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The Thirring–Wess model revisited: a functional integral approach
Annals of Physics 317 (2005) 423–443 www.elsevier.com/locate/aop
The Thirring–Wess model revisited: a functional integral approach L.V. Belvedere, A.F. Rodrigues* Instituto de Fı´sica, Universidade Federal Fluminense, Av. Litoraˆnea, S/N, Boa Viagem, Nitero´i, CEP.24210-340, Rio de Janeiro, Brazil Received 21 October 2004; accepted 7 December 2004 Available online 1 February 2005
Abstract We consider the Wess–Zumino–Witten theory to obtain the functional integral bosonization of the Thirring–Wess model with an arbitrary regularization parameter. Proceeding a systematic of decomposing the Bose field algebra into gauge-invariant- and gauge-non-invariant field subalgebras, we obtain the local decoupled quantum action. The generalized operator solutions for the equations of motion are reconstructed from the functional integral formalism. The isomorphism between the QED2 (QCD2) with broken gauge symmetry by a regularization prescription and the Abelian (non-Abelian) Thirring–Wess model with a fixed bare mass for the meson field is established. 2004 Elsevier Inc. All rights reserved. PACS: 199811.15.Ex; 199811.15.Tk Keywords: Regularization; Bosonization; QED2; QCD2
1. Introduction The Thirring–Wess (TW) model [1] was considered in [2–7] in order to investigate the way in which QED2 can be understood as a limit of a vector meson theory when *
Corresponding author. Fax: +5521 26295887. E-mail addresses: [email protected]ff.br (L.V. Belvedere), armfl[email protected]ff.br (A.F. Rodrigues).
0003-4916/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2004.12.003
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the bare mass (mo) of the meson tends to zero. The standard TW model, in which the gauge invariant regularization prescription is adopted, corresponds to quantize the model absorbing all effects of the gauge symmetry breakdown in the bare mass of the meson theory. The computation of the vector current is performed using the gauge invariant regularization for the point-splitting limiting procedure, which corresponds to the regularization parameter a = 2. Within a formulation in a positive-definite metric Hilbert space, the limit mo fi 0 does not exist for the Fermi field operator (w) and vector field operator ðAl Þ themselves [7]. The zero mass limit is not well defined for the general Wightman functions that provide a representation of the gauge non-invariant field algebra of the TW model. However, the gauge invariant Wightman functions of the QED2 are obtained as the zero mass limit of the Wightman functions of the gauge invariant field subalgebra of the standard TW model. More recently, the QED2 with broken gauge symmetry, by the use of an arbitrary regularization parameter a, has been considered in [9] and entitled ‘‘anomalous’’ vector Schwinger model. The standard QED2 should corresponds to the gauge invariant regularization a = 2. However, also in this case, the gauge invariant limit a fi 2 does not exist for the fields w and Al themselves. The limit a fi 2 is well defined only for the gauge invariant subset of Wightman functions. The structural problem associated with the zero bare mass limit mo fi 0 of the Wightman functions for the standard TW model can be mapped into the corresponding problem of perform the limit a fi 2 in the QED2 with gauge symmetry breakdown. One of the purposes of the present work is to fill a gap in the existing literature by discussing structural aspects of the TW model from the functional integral formulation and by extending the analysis to the general TW model regularized with an arbitrary parameter a. To this end, we use the Abelian reduction of the Wess–Zumino–Witten theory to consider the functional integral bosonization of the generalized TW model. To obtain a better insight into the behavior of gauge variant operators, in the present approach we adopt the systematic of decomposing step-by-step the Bose field algebra into gauge-invariant (GI) and gauge-non-invariant (GNI) field subalgebras, such that the effective bosonized quantum action decouples into GI and GNI pieces. The gauge symmetry breakdown is characterized by the presence of a non-canonical free massless Bose field. The vector field with bare mass mo acquires a dynamical mass ~ 2o ¼ m
e2 ða þ 2Þ þ m2o : 4p
ð1:1Þ
Within the present approach we obtain a formulation in an indefinite-metric Hilbert space of states and the Proca equation is satisfied in the weak form. The generalized field operators are reconstructed from the functional integral formalism and are written as gauge transformed fields by an operator-valued gauge transformation. In the indefinite-metric formulation, the GI limit can be performed and we obtain the corresponding field operators of QED2, as obtained by Lowenstein–Swieca [6]. Performing a canonical transformation, the singular gauge part becomes the identity and we obtain the generalized operator solution for the coupled Dirac–Proca equations. For the gauge invariant regularization a = 2, we recover the operator solution of the standard TW model, as obtained by Lowenstein–Rothe–Swieca [6,7]. Since in this
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case the longitudinal current does not carry any fermionic charge selection rule, commutes with itself and commutes with all operators belonging to the field algebra, it is reduced to the identity operator. This leads to a positive-metric Hilbert space of states and the coupled Dirac–Proca equations are satisfied in the strong form. Since in this case the Fermi field operator is given in terms of the charge-carrying fermion operator of the Thirring model, the zero mass limit exists only for the GI field subalgebra. This streamlines the presentation of [6,7]. Another purpose of the present paper is to discuss the isomorphism between the QED2 with broken local gauge invariance (the ‘‘anomalous’’ vector Schwinger model considered in [9]) and the TW model. Using the Wess–Zumino–Witten theory, we show that the effective quantum action of the QED2 quantized with a gauge non-invariant regularization b „ 2, is equivalent to the quantum action of the TW model with a vector field with bare mass e2 ðb aÞ; ð1:2Þ m2o ¼ 4p where a < b is the parameter used in the regularization of the TW model. In this way, the gauge invariant limit b fi 2 for the QED2 with broken gauge symmetry, is mapped into the limit mo fi 0 for the standard TW model (a = 2). As is well known [7,8], the confinement phenomenon in the standard QED2 is associated with the absence of charge sectors. In this way, the conclusion of [9], according with the parameter a controls the screening and confinement properties is nothing but that the TW model exhibits a charge-carrying fermion operator and thus there is no confinement. The introduction of the mass term for the Fermi field of the generalized TW model is also considered. For the GI regularization we recover the operator solution obtained by Rothe–Swieca [7] for the massive TW model. In the Section 5 we consider the functional integral bosonization of the non-Abelian TW model with an arbitrary regularization parameter. In this case, the quantum action of the non-Abelian TW model is mapped into the action of the QCD2 with gauge symmetry breakdown. 