tron, et ., the invarian e of the ground stateunder a group of gauge transformations implies that the orrespondingrepresentation of the exponential (cid:28)eld algebra is not regular. [11℄Proposition 2.1 A va uum state ω on the exponential (cid:28)eld algebra F W , satisfying the Gauss law, de(cid:28)nes a non-regular representation of F W , sin e : ω ( W ( f i )) = 0 , if f i ( x ) = 0 , (2.10) = 1 , if f = 0 . The (cid:28)elds A , formally the generators of the W ( f ) , annot be de(cid:28)nedin the GNS Hilbert spa e de(cid:28)ned by the va uum expe tations and inparti ular the two point fun tion of the gauge potential does not exist,only (the va uum expe tations of ) the exponential fun tions (and of ourse the gauge invariant fun tions) of A an be de(cid:28)ned.In the free ase, i.e. for vanishing gauge oupling onstant, the ex-ponential (cid:28)eld algebra be omes a Weyl (cid:28)eld algebra, generated by theexponentials of A a and of its onjugate momenta E a , and eqs. (2.10)uniquely determine its representation as a non-regular Weyl quan-tization.Proof. For ea h f i there is a one-parameter subgroup U λ su h that U λ f i U − λ = f i , and exp iD U λ ( f i ) ≡ exp ( i U λ ∂ U − λ ( f i )) = 1 ; therefore,by eq. (2.9), one has ω ( W ( f i ) = ω ( V ( U λ ) W ( f i ) V ( U λ ) ∗ ) = e iD U λ ( f i ) ω ( W ( f i )) , and eqs. (2.10) follow.Clearly, the one-parameter groups de(cid:28)ned by W ( f ) annot be weakly ontinuous and therefore the orresponding generators, i.e. the (cid:28)elds A ia ( f ) do not exist as operators in the GNS Hilbert spa e de(cid:28)ned bythe expe tations of F W on ω . The free ase an be worked out alongthe same lines as for the abelian ase. [9℄The Hilbert spa e H of the representation of F W de(cid:28)ned by theva uum state ω , satisfying the Gauss law, ontains a subspa e H ′ ofGauss invariant ve tors V ( U λ ) Ψ = Ψ , ∀ Ψ ∈ H ′ , ∀ U λ . We denote by π the representation of G in H given by the V ( U λ ) .It is worthwhile to remark that the lo al operator v ( U λ ) ≡ V ( U λ ) − is non zero in H , sin e V ( U λ ) implements the time independent gauge0transformations, orresponding to the one-parameter subgroups, whi hare non-trivial on F W . Thus, the assumptions of the Reeh-S hliedertheorem, a ording to whi h a lo al operator whi h annihilates theva uum must vanish, annot be satis(cid:28)ed.One an easily he k that the ru ial point in the proof of the the-orem fails, namely F W ( O ) Ψ , where F W ( O ) is the exponential (cid:28)eldalgebra lo alized in the region O , is not dense in H . In fa t, for any O disjoint from C U λ , by lo ality one has ( v ( U λ ) F W Ψ , F W ( O ) Ψ ) = ( F W Ψ , F W ( O ) v ( U λ ) ∗ Ψ ) = 0 ,v ( U λ ) F W Ψ = 0 . A tually, the Reeh-S hlieder theorem does not apply be ause the rela-tivisti spe tral ondition fails in H . In fa t, the implementers U ( a ) ofthe spa e translations are not weakly ontinuous, sin e, as in the proofof Proposition 2.1, ω ( W ( f ) U ( a ) W ( − f )) = ω ( W ( f ) W ( − f a )) == e iD U λ ( f − f a ) ω ( W ( f ) W ( − f a )) , so that the right hand side vanishes if a = 0 and it is = 1 , otherwise.The spe tral ondition for the Fourier transforms of the matrix elementsof U ( a , t ) = U ( a ) U ( t ) is violated if there is strong ontinuity in t , sin eit would imply that, after smearing in time, the Fourier transform in a is (cid:28)nite measure and therefore ontinuity in a of the above matrixelements.It is worthwhile to remark that the one-parameter groups V ( U λ ) arenot assumed to be weakly ontinuous in λ ; a tually, ontinuity annothold if the global gauge group is simple and has rank at least two(as in the ase of olor SU (3) ), sin e then one obtains the vanishing of ω ( W ( f ) V ( U λ ) W ( − f )) , for λ = 0 , f ic ( x ) = δ ca f i ( x ) , U λ f i U − λ = f i ( x ) T b , [ T a , T b ] = 0 , from the invarian e of ω under the subgroup generatedby T a , as in the proof of Proposition 2.1. Thus, in this ase the Gausslaw onstraint an only be imposed in the exponential form.13 Topology of the gauge groupBy de(cid:28)nition, eq. (2.7), we onsider lo alized gauge fun tions; they ob-viously extend to the one point ompa ti(cid:28) ation of R , ˙ R , whi h isisomorphi to the three-sphere S , U ( x ) : ˙ R ∼ S → G . Su h maps fall into disjoint homotopy lasses labeled by (cid:16)winding(cid:17)numbers nn ( U ) = (24 π ) − Z d x ε ijk Tr ( U i ( x ) U j ( x ) U k ( x )) ≡ Z d x n U ( x ) , where U i ( x ) ≡ U ( x ) − ∂ i U ( x ) .The gauge transformations with n = 0 are alled large gauge trans-formations. Those with zero winding number are alled small; sin ethey are ontra tible to the identity, they are produ ts of U ( x ) whi hare lose to the identity (in the C ∞ topology) and therefore are ex-pressible as produ ts of U λ , i.e. they are elements of G . Clearly, allelements of G have zero winding number.The following analysis may be also applied to gauge transformations U ( x ) whi h are only required to have a limit for | x | → ∞ , sin e theyare of ompa t support modulo Gauss transformations U λ with su ha behaviour; only the existen e of the orresponding Gauss operators V ( U λ ) is required, with no additional impli ations.In the above realization of the temporal gauge, the small gaugetransformations are implemented by the unitary operators V ( U λ ) ∈F W ; the next question is the implementability of the large gauge trans-formations and their distin tion from the small on the Gauss invariantstates. In fa t, the non-triviality of the large gauge transformations onthe physi al spa e turns out to be a rather subtle question as displayedby the following Proposition.Proposition 3.1 A lo al (cid:28)eld operator invariant under Gauss gaugetransformations is also invariant under large gauge transformations.Any ve tor Ψ ∈ H ′ , in parti ular the va uum ve tor, de(cid:28)nes a state,i.e. expe tations, on F W invariant also under the large gauge transfor-mations.2Proof. In fa t, given a gauge fun tion U n ( x ) and its spa e translatedby a , U an ( x ) = U n ( x − a ) , the ombined gauge transformations α − U an α U n and α U n α − U an have zero winding number and therefore are small gaugetransformations, say α U and α U ′ respe tively.Then, for any lo al (cid:28)eld operator F invariant under small gauge trans-formations one has α U n ( F ) = α U