2. Functional integral bosonization The Thirring–Wess (TW) model is defined [1] from the classical Lagrangian density,1 l ol ecl Al Þw þ 1m2 Al Al ; L ¼ 14Flm Flm þ wðic ð2:1Þ 2 o where the field-strength tensor Flm is given by Flm ¼ om Al ol Am :
ð2:2Þ
00 01 10 l 5 lm l 5 0 1 m ~ Our conventions are: g = 1, = 01 = = 1, c c = cm, = c ol; ol ¼ lm o , c = c c ; 0 1 0 1 , c1 ¼ ; o± = o0 ± o1; A ¼ A0 A1 ; = clol;For a free massless scalar field, c0 ¼ 1 0 1 0 ~ ~ ¼ ol U. The Fermi field W is the two-component spinor W ¼ w1 . ol U w2 1
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The local gauge invariance is broken by the mass term for the vector field. In the classical level, the local gauge invariance can be restored by performing in (2.1) the limit mo fi 0. This limit corresponds to the classical Lagrangian of the two-dimensional electrodynamics. However, in the quantum level, the zero mass limit for the vector field, even for a gauge invariant regularization prescription, is well defined only for the gauge invariant subset of Wightman functions [7]. 2.1. Decoupled quantum action To obtain the bosonized action, we shall consider the Abelian reduction of the Wess–Zumino–Witten theory [11]. To this end, let us consider the generating functional (in Minkowski space), Z 2 l l Z½#; #; | ¼ exp i d zð#w þ w# þ |l A Þ ; ð2:3Þ where the average is taken with respect to the functional integral measure, Z iS½w;w;A l dl ¼ DAl DwDwe ;
ð2:4Þ
and the Lagrangian density is written as, L ¼ 14Flm Flm þ wy1 Dþ ðAÞw1 þ wy2 D ðAÞw2 þ 12m2o Aþ A ;
ð2:5Þ
where : D ðAÞ ¼ ðio eA Þ:
ð2:6Þ
The vector field can be parametrized in terms of the U (1) group-valued Bose fields (U, V) as follows [13–15]: 1 Aþ ¼ U 1 ioþ U ; e
ð2:7Þ
1 A ¼ V io V 1 e
ð2:8Þ
with U ¼ e2i V ¼e
pffiffi pu
pffiffi 2i pv
;
ð2:9Þ
;
ð2:10Þ
such that, ðAÞw ¼ ðU w Þy ðioþ ÞðU w Þ þ ðV 1 w Þy ðio ÞðV 1 w Þ: w 1 1 2 2
ð2:11Þ
To decouple the Fermi- and vector fields in the Lagrangian (2.1), the spinor components (w1, w2), are parametrized in terms of the Bose fields (U, V) and the free Fermi field components (v1, v2), w1 ¼ U 1 v1 ;
ð2:12Þ
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w2 ¼ V v2 :
427
ð2:13Þ B
To introduce the systematic of decomposing the Bose field algebra I , IB ¼ IB fU ; V g;
ð2:14Þ
into gauge invariant (GI) and gauge non-invariant (GNI) field subalgebras, let us consider a class of local gauge transformations g acting on the Bose fields U, V, as follows: g : U ! g U ¼ Ug;
ð2:15Þ
g : V ! g V ¼ g1 V ;
ð2:16Þ
with gðxÞ ¼ e2i
pffiffi pKðxÞ
:
ð2:17Þ
Under g-transformations the fields (u, v) transform according with, g : u ! u þ K;
ð2:18Þ
g : v ! v K:
ð2:19Þ
The field combination (u + v) is gauge invariant, whereas the combination (u v) is gauge non-invariant. The vector field and the Fermi field transform under g according with, g
1 Al ¼ Al þ giol g1 ; e
ð2:20Þ
g
w1 ¼ ðg U 1 Þv1 ¼ g1 w1 ;
ð2:21Þ
g
w2 ¼ ðg V Þv2 ¼ g1 w2 :
ð2:22Þ
Let us introduce the gauge invariant Bose field G, pffiffi : G ¼ UV ¼ e2i pðuþvÞ ;
ð2:23Þ
such that, g
G ¼ g U g V ¼ UV ¼ G:
In terms of the field G, the field-strength tensor is given by, pffiffiffi p 1 1 1 ðu þ vÞ: F01 ¼ ðo Aþ oþ A Þ ¼ oþ ðG io GÞ ¼ e 2 2e
ð2:24Þ
ð2:25Þ
and the Maxwell action can be written as, Z Z 1 1 2 S M ½UV ¼ 2 d2 zðu þ vÞ2 ðu þ vÞ ¼ S M ½G ¼ 2 d2 z½oþ ðGio G1 Þ 8e 2~ lo Z 1 2 d2 z½o ðG1 ioþ GÞ ; ¼ 2 ð2:26Þ 8e
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where we have defined e2 ~2o ¼ : ð2:27Þ l 4p Let us return to the functional integration in the generating functional. Introducing the identities, Z 1 ¼ DU ½det Dþ ðU ÞdðeAþ U 1 ioþ U Þ; ð2:28Þ 1¼
Z
DV ½det D ðV ÞdðeA V io V 1 Þ;
ð2:29Þ
the change of variables fAþ ; A g ! fU ; V g is performed by integrating over the vector field components A . Performing the Fermi field rotations (2.12) and (2.13), and taking due account of the Jacobians in the change of the integration measure in the generating functional, we obtain the effective integration measure [13–15], 0
dl ¼ D vDvDU DV eiS½U ;V ;v;v eiS ½U ;V ;a;mo ; where, S½U ; V ; v; v ¼
Z
d2 z vi v þ S M ½UV ;
ð2:30Þ
ð2:31Þ
and S 0 [U, V, a, mo] is the GNI contribution, which is given in terms of the Wess–Zumino–Witten (WZW) functionals C [U], C [V], Z 1 S 0 ½U ; V ; a; mo ¼ C½U C½V 2 ða~ l2o þ m2o Þ d2 zðU 1 oþ U ÞðV o V 1 Þ: 2e ð2:32Þ The term carrying the regularization parameter a in the action (2.32) corresponds to the Jackiw–Rajaraman action, Z a 2 ~o d2 zAþ A ; S JR ¼ l ð2:33Þ 2 which in an anomalous model, such as chiral QED2, characterizes the quantization ambiguity [10]. The value a = 2 corresponds to the GI regularization. The WZW functionals enter (2.32) with negative level [11]. In the Abelian case, the WZW functional reduces to the action of a free massless scalar field, Z 1 d2 zðol hÞðol h1 Þ: C½h ¼ ð2:34Þ 8p Using the Abelian reduction of the Polyakov–Wiegmann (PW) identity [12], Z 1 d2 zðg1 oþ gÞðho g1 Þ; C½gh ¼ C½g þ C½h þ ð2:35Þ 4p the total effective action can be written as, S½U ; V ; v; v ¼ S½ v; v; U ; V þ S 0 ½U ; V ;
ð2:36Þ
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where ð0Þ
S½ v; v; U ; V ¼ S F ½ v; v þ S M ½UV ; S 0 ½U ; V ¼
ð2:37Þ
1 1 f~ l2o a þ m2o gC½UV þ 2 f~ l2 ða 2Þ þ m2o gðC½V þ C½U Þ: 2 2~ lo 2~ lo o
ð2:38Þ The standard TW model corresponds to a = 2 and in the limit mo fi 0, the gauge invariance is restored, S 0 ½U ; V ! S 0 ½G;
ð2:39Þ
such that the action (2.36) reduces to the action of the standard QED2, v; v: S½U ; V ; v; v ! SQED ½G;
ð2:40Þ
For a gauge non-invariant regularization, in the limit mo fi 0, we obtain the action for the QED2 with broken gauge symmetry, b 1 ð2:41Þ S 0 ½U ; V ; b; 0QED ¼ C½UV þ ðb 2ÞðC½V þ C½U Þ: 2 2 where b is the corresponding regularization parameter. The actions (2.38) and (2.41) are equivalent, S 0 ½U ; V ; b; 0 S 0 ½U ; V ; a; mo ;
ð2:42Þ
provided that b > a and ~2o ðb aÞ: m2o ¼ l
ð2:43Þ
This implies that the QED2 quantized with a GNI regularization b „ 2 (QED2 with broken gauge symmetry) is isomorphic to the TW model with a regularization parameter a and with a fixed bare mass for the vector field given by (2.43). To proceed further the decomposition of the Bose field algebra IB fU ; V g into GI e and GNI field subalgebras, let us introduce the gauge invariant pseudo-scalar field /, : 1 e¼ ðu þ vÞ; ð2:44Þ / 2 and the gauge non-invariant scalar field f, : f ¼ 12ðu vÞ:
ð2:45Þ
The Bose fields (U, V), defined by (2.9) and (2.10), can be decomposed in terms of the e and the gauge non-invariant field h [f] as gauge invariant field g½ / U ¼ gh;
ð2:46Þ
V ¼ gh1 ;
ð2:47Þ
where g ¼ e2i h¼e
pffiffi ~ p/
pffiffi 2i pf
;
ð2:48Þ
:
ð2:49Þ
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The gauge invariant field G can be rewritten as pffiffi ~ G ¼ UV ¼ g2 ¼ e4i p/ :
ð2:50Þ
The vector field can be decomposed in the standard form in terms of GI and GNI contributions, pffiffiffi p ~ e fol / þ ol fg; Al ¼ ð2:51Þ e and the Maxwell action (2.26) is now given by Z 1 e 2: S M ½g ¼ 2 d2 zð /Þ 2~ lo
ð2:52Þ
The Bose field algebra can be decomposed into IB fU ; V g ¼ IBGI fgg IBGNI fhg;
ð2:53Þ
and the effective quantum action S½U ; V ; v; v can be rewritten in terms of the fields g and h, S½U ; V ; v; v ¼ S½g; h; v; v ¼ S½g; v; v þ S 0 ½g; h;
ð2:54Þ
where ð0Þ
S½g; v; v ¼ S F ½ v; v þ S M ½g; S 0 ½g; h ¼
ð2:55Þ
1 1 f~ l2 a þ m2o gC½g2 þ 2 f~ l2 ða 2Þ þ m2o gfC½gh1 þ C½ghg; 2~ l2o o 2~ lo o ð2:56Þ
with the Maxwell action given by Z 2 1 S M ½g ¼ d2 z oþ ðgio g1 Þ : 2 8p~ lo
ð2:57Þ
Using the P–W identity, the GNI action S 0 [g, h] given by (2.56) decouples into, S 0 ½g; h ¼ S½g þ S½h ¼
~2 ~ 2o m m C½g þ C½h; ~2o ~2o l l
ð2:58Þ
where ~2o ða þ 2Þ þ m2o ; ~2 ¼ l m
ð2:59Þ
~2o ða 2Þ þ m2o : ~ 2o ¼ l m
ð2:60Þ
e h [f], decouple in the action (2.58) and the GNI field h [f] is a free The fields g½ /, massless non-canonical scalar field. The total partition function factorizes as Z Z DgD vDv expfiS½g þ S½g; v; vg ¼ Zh Zv;v;g : Z¼ Dh expfiS½hg ð2:61Þ
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The partition function Zh characterizes the local gauge symmetry breakdown. Although the partition function can be factorized, the generating functional, and thus the Hilbert space of states H, cannot be factorized H 6¼ Hv;v;g Hh :
ð2:62Þ
~ o ¼ 0, that is obThe bonafide gauge invariant vector model corresponds to m tained with mo = 0 and the gauge invariant regularization a = 2. In this case, all reference to the field h [f] has disappeared from the effective quantum action (2.58), which is given in terms of the gauge invariant field g. In this case, the field f is not a dynamical degree of freedom and corresponds to a pure c-number gauge excitation. The commutator (2.85) and the corresponding Hamiltonian of the effective bosonized quantum action are singular for m~o = 0. For these critical values of the parameters, the GNI action vanishes, S½h ¼
~ 2o m C½h ! 0; ~2o l
ð2:63Þ
~ o ¼ 0, the gauge invariance is formally restored and the field h [f] implying that for m is not a dynamical degree of freedom. At this critical point the constraint structure of ~ o ! 0 the field h becomes a the model change. Since, in the gauge invariant limit m pure gauge excitation, the corresponding partition function reduces to a gauge volume, Z lim Zh ! Dh ¼ V gauge : ð2:64Þ ~ o !0 m
The ‘‘anomalous vector Schwinger model’’ considered in [9] is obtained considering mo = 0 and b „ 2. The GNI action is now given by S 0 ½g; h ¼ ðb þ 2ÞC½g þ ðb 2ÞC½h;
ð2:65Þ
and corresponds to the TW model with bare mass for the vector field given by (2.43). 2.2. Local action The Maxwell action (2.57) is non-local due to the quartic-self-interaction of the field g. To dequartize the action of the gauge invariant field g, let us consider the functional integral over the field g in the partition function Zv;v;g . To begin with, let us introduce an auxiliary gauge invariant field X, such that, (Z ( )) Z 2 ~ m 1 1 2 2 Dgexp i d z ½oþ ðgio g1 Þ oþ go g1 8p l ~2o 8p~ l2o (Z ( )) Z ~2 1 2 1 1 m 2 1 1 oþ go g DXDgexp i d z X þ pffiffiffi X½o ðg ioþ gÞ : 2 8p l ~o 2 pl ~2o ð2:66Þ
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Making the change of variables, : o X ¼ ðxio x1 Þ; we can write, 1 X ¼ o1 ; ðxio x Þ: Rescaling the exponential field x, pffiffiffi ~o 1 2 pl 1 ioþ x; x ioþ x ¼ x 2 ~ m
ð2:67Þ ð2:68Þ
ð2:69Þ
we obtain from (2.66) after an integration by parts, ( Z ( Z ~ 4 1 ~ 2 1 1 m 1 m 2 1 2 1 ioþ xÞ DxDg exp i d z ½o ð xio Þ þ ðg ioþ g x x 8p l 8p l ~2o ~2o )) ~ 2 1 1 m 1 1 1 x ioþ xÞð Þ xio x Þ : ðgio g xio ðx ð2:70Þ 8p l ~2o Defining a new gauge invariant field h, : 1 ioþ x; h1 ioþ h ¼ g1 ioþ g x
ð2:71Þ
: x 1 ; hio h1 ¼ gio g1 xio
ð2:72Þ
the field g can be factorized as, g ¼ hx:
ð2:73Þ
The total effective action is now written in terms of the gauge invariant fields h, x, the gauge non-invariant field h and the free Fermi field v. To obtain the complete bosonized action, we introduce the bosonized form for the action of the free Fermi field [16], Z 1 ð0Þ d2 zol uol u; S F ¼ C½fu ¼ ð2:74Þ 2 with fu ¼ e2i
pffiffi pu
;
ð2:75Þ
and the Mandelstam representation for the bosonized free massive Fermi field, Z 1 n p o
j 12 pffiffiffi 5 o 5 1 0 1 ~ ðxÞ þ ~ ðx ;z Þ :; vðxÞ ¼ exp i c : exp i p c u dz o0 u ð2:76Þ 4 2p x1 where jo is an arbitrary finite mass scale. The total local action is given in the bosonized form by, ~2 m h ¼ C½fu C½h þ C½x þ 2o C½h S½f ; g; h ¼ S½f ; h; x; ~o l 4 Z ~ 1 m xÞ 2; ð2:77Þ d2 z½o1 ðxio 2 8p l ~o
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as, where the field h is quantized with negative metric. Parametrizing the fields h and x pffiffi ð2:78Þ hðxÞ ¼ e2i p~gðxÞ ; pffiffiffi l ~ ~ ; ð2:79Þ xðxÞ ¼ exp 2i p o2 RðxÞ ~ m performing the field scaling, l 0 ~ g! o~ g; ~ m ~ !m ~ 0; ~R R
ð2:80Þ ð2:81Þ
and streamlining the notation by dropping primes everywhere, we obtain for the e gauge invariant field /, ~o e ¼l ~ ð~ g þ RÞ; / ~ m pffiffiffi l ~o ~ ¼ exp 2i p ð~ g þ RÞ : g ¼ hx ~ m
ð2:82Þ ð2:83Þ
The effective bosonized total Lagrangian density is then given by, ~2 ~2 1 ~ 2o 1 1 1m 2 2 ~ 2m ~ Þ2 þ ðol RÞ R ðol ~ L ¼ ðol u ðol fÞ : gÞ þ 2 2 2 2 l 2 ~2o
ð2:84Þ
The field ~ g is quantized with negative metric. The gauge non-invariant field f is a non-canonical free massless decoupled field, ½fðxÞ; fðyÞ ¼
~2o l Dðx y; 0Þ: ~ 2o m
ð2:85Þ
For massless Fermi fields, although the total partition function factorizes into free field partition functions, Z ¼ Zf Zh Zx Zh ;
ð2:86Þ
the generating functional does not factorizes.