an α U ( F ) = α U an ( F ) , and for | a | su(cid:30) iently large, by lo ality α U an ( F ) = F .Quite generally, for any (lo al) operator F ∈ F W one has, for | a | su(cid:30)- iently large, α U n ( F ) = α U ′ α U an ( F ) = α U ′ ( F ); by the Gauss invarian e of ω , this implies ω ( α U n ( F )) = ω ( F ) . By a standard argument, the invarian e of the va uum under thelarge gauge transformations implies that the gauge transformations areimplemented by unitary operators; they an be hosen to represent thegauge group G , to oin ide the V ( U λ ) , for all Gauss transformations,and to transform ovariantly under spa e translations.Furthermore, if, as we assume, the dynami s is generated by lo algauge invariant Hamiltonians, the implementers ommute with the timetranslations. The implementers are unique, up to phases, if the (cid:28)eldalgebra is irredu ible in H ; in this ase the implementers of the largegauge transformations are multiples of the identity in H ′ .The above results indi ate that the distin tion between the smalland the large transformations at the level of physi al states is problem-ati and a ru ial question is the impli ation, if any, of the non-trivialtopology of the gauge group on the physi al states, more generally onthe representations of the observable algebra.One of the standard (and histori ally the (cid:28)rst) arguments in favorof a non-trivial role of the topology of the gauge transformations re-lies on semi lassi al approximations, in terms of boundary onditionsof the lassi al (cid:28)eld on(cid:28)gurations and their e(cid:27)e t on the (eu lidean)fun tional integral. This approa h has been ex ellently reviewed byColeman, [5℄ who also pointed out the limitation of su h arguments,sin e smooth (cid:28)eld on(cid:28)gurations, in parti ular those of (cid:28)nite a tion,have zero fun tional measure.3The relevan e of the lassi(cid:28) ation of the (smooth) (cid:28)eld on(cid:28)gura-tions in terms of their pure gauge behaviour at spa e in(cid:28)nity, and of the orresponding winding number, is therefore a non-trivial mathemati alproblem, espe ially in the in(cid:28)nite volume limit. The topologi al las-si(cid:28) ation is done in (cid:28)nite volume, but in the in(cid:28)nite volume limit anon-trivial instanton density implies, in the dilute gas approximation,that the fun tional measure is on entrated on on(cid:28)gurations with di-vergent topologi al number. The asso iation of the non-trivial topologywith the existen e of instanton solutions, whi h minimize the lassi ala tion and as su h do not have ompa t support, is probably the reasonwhy lo alized large gauge transformations have not been onsidered inthe literature.As we shall see below, the presen e of fermions plays a ru ial rolefor the non-trivial e(cid:27)e ts of the topology of the gauge group.Another argument for the physi al onsequen es of the gauge grouptopology has been proposed in terms of the topologi al harge. [7℄ Theso- alled topologi al urrent is formally de(cid:28)ned by C µ ( x ) = − (16 π ) − ε µνρσ Tr ( F νρ ( x ) A σ ( x ) − A ν ( x ) A ρ ( x ) A σ ( x )) . (3.1) ∂ µ C µ ( x ) = − (16 π ) − Tr ∗ F µν ( x ) F µν ( x ) ≡ P , where A µ = (0 , A i ) , ∗ F µν ≡ ε µνρσ F ρσ . In the mathemati al literature,for lassi al (cid:28)elds, P is alled the (cid:16)Pontryagin density(cid:17) and C µ the(cid:16)Chern-Simons se ondary hara teristi lass(cid:17).At the lassi al level, one has [7℄ the following transformation lawof C ( x ) under gauge transformations α U , de(cid:28)ned by eqs. (2.7), α U ( C ( x )) = C ( x ) − (8 π ) − ∂ i [ ε ijk Tr ( ∂ j U ( x ) U ( x ) − A k ( x ))] + n U ( x ) . (3.2)Therefore, at the lassi al level the spa e integral of C ( x , x ) is invari-ant under small transformations, but it get shifted by n under gaugetransformations with winding number n .For the quantum ase one meets non-trivial onsisten y problems.First of all the formal expression in the right hand side of eq. (3.3)requires a point splitting regularization. It is reasonable to assumethat this an be done by keeping the transformation properties of theformal expression under large gauge transformations, eq. (3.4).4The next problem is the spa e integral of C ( x ) . The spa e integralsof harge densities, even for onserved urrents, are known to divergeand suitable regularizations are needed, in luding a time smearing (seeeq. (3.5) below). In the ase of onserved urrents, under some general onditions one may obtain the onvergen e of a suitably regularizedintegral of the harge density, in matrix elements on states with suitablelo alization; [12℄ [13℄ [14℄ [15℄ but in the general ase the problem seemsto be open.A tually, in the quantum temporal gauge an even more serious prob-lem arises by the gauge dependen e of C µ ( x ) . By Proposition 2.1, theregularized spa e integral of C ( x ) C ( f R α R ) ≡ Z d x f R ( x ) α R ( x ) C ( x ) , (3.3)where f R ( x ) = f ( | x | /R ) , f ( x ) = 1 , for | x | ≤ , = 0 , for | x | ≥ ε , α R ( x ) = α ( x /R ) /R , R dx α ( x ) = ˜ α (0) = 1 , annot exist as an op-erator in H , only its exponential V C ( f R α R ) , formally exp [ i C ( f R α R )] ,may be de(cid:28)ned. Furthermore, as shown by the following Proposition,su h exponentials have vanishing expe tation on Gauss invariant states,i.e. their restri tion to the physi al states vanishes.Proposition 3.2 The operators V C ( λf R α R ) , formally the exponentials exp [ iλ C ( f R α R )] of the regularized spa e integrals C ( f R α R ) of C ( x ) ,and therefore assumed to transform under gauge transformations assu h exponentials, annot be weakly ontinuous in λ and therefore the(cid:28)eld C ( f R α R ) annot be de(cid:28)ned.Furthermore, the V C ( f R α R ) satisfy, for all Gauss invariant ve tors Ψ , Φ , (Ψ , V C ( f R α R ) Φ) = 0 . (3.4)Proof. In fa t, if C ( f ) , f ∈ D ( R ) , exists, by using the Gauss gaugeinvarian e of the va uum state ω , the vanishing of ω ( A k ) by rotationalinvarian e and eq. (3.4), one has ω ( C ( f )) = ω ( V ( U λ )) C ( f ) V ( U λ ) − ) == ω ( C ( f )) + Z d xf ( x , t ) n U λ ( x ) . f there is at least one U λ ( x ) su h that the last term onthe right hand side does not vanish, one gets a ontradi tion. Thus,only the exponential of C ( f ) an be de(cid:28)ned.Moreover, given f R one an (cid:28)nd a small gauge transformation