3. Field operators and Hilbert space The Bose fields (U, V) are given by, pffiffiffi pffiffiffi l ~ ~ ~ h½f; U ¼ expf2i p/gh½f ¼ exp 2i p o ð~ g þ RÞ ~ m pffiffiffi pffiffiffi l ~o 1 ~ ~ V ¼ expf2i p/gh ½f ¼ exp 2i p ð~ g þ RÞ h1 ½f: ~ m
ð3:1Þ ð3:2Þ
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The bosonized form of the Fermi field operator is given in terms of the free Fermi field as,
j pffiffiffi l ~o 5 p 5 o ~ wðxÞ ¼ expfi c g : exp 2i p c f~gðxÞ þ RðxÞg : ~ 4 2p m Z 1 pffiffiffi 5 0 1 1 ~ ~ o0 uðx ; z Þ dz g : h1 ½f; ð3:3Þ : exp i pfc uðxÞ þ x1
and the vector field is given by, Al ¼
1~ ~ 1 ol ð R þ ~ gÞ þ hiol h1 : ~ m e
ð3:4Þ
The divergent part h [f] has the form of a gauge term. The bosonized expressions for fields (3.3) and (3.4) correspond to those of the Schwinger model gauged by a diver~ o ! 0. gent operator-gauge transformation as long as m The vector current is computed with the standard point-splitting limit procedure, . .. 1 J l ðxÞ ¼ ..wðxÞc l wðxÞ. ¼ Z ðÞ
Z xþ l lo Al ðzÞ dz wðxÞ V :E:V : : wðx þ Þcl exp ia~ ð3:5Þ x
In terms of the GI and GNI field combinations (u ± v), the vector current is given by, ~ ~ e e l l J l ¼ |l o ða þ 2Þ~ ol ðu þ vÞ o ða 2Þol ðu vÞ; 2 2 2 2
ð3:6Þ
where |l is the conserved current associated with the free Fermi field v, . . 2 ~: ol u |l ¼ .. vcl v.. ¼ pffiffiffi ~ p
ð3:7Þ
The bosonized form of the vector current is given by, ~ m2o Þ ~ ~ e ðm ~þ ~ o ðm ~ 2o m2o Þol f: J l ¼ 2~ ol ð R þ ~ ol u lo ~ gÞ l ~ 2 m
ð3:8Þ
The conserved current that acts as the source of Flm is given by ~2 ~ ~2 e e m m Kl ¼ m2o Al J l ¼ |l þ ol ðu þ vÞ þ o ol ðu vÞ; 2 2 2~ lo 2~ lo which can be written as, ~ þ Ll ; ~~ Kl ¼ m ol R
ð3:9Þ
ð3:10Þ
where Ll is a longitudinal current of zero norm given by, Ll ¼ ol L;
ð3:11Þ
with the potential L given by, ~ þ L ¼ 2~ lo u þ mg
~ 2o m f: ~o l
ð3:12Þ
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The longitudinal current Ll commutes with itself and thus generates zero-norm states from the vacuum, hWo jLl ðxÞLm ðyÞjWo i ¼ 0:
ð3:13Þ
Due to the presence of the longitudinal current Ll, the Proca equation is not satisfied as an operator identity, e ð3:14Þ om Fml þ m2o Al J l ¼ Ll : 2 w; Al g generate a local field algebra I. These field operThe fundamental fields fw; ators constitute the intrinsic mathematical description of the model and whose Wightman functions define the model. The field algebra I is represented in the indefinitemetric Hilbert space of states H, H ¼ IjWo i: ð3:15Þ In a gauge invariant model, the field f is not a dynamical degree of freedom. In the indefinite metric formulation, the GaussÕ law is satisfied in weak form, om Fml eJ l ¼ Ll ; ð3:16Þ where Ll is the zero-norm longitudinal piece of the vector current that acts as the source in the GaussÕ law. The local gauge transformations of the intrinsic fields w; Al g are implemented by the longitudinal current Ll. The physical gauge fw; invariant field algebra Iphys , is a subalgebra of the intrinsic field algebra I that commutes with the longitudinal current, Iphys I;
ð3:17Þ
½Iphys ; Ll ¼ 0:
ð3:18Þ
The physical Hilbert space Hphys ¼ Iphys jWo i; is a subspace of the Hilbert space H ¼ IjWo i, Hphys H:
ð3:19Þ ð3:20Þ
In a model with gauge symmetry breakdown, the field f is a dynamical degree of freedom. The intrinsic field algebra commutes with the longitudinal current, ½I; Ll ¼ 0: ð3:21Þ w; Al g are singlet under gauge transformations genThis implies that the fields fw; erated by the longitudinal current and thus are physical operators. The intrinsic field algebra is by itself the physical field algebra, I Iphys :
ð3:22Þ
3.1. The QED2 limit It is very instructive to make some comments about the gauge invariant limit ~ o ! 0. From the Lagrangian (2.84), written in terms of the non-canonical field f, m we obtain in the zero mass limit,
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Lm~ o !0 ! LQED :
ð3:23Þ
In this indefinite-metric Hilbert space formulation, the GI limit can be performed in the operator field algebra written in terms of the non-canonical free field f. The operator solution of the QED2, as given by Lowenstein–Swieca [6], can be formally obtained from (3.3), (3.4), (3.10), and (3.11). In the gauge invariant limit, the field f decouples from the quantum action corresponding to the Lagrangian (2.84) and thus it is not a dynamical degree of freedom. The field f becomes a c-number, f (x) fi K (x), and we get, pffiffiffi pffiffiffi ~ wðxÞ ¼: expfi pc5 f~ gðxÞ þ RðxÞgg : vðxÞ expfi pKðxÞg; ð3:24Þ pffiffiffi pffiffiffi p~ ~ p ol ð R þ ~ ol KðxÞ; Al ¼ gÞ þ e e
ð3:25Þ
1 ~ þ Ll ; ol R J l ¼ pffiffiffi ~ p
ð3:26Þ
where Ll is the longitudinal current of zero norm, e Ll ¼ pffiffiffi ol ðu þ gÞ: p Taking this into account, we obtain for the generating functional, #; |l Z½#; ~ o !0 ! Z½#; #; |l QED : m
ð3:27Þ
ð3:28Þ
It must be stressed that this gauge invariant limit can be formally performed only in this level. It cannot be performed in the general Wightman functions due to the singular commutation relation for the field f. This limit is well defined only for the subset of Wightman functions that are independent of the field f. By considering the quantum action and the field operators written in terms of the non-canonical field f, to take the limit mo fi 0 is equivalent to start from the beginning with a zero bare mass for the vector meson. As a matter of fact, in this indefinite-metric Hilbert space formulation, the GI limit can be formally performed as above, since the Fermi field operator is written in terms of the free Fermi field and not in terms of the charge-carrying Fermi field operator of the Thirring model. The intrinsic field algebra I is the physical field algebra, which is singlet under gauge transformations generated by the longitudinal current.