U ( x ) ,with U ( x ) = U ( x ) U ( x ) , n U + n U = 0 , n U = 0 ,f R U = , f R U = U . Then, ∂ i f R ∂ j U = 0 and the se ond term on the right hand side ofeq. (3.4) vanishes; furthermore R d x n U ( x ) f R ( x ) = n U . Hen e, onehas (Ψ , V C ( f R α R ) Φ) = (Ψ , V ( U ) V C ( f R α R ) V ( U ) − Φ) == e in U (Ψ , V C ( f R α R ) Φ) and eq. (3.4) follows.In on lusion, one annot dire tly exploit the non-invarian e of C ( x ) for proving a non-trivial a tion of the large gauge transforma-tions in H ′ ; it is essential to take into a ount the non regularity of exp i C ( f R α R ) , its non observability and the non-existen e of the limit R → ∞ .4 Chiral symmetry and solution of the U (1) problemThe situation hanges substantially in the presen e of massless fer-mions, sin e the role of the topologi al urrent is taken by a onserved urrent; hen e, there is a symmetry asso iated to it and the ru ialpoint is its relations with the implementers of the large gauge transfor-mations.In this ase, the Lagrangean, eq. (2.1) gets modi(cid:28)ed by the additionof the (gauge invariant) fermion Lagrangean and the Gauss operatorsbe ome G a = ( D · E ) a − j a , j aµ = ig ¯ ψγ µ t a ψ. The time independent gauge transformations of the fermion (cid:28)elds inthe fundamental representation of the gauge group are α U ( ψ ( x )) = U ( x ) ψ ( x ) . β λ , λ ∈ R , β λ ( ψ ) = e λγ ψ, β λ ( ¯ ψ ) = ¯ ψ e λγ , γ ∗ = − γ , β λ ( A ) = A . Correspondingly, there is a onserved urrent j µ = ig ¯ ψγ γ µ ψ , the gaugeinvariant fermion axial urrent.In the quantum ase, a gauge invariant point splitting regularizationis needed for the de(cid:28)nition of j µ and this inevitably leads to an anomaly, ∂ µ j µ = − ∂ µ C µ = − P . The onserved axial urrent is now the gauge dependent urrent J µ ( x ) = j µ ( x ) + 2 C µ , its onservation being equivalent to the anomaly equation for j µ .For the dis ussion of the Weyl quantization, we take as lo al expo-nential (cid:28)eld algebra F W the algebra generated by the operators W ( f ) ,by the gauge invariant bilinear fun tions of the fermion (cid:28)elds and bythe unitary operators V ( f ) , f ∈ S ( R ) , formally the exponential of J ( f ) .As shown by Bardeen [8℄ on the basis of perturbative renormaliza-tion in lo al gauges, the above (time independent) hiral transforma-tions of the fermion (cid:28)elds are generated by the quantum (cid:28)eld operator J µ ( x ) and not by the gauge invariant non onserved urrent j µ ; the ontinuity equation of J µ plays a ru ial role in Bardeen analysis.This justi(cid:28)es our assumption that the exponential (cid:28)eld algebra on-tains the (formally de(cid:28)ned) exponential of the smeared (cid:28)eld J µ and inparti ular of the regularized integral of the harge density J ( f R α R ) ;the orresponding one-parameter group of unitary operators is denotedby V R ( λ ) = V ( λ f R α R ) .With the same motivations given before, eqs. (2.7) are assumed tohold together with the following transformation law of V R ( λ ) undergauge transformations: for R large enough so that f R ( x ) = 1 on thelo alization region of U n ( x ) , one has α U n ( V R ( λ )) = e i n λ V R ( λ ) . (4.1)7This relation formally re(cid:29)e ts the transformation properties of C ( f ) and the gauge invarian e of j . By the proof of Proposition 3.3, V R ( λ ) is not weakly ontinuous in λ and its formal generator J ( f R α R ) doesnot exist. However, V R ( λ ) a t as lo al implementers of β λ : in fa t,sin e J µ is onserved, most of the standard wisdom is available [16℄ andone has lim R →∞ V R ( λ ) F V R ( − λ ) = β λ ( F ) , ∀ F ∈ F W . (4.2)It is important to stress that, thanks to lo ality, the above limitis rea hed for (cid:28)nite values of R , and that it preserves lo ality andgauge invarian e. Thus, ontrary to what is stated in the literature, thepresen e of the hiral anomaly does not prevent the hiral symmetryfrom being a well de(cid:28)ned time independent automorphism of the (cid:28)eldalgebra F W of its gauge invariant (observable) subalgebra.The loss of hiral symmetry is therefore a genuine phenomenon ofspontaneous symmetry breaking and the onfrontation with the Gold-stone theorem be omes a ru ial issue, the so- alled U ( ) problem.The absen e of parity doublets requires that the hiral symmetrybe broken and the U (1) problem amounts to explaining the absen e ofthe orresponding Goldstone massless bosons.As dis ussed above, one of the basi assumptions of the Goldstonetheorem, namely the existen e of an automorphism of the algebra ofobservables, whi h ommutes with spa e and time translations is satis-(cid:28)ed.The se ond ru ial property, needed the proof of the theorem, isthe lo al generation of the symmetry by a onserved urrent at least inexpe tations on the va uum state, i.e. ddλ < β λ ( A ) > λ =0 = < δ ( A ) > = i lim R →∞ < [ J ( f R α R ) , A ] > . (4.3)Sin e the hiral automorphism β λ is C ∞ in λ its generator δ is wellde(cid:28)ned, but the problem is its relation with the formal generator of theunitary one-parameter group de(cid:28)ned by the V R ( λ ) .As a matter of fa t, even if β λ an be des ribed by the a tion of thelo al operators V R ( λ ) , eq. (4.2), the non-regularity of the one-parameterunitary group V R ( λ ) , prevents the existen e of the orresponding gener-ator J ( f R α R ) , so that one annot write the symmetry breaking Wardidentities and obtain the Goldstone energy-momentum spe trum.8Quite generally, one hasProposition 4.1 If ω is a gauge invariant va uum state and A is anobservable symmetry breaking order parameter, i.e. ω ( β λ ( A )) = ω ( A ) = 0 , (the standard andidate being ¯ ψψ ) then the expe tations ω ( J ( f R α R ) A ) annot be de(cid:28)ned and eq. (4.3) does not hold.Proof. In fa t, otherwise, for R su(cid:30) iently large, one would have ω ( J ( f R α R ) A ) = ω ( α U n ( J ( f R α R ) A )) = ω ( J ( f R α R ) A ) + 2 n ω ( A ) , (4.4)i.e. a ontradi tion.Clearly, by Proposition 3.1, the above Proposition applies to theGauss invariant state Ψ . The impossibility of writing expe tationsinvolving J µ on a gauge invariant va uum state, solves the problemsraised by R.J. Crewther in his analysis of hiral Ward identities. [17℄It is worthwhile to remark that, for the evasion of the Goldstone the-orem dis ussed above, the o urren e of the so- alled hiral anomaly(whi