4. The positive-definite-metric formulation The bosonization in the indefinite-metric formulation introduces a larger Bose field algebra IB that contains more degrees of freedom than those needed for the description of the model, I IB . To extract the redundant degrees of freedom, as well as, to obtain the solution of the equations of motion as operator identities in a positive definite-metric Hilbert space of states, we introduce two free Bose fields ~ NÞ, ~ by the following canonical transformation: ðU;
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pffiffiffi l ~ b ~ pffiffiffi ~ þ2 p o~ U ¼ pu g; ~ 2 m
ð4:1Þ
pffiffiffi l pffiffiffi ~ b~ ~ þ p~ N¼ 2 p ou g: ~ 2 m
ð4:2Þ
~ and N ~ are The negative metric quantization for the field ~g ensures that the fields U independent degrees of freedom, ~ ~ ½UðxÞ; NðyÞ ¼0
8ðx; yÞ:
ð4:3Þ
Imposing canonical commutation relations for the fields U and N, we get 2 ~o b2 m ¼ > 0; ð4:4Þ 4p ~2 m ~ is quantized with negative metric. Defining the canonical-free field, and the field N n¼
~o m f; ~o l
ð4:5Þ
the Fermi field and the vector field operators (3.3) and (3.4) can be re-written as ‘‘gauge transformed fields,’’ ^ w ¼ wq;
ð4:6Þ
1 Al ¼ A^l þ qiol q1 : e
ð4:7Þ
Here, the operator q is a pure gauge excitation given by, pffiffiffi l ~o q ¼: exp 2i p ðN nÞ :; ~o m and
~ ~ l ^ ¼: exp 2ic5 pffiffiffi w : WU~ ; p oR ~ m ~o ~ 1 ~ 2l ol R U ; A^l ¼ ~ ~ ~o m m
ð4:8Þ
ð4:9Þ
ð4:10Þ
Here W is the charge-carrying Fermi field operator of the Thirring model, which is given by the Mandelstam representation, Z
l 12 p b~ 2p þ1 ~ 0 1 1 i WðxÞ ¼ o expfi c5 g : exp ic5 UðxÞ o0 Uðx ; z Þ dz : : 4 2 b x1 2p ð4:11Þ The vector current Jl, given by (3.10), can be re-written as ~ 2 m2o Þ ~ ~ 2~ e ðm l m2 ~ Jl ¼ ol R þ o o ~ ol U: ~ ~ om ~ 2 m m
ð4:12Þ
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and the current Kl is given by, ~ þ Ll ; ~~ Kl ¼ m ol R
ð4:13Þ
where the longitudinal current is now given by, ~o m ~ o ol ðN nÞ ¼ pffiffiffi qol q1 : ‘ l ¼ ol L ¼ m ~o 2 pl
ð4:14Þ
The longitudinal current carries no fermion selection rule since it is independent of the vector current of the Thirring model, J Th l ¼
2~ lo m2o ~ ~ ol U: ~ om ~ m
ð4:15Þ
For the GI regularization a = 2, we obtain, b2 m2 ¼ e2 o ; 4p p þ m2o
ð4:16Þ
b ~ ~ J Th ð4:17Þ l ¼ pffiffiffi ol U; p and the field operators given by (4.6) and (4.7) correspond to those operators obtained by Lowenstein–Rothe–Swieca [6,7] gauged by a singular operator gauge transformation. The operator q has zero scale dimension, commutes with itself and thus generates infinitely delocalized states that leads to constant Wightman functions hWo jq ðx1 Þ q ðxn Þqðy 1 Þ qðy n ÞjWo i ¼ 1:
ð4:18Þ
The spurious operator q does not carry any fermionic charge selection rule, and since it commutes with all operators belonging to the field algebra I, it is reduced to the identity operator in H. The position independence of the state q (x)Wo can be seen by computing the general Wightman functions involving the operator q and all operators belonging to the local field algebra I. Thus, for any operator R w; Al g, the poOðfz Þ ¼ OðzÞf ðzÞ d2 z 2 I, of polynomials in the smeared fields fw; sition independence of the operator q can be expressed in the weak form as hWo ; q ðx1 Þ q ðx‘ Þqðy 1 Þ qðy m ÞOðfz1 ; . . . ; fzn ÞWo i ¼ Wðz1 ; . . . ; zn Þ hWo ; Oðfz1 ; . . . ; fzn ÞWo i 8Oðfz Þ 2 I;
ð4:19Þ
where Wðz1 ; . . . ; zn Þ is a distribution independent of the space-time coordinates (xi, yj). Within the functional integral approach, the Hilbert space of states is constructed from the generating functional (2.3), which can be rewritten as, Z ^ þ wq ^ # þ |l ð A #; |l ¼ exp i d2 z #wq ^ l þ 1 q1 iol qÞ ; Z½#; ð4:20Þ e where the average is taken with respect to the functional integral measure, Z Z ð0Þ ~ ~ Re ~ iS½R~ U dl ¼ DNDneiS ½N;n DUD ;
ð4:21Þ
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and the actions appearing in (4.21) are given in terms of the corresponding bosonized Lagrangian densities, 2
2
Lð0Þ ¼ 12ðol NÞ þ 12ðol nÞ ;
ð4:22Þ
~2 ~2 1 ~ 2 þ 1 ðol RÞ ~ 2m R: L ¼ ðol UÞ 2 2 2
ð4:23Þ
Since the free massless fields N and n decouples in the quantum action, the partition function factorizes, ð0Þ
ð0Þ
ð0Þ
ð0Þ
Z ¼ Zn ZN ZU ZR :
ð4:24Þ
In the computation of general correlation functions from the generating functional (4.20), due to the opposite metric quantization, the space-time contributions of the functional integration over the field n cancel those contributions coming from the integration over the field N. In this way, the field q becomes the identity with respect to the functional integration and we get the identities, ^ w w;
Al A^l :
ð4:25Þ
c is builded from the generating funcThe positive-definite metric Hilbert space H tional, Z
^ 2 l l l^ ^ c Z½#; #; | Z½#; #; | ¼ exp i d z #w þ w# þ | Al ; ð4:26Þ where the average is taken with respect to the functional integral measure Z ~ c ¼ DUD ~ Re ~ iS½R~ U : dl
ð4:27Þ
c corresponds to the quotient space The Hilbert space H c¼ H; H H0
ð4:28Þ
where H0 is the zero-norm space. The fields (4.6) and (4.7) provide the operator solution for the coupled Dirac–Proca equations icl ol wðxÞ þ ecl : Al ðxÞwðxÞ :¼ 0;
ð4:29Þ
e om Fml þ m2o Al J l ¼ 0: 2
ð4:30Þ
For the gauge invariant regularization a = 2 and mo „ 0, we recover from (4.25) the operator solution for the TW model obtained by Lowenstein–Rothe–Swieca [6,7]. For mo = 0 and the gauge non-invariant regularization a „ 2, we obtain the operator solution of QED2 with broken gauge symmetry. In the positive-metric Hilbert space formulation, the QED2 limit does not exist for the Fermi field and vector field themselves. In this case, the equations of motion are satisfied as operator identities and the Fermi field is a charge-carrying operator,
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½J 0TW ðxÞ;
WðzÞx0 ¼z0 ¼ e
m2o dðx1 z1 ÞWðzÞ: ~ 2o m
ð4:31Þ
As stressed in [7], if this limit were to exist, we would obtain for the QED2 a local charge-carrying Fermi field operator, which is incompatible with the MaxwellÕs equation being satisfied in the strong form. Nevertheless, this limit is well defined for the gauge invariant field subalgebra, as for instance, 1 ~ ol R; J l ! pffiffiffi ~ p
ð4:32Þ
e ~ Flm ! lm pffiffiffi R: p
ð4:33Þ
4.1. Massive fermions The introduction of a mass term for the Fermi field gives a contribution to the action, ¼ M o fvy v ðUV Þ þ vy v ðUV Þ1 g ¼ M o vy v g2 þ vy v g2 : M ¼ M o ww 1 2
2 1
1 2
2 1
ð4:34Þ Using the decomposition for the GI field g and the bosonized form for the free Fermi field and the bosonized chiral density of the free Fermi field,
j pffiffi o v 1 v2 ¼ ð4:35Þ : e2i pu :; 2p we get,
pffiffi M o jo 2ipffiffipu~ 2 þ e2i pu~ ½xh 2 Þ ðe ½xh 2p pffiffiffi pffiffiffi l ~ M o jo ~ : ~ þ 4 p o ð~ cos 2 pu g þ RÞ ¼ ~ p m
M¼
ð4:36Þ
Performing the canonical transformation (4.1) and (4.2), the mass term can be written as pffiffiffi l ~o ~ M o jo ~ M¼ cos 2bU þ 4 p R : ð4:37Þ ~ p m ~ Notice that, even for a massive fermion field, the field f (n) is a free field. The fields U ~ and R are coupled by the sine-Gordon interaction and the total Lagrangian density is now given by 1 1 1 2 2 ~ 2 þ 1 ðol RÞ ~ 2 L ¼ ðol NÞ þ ðol nÞ þ ðol UÞ 2 2 2 2 pffiffiffi ~o ~ 2 ~ 2 M o jo m ~ þ4 pl ~ : cos 2bU R R ~ 2 p m
ð4:38Þ
For a = 2 we recover from (4.38) the bosonized Lagrangian density for the massive TW model, as obtained by Rothe–Swieca [7].
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5. A glance into the non-Abelian model The classical Lagrangian density defining the non-Abelian TW model is given by icl ol ecl Al w þ 1m2 trAl Al ; ð5:1Þ L ¼ LYM þ w 2 o where the Yang–Mills Lagrangian is given by, LYM ¼ 14trFlm Flm :
ð5:2Þ
Let us apply the Wess–Zumino–Witten theory to obtain the effective bosonic action. To this end, we shall consider the change of variables (2.7), (2.8), (2.12), and (2.13), in terms of the Lie-algebra-valued Bose fields (U, V). The partition function can be factorized as [13,14], ð0Þ
ð0Þ
Z ¼ ZF Zgh Z;
ð5:3Þ
ð0Þ
where ZF is the partition function of free fermions, Z R ð0Þ i d2 z vi v ZF ¼ DvD ve ;
ð5:4Þ
ð0Þ
Zgh is the partition function of free ghosts associated with the change of variables (2.7) and (2.8), Z Z Z ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ Dbð0Þ Zgh ¼ Dbþ Dcþ exp i d2 z tr bð0Þ io c þ þ Dc Z 2 ð0Þ ð0Þ exp i d z tr b ioþ c ; ð5:5Þ and ¼ Z
DU DV expfiS YM ½UV g exp ifC V C½UV þ C½U 1 þ C½V g Z 1 l2o a þ m2o g d2 z tr½ðU1 oþ UÞðVo V1 Þ ; 2 f~ 2e Z
with the Yang–Mills action given by, Z 1 1 2 S YM ½UV ¼ 2 d2 z tr ½oþ ðGio GÞ : 4e 2 In the non-Abelian case, the WZW functional is given by [11,16], C½g ¼ S P rM ½g þ S WZ ½g; where SPrM[g] is the principal sigma model action, Z 1 d2 x tr ½ðol gÞðol g1 Þ; S P rM ½g ¼ 8p and the functional SWZ[g] is the Wess–Zumino action, Z 1 S WZ ½g ¼ d3 xeijk tr½ð^ g1 oi g^Þð^ g1 oj g^Þð^ g1 ok g^Þ: 12p S B
ð5:6Þ
ð5:7Þ ð5:8Þ
ð5:9Þ
ð5:10Þ
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Using the PW identity, the total partition function can be rewritten as, Z ð0Þ ð0Þ DU DV expfiS YM ½UV g Z ¼ ZF Zgh ( 1 1 exp i 2 f~ l2o ða þ 2C V Þ þ m2o gC½UV þ i 2 f~ l2 ða 2Þ þ m2o gC½V 2~ lo 2~ lo o ) 1 2 1 2 iC½U þ i 2 f~ l a þ mo gC½U : ð5:11Þ 2~ lo o From the partition function (5.11), we read off the equivalence of the QCD2 provided with a GNI regularization b „ 2 and the non-Abelian TW model presenting a fixed ~2o ðb aÞ for the vector field. Similar to the Abelian case, this isobare mass m2o ¼ l morphism also holds in the non-Abelian model with massive Fermi fields.
6. Conclusion We have re-analyzed the TW model from the functional integral approach using the Wess–Zumino–Witten theory. The present approach give us easiness to read off the equivalence of the QED2 (QCD2) with gauge symmetry breakdown and the TW model (non-Abelian TW model). w; Al g are singlet under gauge In the indefinite-metric formulation, the fields fw; transformations generated by the longitudinal current. This implies that the field algebra is by itself the physical field algebra. The GI limit can be performed in the field operators written in terms of the non-canonical free field f, leading to the Lowenstein–Rothe–Swieca solution for the QED2. The positive-definite formulation is obtained by performing a canonical transformation that maps the redundant Bose degrees of freedom into a zero-norm gauge excitation. The fermion field operator is now written in terms of the charge-carrying fermion field of the Thirring model. In this case, the QED2 limit is defined only for the GI field subalgebra. The ‘‘anomalous’’ vector Schwinger model considered in [9] is nothing but the Thirring–Wess model. As a matter of fact, the structural physical aspect underneath the conclusion given in [9], according with the parameter a apparently controls the screening and confinement properties in the ‘‘anomalous’’ vector Schwinger model, is that the Hilbert space of states of the TW model exhibits charge sectors and thus there is no confinement at all.
Acknowledgment One of us (L.V.B.) wishes to thank R.L.P.G. Amaral for some very clarifying conversations.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
W.E. Thirring, J.E. Wess, Ann. Phys. 27 (1964) 331. C.R. Hagen, Nuovo Cimento B 51 (1967) 169. N. Nakanishi, Progr. Theor. Phys. 58 (1977) 1580. M. Bander, Phys. Rev. D 13 (1976) 1566. M. Kaku, Phys. Rev. D 12 (1975) 2330. J. Lowenstein, J.A. Swieca, Ann. Phys. (NY) 68 (1971) 172. K.D. Rothe, J.A. Swieca, Phys. Rev. D 15 (1977) 1675. J.A. Swieca, Fortschritte der Physik 25 (1977) 303. S.A. Dias, R.A. Casana, Int. J. Mod. Phys. A 15 (2000) 4603. R. Jackiw, R. Rajaraman, Phys. Lett. 54 (1985) 1219. E. Witten, Commun. Math. Phys. 92 (1984) 455. A.M. Polyakov, P. Wiegmann, Phys. Lett. B 131 (1983) 121; Phys. Lett. B 141 (1984) 224. R.L.P.G. Amaral, L.V. Belvedere, K.D. Rothe, F.G. Sholtz, Ann. Phys. 262 (1998) 1. L.V. Belvedere, R.L.P.G. Amaral, K.D. Rothe, Ann. Phys. 14 (1999) 1163. L.V. Belvedere, J. Phys. A 33 (2000) 2755. E. Abdalla, M.C.B. Abdalla, K.D. Rothe, Nonperturbative Methods in Two-dimensional Quantum Field Theory, World Scientific, Singapore, 1991.
The Thirring–Wess model revisited: a functional integral approach L.V. Belvedere, A.F. Rodrigues* Instituto de Fı´sica, Universidade Federal Fluminense, Av. Litoraˆnea, S/N, Boa Viagem, Nitero´i, CEP.24210-340, Rio de Janeiro, Brazil Received 21 October 2004; accepted 7 December 2004 Available online 1 February 2005
Abstract We consider the Wess–Zumino–Witten theory to obtain the functional integral bosonization of the Thirring–Wess model with an arbitrary regularization parameter. Proceeding a systematic of decomposing the Bose field algebra into gauge-invariant- and gauge-non-invariant field subalgebras, we obtain the local decoupled quantum action. The generalized operator solutions for the equations of motion are reconstructed from the functional integral formalism. The isomorphism between the QED2 (QCD2) with broken gauge symmetry by a regularization prescription and the Abelian (non-Abelian) Thirring–Wess model with a fixed bare mass for the meson field is established. 2004 Elsevier Inc. All rights reserved. PACS: 199811.15.Ex; 199811.15.Tk Keywords: Regularization; Bosonization; QED2; QCD2
1. Introduction The Thirring–Wess (TW) model [1] was considered in [2–7] in order to investigate the way in which QED2 can be understood as a limit of a vector meson theory when *
Corresponding author. Fax: +5521 26295887. E-mail addresses: [email protected]ff.br (L.V. Belvedere), armfl[email protected]ff.br (A.F. Rodrigues).