h is present also in the abelian ase) is not enough; the ru ial in-gredient is eq. (4.1), whi h dire tly implies the non-regularity of theunitary operators V R ( λ ) and the non-existen e of the lo al harges J ( f R α R ) in expe tations on a gauge invariant va uum state.5 Topology, hiral symmetry breaking andva uum stru tureThe impli ations of the non-trivial topology of the gauge group on thebreaking of the hiral symmetry ru ially involve the hiral transforma-tions of the implementers of gauge group. Su h hiral transformationsdo not follow merely from eq. (4.1), be ause eq. (4.2) applies to the ele-ments of the lo al (cid:28)eld algebra and its extension to implementers, evenif they are strong limits of elements of F W , is not uniquely de(cid:28)ned.This problem does not arise if there are implementers V ( U n ) belongingto the lo al (cid:28)eld algebra.9Under this assumption, we shall (cid:28)rst prove that the hiral symme-try is broken in any irredu ible (or fa torial) representation of the (cid:28)eldalgebra, as well as in any fa torial representation of the observable alge-bra in the Gauss invariant subspa e H ′ (we re all that a representationis fa torial if the enter of the strong losure onsists of multiples of theidentity there).We shall later show that the standard va uum stru ture is obtainedby analyzing the de omposition of H ′ into fa torial representation of A ( θ se tors), in the ase of a redu ible representation of the (cid:28)eldalgebra de(cid:28)ned by a hiral invariant va uum; a ru ial ingredient for thederivation are the non-trivial hiral transformations of the implementers V ( U n ) , whi h uniquely follow from eqs. (4.1), (4.2), sin e they belongto the (cid:28)eld algebra.The lo al stru ture of the gauge transformations imply that theimplementers V ( U ) ommute with the (cid:28)elds lo alized in spa etime re-gions O disjoint from the spa etime lo alization region C U of U ; thusthe V ( U ) are lo al with respe t to the (cid:28)eld algebra, with lo alizationregion of V ( U ) given by C U . This lo ality property partly motivate theassumption of existen e of lo al implementers, in the following sense.As it is standard, in the following the lo al (cid:28)eld algebras F W ( O ) and their gauge invariant ((cid:16)observable(cid:17)) subalgebras A ( O ) are taken asstrongly losed; F W and A will denote their unions over O . For anybounded region O , the lo al algebra F W ( O ) may be identi(cid:28)ed withthe strong losure of the polynomial algebra generated by the (cid:28)eldexponentials W ( f ) , by the gauge invariant bilinear fun tions of thefermion (cid:28)elds and by the unitary operators V ( f ) ; as it is standard,the enter of F W ( O ) is assumed to be trivial.Then, we assume that if supp U is lo alized in O , the gauge trans-formation α U may be implemented by a unitary operator V ( U ) ∈F W ( O ) . [18℄ Sin e the enter of F W ( O ) is trivial, su h a lo al imple-menter is unique; in parti ular, for Gauss trasformations U λ it redu esto the previously introdu ed Gauss operators.Then, we haveProposition 5.1 The lo al implementers V ( U ) of the gauge transfor-mations are of the form V ( U ) F Ψ = α U ( F ) C U Ψ , where C U om-mutes with F W and belongs to the strong losure of F W . C U n depends0only on n and on the Gauss invariant ve tors one has V ( U n ) Ψ = C n Ψ , ∀ Ψ ∈ H ′ . (5.1)The algebra generated by the C n is abelian and one has C n C m = C n + m . (5.2)Furthermore, one hasi) for any O , there exist lo al operators C n ( O ) belonging to the lo alobservable algebra A ( O ) , satisfying [ C n ( O ) , A ] = 0 and C n ( O ) C m ( O ) = C n + m ( O ) , C n ( O ) Ψ = C n Ψ , ∀ Ψ ∈ H ′ , ∀O , (5.3)ii) s − lim O→ R C n ( O ) = C n P , P H = H ′ . Proof. In fa t, if F ∈ F W ( O ) , given α U n , the transformation α U n α − U an ,with U an ( x ) ≡ U ( x − a ) , is a Gauss gauge transformation and thereforeit is implemented by a produ t V ( a ) of Gauss operators V ( a ) ( U i ) , whi hleave the va uum ve tor invariant. Then, the following strong limitexists and de(cid:28)nes an operator S ( U n ) ≡ s − lim | a |→∞ V ( a ) , whi h satis(cid:28)es S ( U n ) F S ( U n ) − = α U n ( F ) , ∀ F ∈ F W , S ( U n ) Ψ = Ψ . Clearly, C U n ≡ V ( U n ) S ( U n ) − ommutes with F W , belongs to thestrong losure of F W and gives V ( U n ) F Ψ = α U n ( F ) C U n Ψ .Sin e C U n ommutes with F W , they are ompletely hara terized bytheir a tion on Ψ , where they only depend on the topologi al num-ber n , ommute with spa e time translations and satisfy eq. (5.2), be- ause V ( U ′ n ) − V ( U n ) , and in parti ular V ( U an ) − V ( U n ) , are produ tsof Gauss operators, whi h leave Ψ invariant.Any open set ontains an open ylinder O , with base O , O = O × ( t , t ) , we denote by H ′ ( O ) the subspa e of ve tors invariant underall the Gauss operators V ( U λ ) , with supp U λ ⊂ O and by P ( O ) F W ( O ) . Then, as in the proof of Proposition 3.2, one has that ∀ U , V ( U λ ) V ( U ) = V ( U ) V (( U ′ ) λ ) , with supp ( U ′ ) λ = supp U λ ; therefore [ V ( U ) , P ( O ) ] = 0 . (5.4)The operators C n ( O ) ≡ V ( U n ) P ( O ) , with supp U n ⊂ O , only dependon the topologi al number n , sin e gauge transformation U n , U ′ n of thesame homotopi lass supported in O di(cid:27)er by a produ t U G of Gausstransformations with the same support and V ( U G ) P ( O ) = P ( O ) . Thisimplies the (cid:28)rst of eqs. (5.3).Furthermore, sin e ∀ U , V ( U ) V ( U n ) = V ( U ′ n ) V ( U ) , supp U ′ n ⊂ O , byeq. (5.4), one has V ( U ) C n ( O ) = C n ( O ) V ( U ) , i.e. C n ( O ) are gauge invariant and therefore belong to A ( O ) . Clearly,by onstru tion C n ( O ) ommutes with A .Sin e P ( O )Ψ = Ψ , ∀ Ψ ∈ H ′ , and ∩ O H ′ ( O ) = H ′ , the remainingstatements follow.The unitary operators C n are related to the topology of the gaugegroup, but they do not implement the (large) gauge transformationson the (cid:28)eld algebra; they provide a unitary representation of the gaugegroup modulo the subgroup of Gauss transformations, through opera-tors in the ommutant, a tually in the enter, of F W .The important feature of the operators C n ( O ) is that of providinga representation of the group of gauge transformations lo alized in O ,modulo Gauss gauge transformations, through lo al operators whi hbelong to the enter Z ( O ) of the observable algebra lo alized in O .Thus, the non-trivial topology of the gauge group is re(cid:29)e ted by a non-trivial enter of the lo al observable algebras A ( O ) .The above stru ture implies that the non-trivial topology of thegauge group and eq. (4.1) for e the breaking of hiral symmetry.Proposition 5.2 Chiral symmetry β λ is broken in any irredu ible (orfa torial) representation of the (cid:28)eld algebra F W , de(cid:28)ned by a Gaussinvariant va uum, as well as in any fa torial (sub-)representation ofthe lo al observable algebra A in the Gauss invariant subspa e H ′ .2Proof. If the hiral symmetry β λ is unbroken in H , then there is aone-parameter group of unitary operators U ( λ ) , λ ∈ R satisfying β λ ( F ) = U ( λ ) F U ( − λ ) , ∀ F ∈ F W , U ( λ ) Ψ = Ψ . (5.5)Su h an a tion of U ( λ ) provides the unique strongly ontinuous exten-sion of β λ to ¯ F W . Therefore, sin e the operators S ( U n ) are strong limitsof Gauss operators, whi h are invariant under hiral transformations,by eqs. (4.1), (4.2), one has β λ ( S ( U n )) = S ( U n ) . On the other hand, by the lo alization of the V ( U n ) , eq. (4.2) appliesand then eq. (4.1) (for R su(cid:30) iently large) gives U ( λ ) V ( U n ) U ( − λ ) = β λ ( V ( U n )) = e i nλ V ( U n ); (5.6)therefore β λ ( C n ) = U ( λ ) C n U ( − λ ) = e i nλ C n . (5.7)This is in onsistent with an irredu ible (or fa torial) representation of F W , where the C n are multiples if the identity.Similarly, as a onsequen e of eqs. (4.1), (4.2), one has β λ ( P ) = U ( λ ) P U ( − λ ) = P and β λ ( C n P ) = e i nλ C n P . (5.8)Sin e C n P is the limit of elements of the (strongly losed) lo al ob-servable algebras A ( O ) , it belongs to the enter of the representation of A in H ′ ; this implies the instability of any fa torial subrepresentationof A , under β λ .The link between the non-trivial topology of the gauge group andthe labeling of the fa torial representations of the lo al observable al-gebra ( θ se tors) is learly displayed in a (redu ible) representation ofthe (cid:28)eld algebra de(cid:28)ned by a hirally invariant va uum state. Su h aninvarian e arises in an analysis based on the fun tional integral formula-tion and semi lassi al onsiderations, [4℄ [5℄ as well as in rigorous treat-ments of soluble models (in primis the S hwinger model [19℄ [20℄ [21℄).3In general, one obtains hirally invariant orrelation fun tions by us-ing hirally invariant boundary onditions in the fun tional integral in(cid:28)nite volume. [22℄ [1℄We shall therefore onsider the ase in whi h hiral symmetry is im-plemented in H by a one-parameter group of unitary operators U ( λ ) ,i.e. eqs. (5.5) hold.Proposition 5.3 Under the above assumptions the fa torial subrep-resentations, π θ , of A in H ′ are labeled by an angle θ ( θ se tors): π θ ( C n ) = e i nθ , θ ∈ [0 , π ) , (the orresponding groundstates are alled θ va ua).Proof. By eq. (5.2), C n = C n and by eq. (5.7) the spe trum of C is σ ( C ) = { e i θ ; θ ∈ [ 0 , π ) } .This is also the spe trum of the operator C P in H ′ , by eq. (5.8).The Hilbert spa e H ′ has a entral de omposition over the spe trum of C P in H ′ . Thus, one has H ′ = Z θ ∈ [0 , π ) dθ H θ , C n H θ = e i nθ H θ , sin e the spe tral measure an be taken invariant under translationsby eq. (5.7) and by the hiral invarian e of the va uum. By eq. (5.7), U ( α ) intertwines between the se tors U ( α ) H θ = H θ + α (5.9)and satis(cid:28)es U ( π ) = ( − F , with F the fermion number ( = 0 , byour de(cid:28)nition of F W ). Sin e the hiral symmetry ommutes with timetranslations, the spe trum of the Hamiltonian is the same in all θ se torsand (cid:16)all the θ va ua have the same energy(cid:17). The same de ompositionapplies to the representation of F W in H .Su h a pi ture is exa tly the same as in the quantum me hani almodel of QCD stru tures dis ussed in Ref. [22℄Equation (5.6) provides a orre t derivation of the equation [ V ( U n ) , Q ] = 2 n, (5.10)whi h is at the basis of most of the standard dis ussions of hiral sym-metry breaking in QCD. The standard derivation assumes that the4gauge transformations of the axial harge density extend to its spa eintegral, giving the transformation properties of the hiral harge Q .Our derivation of the relation between hiral symmetry and large gaugetransformations, eq. (5.6), does neither require the (usually assumed) onvergen e of the spa e integral of J to Q , nor that of its exponential,whi h are in ompatible with Proposition 3.3.It is worthwhile to stress that the breaking of hiral symmetry isgoverned by quite a di(cid:27)erent me hanism with respe t to the Goldstoneor the Higgs me hanism. In all the three ases the symmmetry om-mutes with spa etime translations. However, in the Goldstone ase,the symmetry breaking order parameter, typi ally an observable oper-ator, has strong enough lo alization properties (preserved under timeevolution) and its transformations under the symmetry are generatedby a lo al onserved urrent. In the Higgs ase, in positive gauges likee.g. the Coulomb gauge, the symmetry breaking order parameter is notan observable and it has a non-lo al time evolution, so that the (timeindependent) symmetry is not generated at all times by the asso iated onserved lo al Noether urrent.In the axial U (1) ase of QCD, ontrary to statements appearedin the literature, the hiral transformations de(cid:28)ne a time independentsymmetry of the observables. The Goldstone theorem, i.e. the presen eof asso iated massless Goldstone bosons, is evaded by the impossibil-ity of writing the orresponding symmetry breaking Ward identities,sin e the asso iated onserved Noether urrent does not exist, only itsexponentials do (non-regular representation of the (cid:28)eld algebra).A tually, the hiral symmetry annot be lo ally generated by uni-tary operators in any fa torial representation of the observable algebra,be ause the lo al observable algebras have a enter whi h is not leftpointwise invariant under the hiral symmetry.In on lusion, the non-regular Weyl quantization provides a strat-egy for putting the derivation of the va uum stru ture and the hiralsymmetry breaking in the temporal gauge of QCD in a more a eptableand onvin ing mathemati al setting.56 Regular temporal gaugeAs dis ussed in the abelian ase, one may look for an