0003-4916/$ - see front matter 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2004.12.003
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the bare mass (mo) of the meson tends to zero. The standard TW model, in which the gauge invariant regularization prescription is adopted, corresponds to quantize the model absorbing all effects of the gauge symmetry breakdown in the bare mass of the meson theory. The computation of the vector current is performed using the gauge invariant regularization for the point-splitting limiting procedure, which corresponds to the regularization parameter a = 2. Within a formulation in a positive-definite metric Hilbert space, the limit mo fi 0 does not exist for the Fermi field operator (w) and vector field operator ðAl Þ themselves [7]. The zero mass limit is not well defined for the general Wightman functions that provide a representation of the gauge non-invariant field algebra of the TW model. However, the gauge invariant Wightman functions of the QED2 are obtained as the zero mass limit of the Wightman functions of the gauge invariant field subalgebra of the standard TW model. More recently, the QED2 with broken gauge symmetry, by the use of an arbitrary regularization parameter a, has been considered in [9] and entitled ‘‘anomalous’’ vector Schwinger model. The standard QED2 should corresponds to the gauge invariant regularization a = 2. However, also in this case, the gauge invariant limit a fi 2 does not exist for the fields w and Al themselves. The limit a fi 2 is well defined only for the gauge invariant subset of Wightman functions. The structural problem associated with the zero bare mass limit mo fi 0 of the Wightman functions for the standard TW model can be mapped into the corresponding problem of perform the limit a fi 2 in the QED2 with gauge symmetry breakdown. One of the purposes of the present work is to fill a gap in the existing literature by discussing structural aspects of the TW model from the functional integral formulation and by extending the analysis to the general TW model regularized with an arbitrary parameter a. To this end, we use the Abelian reduction of the Wess–Zumino–Witten theory to consider the functional integral bosonization of the generalized TW model. To obtain a better insight into the behavior of gauge variant operators, in the present approach we adopt the systematic of decomposing step-by-step the Bose field algebra into gauge-invariant (GI) and gauge-non-invariant (GNI) field subalgebras, such that the effective bosonized quantum action decouples into GI and GNI pieces. The gauge symmetry breakdown is characterized by the presence of a non-canonical free massless Bose field. The vector field with bare mass mo acquires a dynamical mass ~ 2o ¼ m
e2 ða þ 2Þ þ m2o : 4p
ð1:1Þ
Within the present approach we obtain a formulation in an indefinite-metric Hilbert space of states and the Proca equation is satisfied in the weak form. The generalized field operators are reconstructed from the functional integral formalism and are written as gauge transformed fields by an operator-valued gauge transformation. In the indefinite-metric formulation, the GI limit can be performed and we obtain the corresponding field operators of QED2, as obtained by Lowenstein–Swieca [6]. Performing a canonical transformation, the singular gauge part becomes the identity and we obtain the generalized operator solution for the coupled Dirac–Proca equations. For the gauge invariant regularization a = 2, we recover the operator solution of the standard TW model, as obtained by Lowenstein–Rothe–Swieca [6,7]. Since in this
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case the longitudinal current does not carry any fermionic charge selection rule, commutes with itself and commutes with all operators belonging to the field algebra, it is reduced to the identity operator. This leads to a positive-metric Hilbert space of states and the coupled Dirac–Proca equations are satisfied in the strong form. Since in this case the Fermi field operator is given in terms of the charge-carrying fermion operator of the Thirring model, the zero mass limit exists only for the GI field subalgebra. This streamlines the presentation of [6,7]. Another purpose of the present paper is to discuss the isomorphism between the QED2 with broken local gauge invariance (the ‘‘anomalous’’ vector Schwinger model considered in [9]) and the TW model. Using the Wess–Zumino–Witten theory, we show that the effective quantum action of the QED2 quantized with a gauge non-invariant regularization b „ 2, is equivalent to the quantum action of the TW model with a vector field with bare mass e2 ðb aÞ; ð1:2Þ m2o ¼ 4p where a < b is the parameter used in the regularization of the TW model. In this way, the gauge invariant limit b fi 2 for the QED2 with broken gauge symmetry, is mapped into the limit mo fi 0 for the standard TW model (a = 2). As is well known [7,8], the confinement phenomenon in the standard QED2 is associated with the absence of charge sectors. In this way, the conclusion of [9], according with the parameter a controls the screening and confinement properties is nothing but that the TW model exhibits a charge-carrying fermion operator and thus there is no confinement. The introduction of the mass term for the Fermi field of the generalized TW model is also considered. For the GI regularization we recover the operator solution obtained by Rothe–Swieca [7] for the massive TW model. In the Section 5 we consider the functional integral bosonization of the non-Abelian TW model with an arbitrary regularization parameter. In this case, the quantum action of the non-Abelian TW model is mapped into the action of the QCD2 with gauge symmetry breakdown. 2. Functional integral bosonization The Thirring–Wess (TW) model is defined [1] from the classical Lagrangian density,1 l ol ecl Al Þw þ 1m2 Al Al ; L ¼ 14Flm Flm þ wðic ð2:1Þ 2 o where the field-strength tensor Flm is given by Flm ¼ om Al ol Am :
ð2:2Þ
00 01 10 l 5 lm l 5 0 1 m ~ Our conventions are: g = 1, = 01 = = 1, c c = cm, = c ol; ol ¼ lm o , c = c c ; 0 1 0 1 , c1 ¼ ; o± = o0 ± o1; A ¼ A0 A1 ; = clol;For a free massless scalar field, c0 ¼ 1 0 1 0 ~ ~ ¼ ol U. The Fermi field W is the two-component spinor W ¼ w1 . ol U w2 1
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The local gauge invariance is broken by the mass term for the vector field. In the classical level, the local gauge invariance can be restored by performing in (2.1) the limit mo fi 0. This limit corresponds to the classical Lagrangian of the two-dimensional electrodynamics. However, in the quantum level, the zero mass limit for the vector field, even for a gauge invariant regularization prescription, is well defined only for the gauge invariant subset of Wightman functions [7]. 2.1. Decoupled quantum action To obtain the bosonized action, we shall consider the Abelian reduction of the Wess–Zumino–Witten theory [11]. To this end, let us consider the generating functional (in Minkowski space), Z 2 l l Z½#; #; | ¼ exp i d zð#w þ w# þ |l A Þ ; ð2:3Þ where the average is taken with respect to the functional integral measure, Z iS½w;w;A l dl ¼ DAl DwDwe ;
ð2:4Þ
and the Lagrangian density is written as, L ¼ 14Flm Flm þ wy1 Dþ ðAÞw1 þ wy2 D ðAÞw2 þ 12m2o Aþ A ;
ð2:5Þ
where : D ðAÞ ¼ ðio eA Þ:
ð2:6Þ
The vector field can be parametrized in terms of the U (1) group-valued Bose fields (U, V) as follows [13–15]: 1 Aþ ¼ U 1 ioþ U ; e
ð2:7Þ
1 A ¼ V io V 1 e
ð2:8Þ
with U ¼ e2i V ¼e
pffiffi pu
pffiffi 2i pv
;
ð2:9Þ
;
ð2:10Þ
such that, ðAÞw ¼ ðU w Þy ðioþ ÞðU w Þ þ ðV 1 w Þy ðio ÞðV 1 w Þ: w 1 1 2 2
ð2:11Þ
To decouple the Fermi- and vector fields in the Lagrangian (2.1), the spinor components (w1, w2), are parametrized in terms of the Bose fields (U, V) and the free Fermi field components (v1, v2), w1 ¼ U 1 v1 ;
ð2:12Þ
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w2 ¼ V v2 :
427
ð2:13Þ B