alternative realiza-tion of the temporal gauge, by weakening the ondition of Gauss gaugeinvarian e of the va uum, so that the orresponding orrelation fun -tions of gauge dependent (cid:28)elds and not only those of their exponentialsmay be de(cid:28)ned.To be more pre ise, as before, one introdu es a lo al (cid:28)eld algebra F , generated by A ( f ) , f ia ∈ S ( R ) , by the fermion (cid:28)elds, by theirgauge invariant bilinears, by the axial urrent J µ , and by lo al operators V ( U ) , whi h implement the time independent gauge transformations α U , eq. (2.7), represent the the group G and satisfy α U n ( J ( f R α R )) = J ( f R α R ) + 2 n. (6.1)We denote by A the gauge invariant (observable) subalgebra of F andby V G a generi monomial of the Gauss operators V ( U λ ) .A regular quantization of the temporal gauge is de(cid:28)ned by a(linear hermitian) va uum fun tional ω on F , whi h is invariant underspa e-time translations and rotations and su h that its restri tion tothe observable algebra A satis(cid:28)es positivity, Lorentz invarian e and therelativisti spe tral ondition.>From a onstru tive point of view, su h a realization of the tem-poral gauge may be related to a fun tional integral quantization witha fun tional measure given by the Lagrangean of eq. (2.1) with theaddition of the fermioni part (see Se tion 4). The invarian e of theLagrangean with respe t to the residual gauge group after the gauge(cid:28)xing A = 0 , does not imply the orresponding residual gauge in-varian e of the orrelation fun tions of F , as dis ussed in the abelian ase, [9℄ [23℄ sin e an infrared regularization is needed whi h breaksthe residual gauge invarian e. Therefore, the Gauss onstraint does nothold anymore.The orrelation fun tions of F given by an ω with the above prop-erties de(cid:28)ne a ve tor spa e D = F Ψ , with Ψ the ve tor representing ω , and an inner produ t on it < . , . > , whi h is assumed to be leftinvariant by the operators V ( U ) .It is further assumed that ω satis(cid:28)es the following weak Gaussinvarian e: ω ( A V G ) = ω ( A ) , ∀ A ∈ A , ∀ V G , < A Ψ , V G Ψ > = < A Ψ , Ψ >, ∀ A ∈ A , ∀ V G . (6.2)It follows that the ve tors of the the subspa e D ′ ≡ A Ψ are weaklyGauss invariant in the sense of eq. (6.2) and furthermore the spa etime translations U ( a ) leave D ′ invariant. Thus, ω de(cid:28)nes a va uumrepresentations of A in whi h the Gauss law holds.The weak form of Gauss gauge invarian e of the va uum fun tionalallows for the existen e of the (cid:28)elds of F as operators on D , but theinner produ t annot be semide(cid:28)nite on D (by the argument of Propo-sition 2.1). The subspa e of ve tors Ψ ∈ D ′ with null inner produ t, < Ψ , Ψ > = 0 , is denoted by D ′′ .Now, there is a substantial di(cid:27)eren e in the realization of the hi-ral symmetry, with respe t to the representation de(cid:28)ned by a Gaussinvariant va uum. Thanks to the weak form of the Gauss gauge invari-an e, the (smeared) onserved urrent J µ may be de(cid:28)ned as an operatorin D = F Ψ and the standard wisdom applies; in parti ular, for thein(cid:28)nitesimal variation δ F of the (cid:28)elds under hiral transformations,following Bardeen, one has δ A = i lim R →∞ [ J ( f R α R ) , A ] , ∀ A ∈ A . (6.3)In general, the representation π (0) of the observable algebra de(cid:28)ned bythe va uum ve tor Ψ may not be irredu ible and therefore in orderto dis uss the breaking of the hiral symmetry one must de omposeit into irredu ible representations. Even if ω ( δ A ) = 0 , a symmetrybreaking order parameter may appear in the irredu ible omponents of π (0) . Furthermore, su h a de omposition of the va uum fun tional on A does not a priori extend to a de omposition of the va uum expe -tations < Ψ , J ( f R α R ) A Ψ > , sin e J ( f R α R ) is not gauge invariant.Thus, one of the basi assumptions of the Goldstone theorem may failand hiral symmetry breaking may not be a ompanied by masslessGoldstone bosons.More de(cid:28)nite statements an be made under the following reason-able assumption, hereafter referred to as the existen e of lo al imple-menters of the gauge transformations:7i) the subspa e D ′ generated by the ve tors V ( U ) D ′ , with U runningover G , satis(cid:28)es the weak Gauss onstraint and semi-de(cid:28)niteness of theinner produ t,ii) if supp U ⊆ O , then, V ( U ) an be obtained as a (cid:16)weak(cid:17) limit ofpolynomials F n of A ai and ψ lo alized in O , in the following sense < Ψ , V ( U )Φ > = lim n →∞ < Ψ , F n Φ >, ∀ Ψ , Φ ∈ D . (6.4)Property i) is supported by the fa t that the states de(cid:28)ned by theve tors V ( U ) A Ψ , A ∈ A , are weakly Gauss invariant and positive; infa t ∀ A, B, C ∈ A , < A V ( U ) B Ψ , V G V ( U ) C Ψ > = < α U ( A ) B Ψ , V ′ G C Ψ > == < A B Ψ , C Ψ > = < A V ( U ) Ψ , V ( U ) C Ψ > . The stability under V ( U ) of a weakly Gauss invariant subspa e, whi hin ludes D ′ , is automati ally satis(cid:28)ed if su h a subspa e may be sele tedby a gauge ovariant subsidiary ondition. Weak Gauss invarian e of D ′ is also implied by the following stronger form of the weak Gaussinvarian e of the va uum fun tional ω ( A V ( U λ ) V ( U )) = ω ( A V ( U )) , ∀ A ∈ A , ∀U λ , U . (6.5)Property ii) is supported by the lo alization of the gauge transforma-tions so that the V ( U ) are lo al relative to the (cid:28)eld algebra, withlo alization region given by the support of the orresponding gaugetransformation.The (cid:28)elds F whi h leave D ′ invariant also leave the subspa e D ′′ ofnull ve tors of D ′ invariant and therefore de(cid:28)ne unique gauge invariantoperators ˆ F in the (cid:16)physi al(cid:17) quotient spa e D phys ≡ D ′ / D ′′ , whi his the analog of the Gauss invariant subspa e H ′ of the non-regularrealization of the temporal gauge. Thus, to all e(cid:27)e ts su h (cid:28)elds anbe onsidered as observable (cid:28)elds; in the following we shall take asobservable algebra lo alized in O , ˆ A ( O ) , the algebra of operators in D phys generated by (cid:28)elds lo alized in O whi h leave D ′ invariant and asobservable algebra ˆ A ≡ ∪ O ˆ A ( O ) . [24℄ In parti ular, the lo al operators V ( U n ( O )) are weakly gauge invariant and therefore they de(cid:28)ne uniqueoperators ˆ T U n ( O ) ∈ ˆ A ( O ) in D phys .8By the same arguments dis ussed before, the ˆ T U n depend only on n , are invariant under spa e time translations and satisfy ˆ T n ˆ T m = ˆ T n + m , ˆ T = . (6.6)Moreover, sin e one may write V ( U n ) = V ( U an ) V G , for any lo al F ,whi h leaves H ′ invariant, one has ˆ T n ˆ F = ˆ F ˆ T n . (6.7)This implies that the ˆ T n generated an abelian group G T and belong tothe enter Z ( O ) of ˆ A ( O ) , ∀O .Furthermore, the lo al generation of the in(cid:28)nitesimal hiral trans-formations, eq. (6.3), implies weak ontinuity of the derivation δ onthe lo al (cid:28)eld algebras F ( O ) and by property ii), the in(cid:28)nitesimal hi-ral transformations of the lo al implementers V ( U n ) of the large gaugetransformations are determined by eq. (6.1), i.e. < D , δ ( V ( U n ) D > = lim m →∞ < D , δ ( F m ) D > =lim m →∞ i < D , [ J ( f R α R ) , F m )] D > = i < D , [ J ( f R α R ) , V ( U n )] D > == i n < D , V ( U n ) D >, (6.8)for R su(cid:30) iently large so that f R ( x ) = 1 on the lo alization region of U n . Thus, one has δ ( ˆ T n ) = i n ˆ T n , (6.9)and if the hiral symmetry is unitarily implemented in D phys the ob-servable algebra (in D phys ) has a non-trivial enter Z .Proposition 6.1 Under the above general assumptions, one hasi) the non-trivial topology of the gauge group gives rise to a enter of theobservable algebra (in the physi al spa e D phys ), whi h not left pointwiseinvariant under the hiral symmetry,ii) the hiral symmetry is broken in any fa torial representation of theobservable algebra,iii) the de omposition of the physi al Hilbert spa e H phys ≡ D phys overthe spe trum of ˆ T de(cid:28)nes representations of the observable algebra la-beled by an angle θ ∈ [0 , π ) , giving rise to the θ va ua stru ture,9iv) the expe tations ω θ ( J ( f R α R ) A ) , A ∈ A , with ω θ invariant undergauge transformations, annot be de(cid:28)ned and a ru ial ondition of theGoldstone theorem fails.Proof. Most of the arguments are essentially the same as in the non-regular realization. In parti ular, an unbroken hiral symmetry in afa torial representation of the algebra of observable is in ompatiblewith the non-trivial hiral transformations of its enter.By eq.(6.9), the spe trum of ˆ T is { e i θ , θ ∈ [0 , π ) , and, even if J is well de(cid:28)ned as an operator in D , the existen e of the expe tations ω θ ( J ( f R α R ) A ) would lead to the same in onsisten y as in eq. (4.4).7 The S hwinger model in the temporalgaugeThe general features dis ussed above are exa tly reprodu ed by theS hwinger model in the temporal gauge, usually regarded as a prototypeof the non-perturbative QCD stru tures; in parti ular, the assumptionsabout the lo al implementers of the large gauge transformations hold.The bosonized S hwinger model in the temporal gauge is formallydes ribed by the following Lagrangean density L = ( ∂ ϕ ) − ( ∂ ϕ ) + ∂ ϕA + ( ∂ A ) , (7.1)where ϕ is the pseudos alar (cid:28)eld whi h bosonizes the fermion bilin-ears and therefore is an angular variable, and A is the gauge ve torpotential.The time evolution is formally determined by the following anoni alequations π = ∂ ϕ + A , ∂ A = E, ∂ ϕ = ∆ ϕ, ∂ E = ∂ ϕ. (7.2)1) Representation by a Gauss invariant va uumAs exponential (cid:28)eld algebra, we take the algebra generated by the uni-tary operators V ϕ ( f ) , Z dx f = n, V A ( h ) , V E ( g ) , V π ( g ) , f, g, h, ∈ D ( R ) , (7.3)0formally orresponding to the exponentials e iϕ ( f ) , e iA ( h ) , e iE ( g ) , e iπ ( g ) ,respe tively, and satisfying the Weyl ommutation relations, with aboverestri tion on f , required by the periodi ity of ϕ .The time independent gauge transformations α U ( V A ( h )) = V A ( h + U ∂ U − ) , U ( x ) − ∈ D ( R ) ,α U ( V π ( g )) = V π ( g + U ∂ U ) , (7.4) ϕ and E being left invariant, are generated by the lo al operators V ( U ) ≡ V ϕ ( f ) V E ( − f ) ≡ V ( f ) , f = U ∂ U − , Z dx f = n. (7.5)The gauge fun tions f with R dxf = 0 , i.e. those of the form f = ∂ g,g ∈ D ( R ) , de(cid:28)ne the Gauss transformations and those with R dxf = n , n = 0 , de(cid:28)ne the large gauge transformations and shall be labeled bythe topologi al number n . Clearly, if U is lo alized in O , equivalentlysupp f ⊆ O , then V ( f ) ∈ F W ( O ) , so that our assumption of lo alimplementability is veri(cid:28)ed. The dynami s is de(cid:28)ned by eqs. (7.2) andtherefore e iσ ( f ) ≡ e i ( ϕ − E )( f ) is independent of time.The hiral transformations β λ are de(cid:28)ned by β λ ( V ϕ ( f n )) = e i nλ V ϕ ( f n ) , (7.6)all the other exponential (cid:28)elds being left invariant. Thus, as arguedin general, the anomaly of the gauge invariant axial urrent j µ = ∂ µ ϕ , ∂ µ j µ = ε µ ν ∂ µ A ν does not prevent the hiral symmetry from de(cid:28)ning aone-parameter group of automorphisms of the (exponential) (cid:28)eld alge-bra and of its gauge invariant subalgebra A , lo ally generated by theunitary operators V R ( λ ) ≡ V π ( f R α R ) .The GNS representation of F W by a Gauss invariant state ω is hara terized by a representative ve tor Ψ whi h satis(cid:28)es V ( ∂ g ) Ψ = Ψ , g ∈ D ( R ) . (7.7)The Gauss invarian e of the va uum ve tor is independently requiredby the ondition of positivity of the energy, using the positivity of thestate ω and the invarian e under spa e translations. In fa t, sin e V ( f ) Ψ as an eigenstate of V ( f ) , i.e. V ( f )Ψ = e iλ ( f ) Ψ ; then, by introdu ing V ( t ) ≡ e iα t ( π ( g ) − ∆ A ( g )) = V (0) e it ∆ σ ( g )+ it R dx ∆(1 − ∆) g , one gets ( V (0)Ψ , H V (0)Ψ = i ( d/dt )( V (0)Ψ , V ( t )Ψ ) t =0 == − λ (∆ g ) − Z dx ∆(1 − ∆) g . Therefore, the positivity of the energy ∀ g requires that the fun tional λ (∆ g ) be ≤ , ∀ g , and therefore = 0 , sin e it is linear in g . On theother hand, sin e any f an be de omposed as the sum h R dxf + h R dx xf, +∆ h , with h i ∈ D ( R ) , the invarian e under spa e transla-tions requires λ ( h ) = 0 and one gets λ ( f ) = 0 , ∀ f = ∂ g .By the same argument of Proposition 2.1, the Gauss invarian eof the va uum ve tor implies the vanishing of all the expe tations ω ( F V A ( h )) , with F any element of the gauge invariant subalgebra A of F W , unless h = 0 . Hen e, we are left with the orrelation fun tions ofthe gauge invariant (cid:28)elds.Sin e eqs. (7.2) imply (cid:3) E + E = ∆ σ , and E is a pseudos alar (cid:28)eld,the va uum orrelations fun tions of E are those of a free pseudos alar(cid:28)eld of mass = 1 . By the Gauss invarian e of the va uum, the expe -tations ω ( V ( f n ) α t ( V E ( g )) depend on f n only through the topologi alnumber n and therefore de(cid:28)ne operators T n , whi h are invariant underspa etime translations and satisfy T n T m = T n + m .The residual arbitrariness is therefore that of the representation ofthe abelian algebra G T , generated by the operators T n in the subspa e A Ψ . The θ va ua are hara terized by the expe tations ω θ ( V ( f n ) A ) = ω θ ( T n A ) = e i nθ ω θ ( A ) , ∀ A ∈ A . (7.8)On the other hand, the redu ible representation de(cid:28)ned by a hirallyinvariant va uum is hara terized by the expe tations ω ( V ( f n ) A ) = ω ( T n A ) = δ n ω ( A ) , ∀ A ∈ A , β λ ( A ) = A. It is easy to he k that all the general features of the QCD asedis ussed in Se tions 3-5, in parti ular the evasion of the Goldstone2theorem, the breaking of hiral symmetry in any irredu ible or fa torialrepresentation of the observable algebra, as a onsequen e of the non-trivial topology of the gauge group and the θ va ua stru ture are exa tlyreprodu ed.2) Regular representationAs lo al (cid:28)eld algebra F we take the anoni al algebra generated by the(cid:28)elds A ( h ) , E ( g ) , V ϕ ( f ) , with R dx f = n , ∂ ϕ ( f ) with the equal time ommutation relations [ A ( x , t ) , E ( y , t ) ] = iδ ( x − y ) , [ V ϕ ( f ) , ( ∂ ϕ + A )( g ) ] = i Z dx f g V ϕ ( f ) . The eu lidean fun tional integral orresponding to the Lagrangeanof eq. (7.1) yields well de(cid:28)ned orrelation fun tions of E and ∂ µ ϕ satisfy-ing the (weak) Gauss law onstraint. As before, the two-point fun tion W E ( x ) of E is that of free pseudos alar massive (cid:28)eld. The orrelationfun tions of e iϕ involve a zero mode ϕ and ru ially depend on theboundary onditions (in (cid:28)nite volume). Any boundary ondition satis-fying positivity yields the (weak) Gauss law holds for the expe tationsof all the gauge invariant variables. Periodi boundary onditions in(cid:28)nite volume give hiral invariant orrelation fun tions and thereforefor any polynomial fun tion P , < e iϕ P ( ∂ ϕ, E ) > = δ n < P ( ∂ ϕ, E ) >, (for a general dis ussion of the role of the boundary onditions in QCDand in related models see Ref. [25℄ [1℄ [22℄).An infrared subtra tion is needed for de(cid:28)ning the orrelation fun -tions of A . The most general two-point fun tion W A of A must sat-isfy − ( d /dt ) W A ( x ) = W E ( x ) and therefore has the following form( ω ( k ) ≡ p k + 1 ) π Z dk ω ( k ) − e i ( ω ( k ) x − k x ) − i ( δ ( x ) − e −| x | ) x + B ( x ) x + C ( x ) . (7.9)3Lo ality requires B ( x ) = − B ( − x ) , C ( x ) = C ( − x ) . The term linearin x violate positivity. As dis ussed in the QED ase [9℄ the fun tion C an be removed by an operator time independent gauge transformationand B = 0 if w is invariant under the CP symmetry Γ : Γ( ϕ ( x )) = ϕ ( − x ) , Γ( A ( x )) = A ( − x ) .The two-point fun tion < ∂ ϕ ( x , A ( y , > parametrizes theinfrared regularization of the fun tional integral orresponding to theLagrangean of eq. (7.1) and an be taken to vanish. Then, all thetwo-point fun tions involving ∂ ϕ and A are determined. The orre-sponding n -point-fun tions an be taken as fa torized (Gaussian).Then, we are left with the orrelation fun tions involving the zeromode ϕ , equivalently the orrelation fun tions involving e iσ ( f n ) , n = 0 .By the weak Gauss invarian e, the one-point fun tion < e iσ ( f n ) > ≡ s n only depends on the topologi al number and s n = ¯ s − n , < e iσ ( f n ) e iσ ( f m ) > = < e iσ ( f n + f m ) > = s n + m . Semi-de(cid:28)niteness of the subspa e A Ψ = D ′ implies that the sequen e { s n } is of positive type and (sin e e iσ ( f n ) ommutes with A ) the va uumfun tional on A has the de omposition w ( A ) = Z π dµ ( θ ) ω θ ( A ) , ∀ A ∈ A , (7.10) ω θ ( e iσ ( f n ) A ) = e i nθ w ( A ) , ∀ A = P ( ∂ ϕ, E ) . The representation of A is the same as in positive ase; in fa t, it onlydepends on the weak Gauss invarian e of the va uum.The (non-positive) extension to the gauge (cid:28)eld algebra F is givenby the orrelation fun tions < e iσ ( f n ) A ( z ) ...A ( z k ) >, z i = ( x ( i )1 , x ( i )0 ) .For simpli ity, we onsider the ase of a hirally invariant va uum fun -tional w . Then, all su h orrelation fun tions for n = 0 vanish and s n = δ n , orresponding to dµ ( θ ) = dθ/π .In agreement with the general analysis of Se tions 3,4, the hiralsymmetry annot be lo ally generated in the physi al spa e D phys = D ′ / D ′′ ; in parti ular, the density of the axial urrent J = ∂ ϕ + A ,whi h generates the symmetry on F , annot be de(cid:28)ned there, by theargument of Proposition 4.1. Thus, the breaking of the hiral symmetryin any fa torial representation of the observable algebra does not requirethe existen e of massless Goldstone bosons.4Sin e the orrelation fun tions < e iσ ( h ) P ( ∂ ϕ, E, A ) > fa torize interms of two-point fun tions, whi h satisfy the luster property, thelimits of the orrelation fun tions of e iσ ( f an ) , f an ( x ) ≡ f n ( x − a ) , when | a | → ∞ , exist and de(cid:28)ne the analogs of the operators C n of Se tion 5.Then, by the hiral invarian e of w , one has w ( C n ) = δ n and w maybe de omposed as a dire t integral of (inde(cid:28)nite) fun tionals w θ on F , hara terized by the expe tations w θ ( C n ) = e i nθ , whi h do not lead toa de omposition in whi h the C n are multiples of the identity. Clearly, w θ oin ides with the θ va uum on A : w θ ( A ) = ω θ ( A ) , ∀ A ∈ A , butrepresents a non-positive extension to F , whi h is not invariant underthe gauge transformations.The orresponding hiral symmetry breaking Ward identities w θ ([ J R , e iσ ( f n ) ]) = i n w θ ( e iσ ( f n ) ) = i n e i nθ , J R ≡ J ( f R α R ) , involve orrelation fun tions < J ( x ) e iσ ( f n ) > whi h are independentof time, but the time independent ve tors, playing the role of the Gold-stone bosons, are equivalent to e i nθ Ψ θ in the physi al spa e H θ andtherefore do not give rise to zero energy states di(cid:27)erent from the va -uum. A tually, the possibility of writing a hiral symmetry breakingWard identity in terms of a ommutator with a urrent operator inthe physi al spa e (with the va uum ve tor Ψ θ in their domain) is ex- luded, sin e any su h a ommutator with e iσ ( f n ))