To introduce the systematic of decomposing the Bose field algebra I , IB ¼ IB fU ; V g;
ð2:14Þ
into gauge invariant (GI) and gauge non-invariant (GNI) field subalgebras, let us consider a class of local gauge transformations g acting on the Bose fields U, V, as follows: g : U ! g U ¼ Ug;
ð2:15Þ
g : V ! g V ¼ g1 V ;
ð2:16Þ
with gðxÞ ¼ e2i
pffiffi pKðxÞ
:
ð2:17Þ
Under g-transformations the fields (u, v) transform according with, g : u ! u þ K;
ð2:18Þ
g : v ! v K:
ð2:19Þ
The field combination (u + v) is gauge invariant, whereas the combination (u v) is gauge non-invariant. The vector field and the Fermi field transform under g according with, g
1 Al ¼ Al þ giol g1 ; e
ð2:20Þ
g
w1 ¼ ðg U 1 Þv1 ¼ g1 w1 ;
ð2:21Þ
g
w2 ¼ ðg V Þv2 ¼ g1 w2 :
ð2:22Þ
Let us introduce the gauge invariant Bose field G, pffiffi : G ¼ UV ¼ e2i pðuþvÞ ;
ð2:23Þ
such that, g
G ¼ g U g V ¼ UV ¼ G:
In terms of the field G, the field-strength tensor is given by, pffiffiffi p 1 1 1 ðu þ vÞ: F01 ¼ ðo Aþ oþ A Þ ¼ oþ ðG io GÞ ¼ e 2 2e
ð2:24Þ
ð2:25Þ
and the Maxwell action can be written as, Z Z 1 1 2 S M ½UV ¼ 2 d2 zðu þ vÞ2 ðu þ vÞ ¼ S M ½G ¼ 2 d2 z½oþ ðGio G1 Þ 8e 2~ lo Z 1 2 d2 z½o ðG1 ioþ GÞ ; ¼ 2 ð2:26Þ 8e
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where we have defined e2 ~2o ¼ : ð2:27Þ l 4p Let us return to the functional integration in the generating functional. Introducing the identities, Z 1 ¼ DU ½det Dþ ðU ÞdðeAþ U 1 ioþ U Þ; ð2:28Þ 1¼
Z
DV ½det D ðV ÞdðeA V io V 1 Þ;
ð2:29Þ
the change of variables fAþ ; A g ! fU ; V g is performed by integrating over the vector field components A . Performing the Fermi field rotations (2.12) and (2.13), and taking due account of the Jacobians in the change of the integration measure in the generating functional, we obtain the effective integration measure [13–15], 0
dl ¼ D vDvDU DV eiS½U ;V ;v;v eiS ½U ;V ;a;mo ; where, S½U ; V ; v; v ¼
Z
d2 z vi v þ S M ½UV ;
ð2:30Þ
ð2:31Þ
and S 0 [U, V, a, mo] is the GNI contribution, which is given in terms of the Wess–Zumino–Witten (WZW) functionals C [U], C [V], Z 1 S 0 ½U ; V ; a; mo ¼ C½U C½V 2 ða~ l2o þ m2o Þ d2 zðU 1 oþ U ÞðV o V 1 Þ: 2e ð2:32Þ The term carrying the regularization parameter a in the action (2.32) corresponds to the Jackiw–Rajaraman action, Z a 2 ~o d2 zAþ A ; S JR ¼ l ð2:33Þ 2 which in an anomalous model, such as chiral QED2, characterizes the quantization ambiguity [10]. The value a = 2 corresponds to the GI regularization. The WZW functionals enter (2.32) with negative level [11]. In the Abelian case, the WZW functional reduces to the action of a free massless scalar field, Z 1 d2 zðol hÞðol h1 Þ: C½h ¼ ð2:34Þ 8p Using the Abelian reduction of the Polyakov–Wiegmann (PW) identity [12], Z 1 d2 zðg1 oþ gÞðho g1 Þ; C½gh ¼ C½g þ C½h þ ð2:35Þ 4p the total effective action can be written as, S½U ; V ; v; v ¼ S½ v; v; U ; V þ S 0 ½U ; V ;
ð2:36Þ
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where ð0Þ
S½ v; v; U ; V ¼ S F ½ v; v þ S M ½UV ; S 0 ½U ; V ¼
ð2:37Þ
1 1 f~ l2o a þ m2o gC½UV þ 2 f~ l2 ða 2Þ þ m2o gðC½V þ C½U Þ: 2 2~ lo 2~ lo o
ð2:38Þ The standard TW model corresponds to a = 2 and in the limit mo fi 0, the gauge invariance is restored, S 0 ½U ; V ! S 0 ½G;
ð2:39Þ
such that the action (2.36) reduces to the action of the standard QED2, v; v: S½U ; V ; v; v ! SQED ½G;
ð2:40Þ
For a gauge non-invariant regularization, in the limit mo fi 0, we obtain the action for the QED2 with broken gauge symmetry, b 1 ð2:41Þ S 0 ½U ; V ; b; 0QED ¼ C½UV þ ðb 2ÞðC½V þ C½U Þ: 2 2 where b is the corresponding regularization parameter. The actions (2.38) and (2.41) are equivalent, S 0 ½U ; V ; b; 0 S 0 ½U ; V ; a; mo ;
ð2:42Þ
provided that b > a and ~2o ðb aÞ: m2o ¼ l
ð2:43Þ
This implies that the QED2 quantized with a GNI regularization b „ 2 (QED2 with broken gauge symmetry) is isomorphic to the TW model with a regularization parameter a and with a fixed bare mass for the vector field given by (2.43). To proceed further the decomposition of the Bose field algebra IB fU ; V g into GI e and GNI field subalgebras, let us introduce the gauge invariant pseudo-scalar field /, : 1 e¼ ðu þ vÞ; ð2:44Þ / 2 and the gauge non-invariant scalar field f, : f ¼ 12ðu vÞ:
ð2:45Þ
The Bose fields (U, V), defined by (2.9) and (2.10), can be decomposed in terms of the e and the gauge non-invariant field h [f] as gauge invariant field g½ / U ¼ gh;
ð2:46Þ
V ¼ gh1 ;
ð2:47Þ
where g ¼ e2i h¼e
pffiffi ~ p/
pffiffi 2i pf
;
ð2:48Þ
:
ð2:49Þ
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The gauge invariant field G can be rewritten as pffiffi ~ G ¼ UV ¼ g2 ¼ e4i p/ :
ð2:50Þ
The vector field can be decomposed in the standard form in terms of GI and GNI contributions, pffiffiffi p ~ e fol / þ ol fg; Al ¼ ð2:51Þ e and the Maxwell action (2.26) is now given by Z 1 e 2: S M ½g ¼ 2 d2 zð /Þ 2~ lo
ð2:52Þ
The Bose field algebra can be decomposed into IB fU ; V g ¼ IBGI fgg IBGNI fhg;
ð2:53Þ
and the effective quantum action S½U ; V ; v; v can be rewritten in terms of the fields g and h, S½U ; V ; v; v ¼ S½g; h; v; v ¼ S½g; v; v þ S 0 ½g; h;
ð2:54Þ
where ð0Þ
S½g; v; v ¼ S F ½ v; v þ S M ½g; S 0 ½g; h ¼
ð2:55Þ
1 1 f~ l2 a þ m2o gC½g2 þ 2 f~ l2 ða 2Þ þ m2o gfC½gh1 þ C½ghg; 2~ l2o o 2~ lo o ð2:56Þ
with the Maxwell action given by Z 2 1 S M ½g ¼ d2 z oþ ðgio g1 Þ : 2 8p~ lo
ð2:57Þ
Using the P–W identity, the GNI action S 0 [g, h] given by (2.56) decouples into, S 0 ½g; h ¼ S½g þ S½h ¼
~2 ~ 2o m m C½g þ C½h; ~2o ~2o l l
ð2:58Þ
where ~2o ða þ 2Þ þ m2o ; ~2 ¼ l m
ð2:59Þ
~2o ða 2Þ þ m2o : ~ 2o ¼ l m
ð2:60Þ
e h [f], decouple in the action (2.58) and the GNI field h [f] is a free The fields g½ /, massless non-canonical scalar field. The total partition function factorizes as Z Z DgD vDv expfiS½g þ S½g; v; vg ¼ Zh Zv;v;g : Z¼ Dh expfiS½hg ð2:61Þ
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The partition function Zh characterizes the local gauge symmetry breakdown. Although the partition function can be factorized, the generating functional, and thus the Hilbert space of states H, cannot be factorized H 6¼ Hv;v;g Hh :
ð2:62Þ
~ o ¼ 0, that is obThe bonafide gauge invariant vector model corresponds to m tained with mo = 0 and the gauge invariant regularization a = 2. In this case, all reference to the field h [f] has disappeared from the effective quantum action (2.58), which is given in terms of the gauge invariant field g. In this case, the field f is not a dynamical degree of freedom and corresponds to a pure c-number gauge excitation. The commutator (2.85) and the corresponding Hamiltonian of the effective bosonized quantum action are singular for m~o = 0. For these critical values of the parameters, the GNI action vanishes, S½h ¼
~ 2o m C½h ! 0; ~2o l
ð2:63Þ
~ o ¼ 0, the gauge invariance is formally restored and the field h [f] implying that for m is not a dynamical degree of freedom. At this critical point the constraint structure of ~ o ! 0 the field h becomes a the model change. Since, in the gauge invariant limit m pure gauge excitation, the corresponding partition function reduces to a gauge volume, Z lim Zh ! Dh ¼ V gauge : ð2:64Þ ~ o !0 m
The ‘‘anomalous vector Schwinger model’’ considered in [9] is obtained considering mo = 0 and b „ 2. The GNI action is now given by S 0 ½g; h ¼ ðb þ 2ÞC½g þ ðb 2ÞC½h;
ð2:65Þ
and corresponds to the TW model with bare mass for the vector field given by (2.43). 2.2. Local action The Maxwell action (2.57) is non-local due to the quartic-self-interaction of the field g. To dequartize the action of the gauge invariant field g, let us consider the functional integral over the field g in the partition function Zv;v;g . To begin with, let us introduce an auxiliary gauge invariant field X, such that, (Z ( )) Z 2 ~ m 1 1 2 2 Dgexp i d z ½oþ ðgio g1 Þ oþ go g1 8p l ~2o 8p~ l2o (Z ( )) Z ~2 1 2 1 1 m 2 1 1 oþ go g DXDgexp i d z X þ pffiffiffi X½o ðg ioþ gÞ : 2 8p l ~o 2 pl ~2o ð2:66Þ
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Making the change of variables, : o X ¼ ðxio x1 Þ; we can write, 1 X ¼ o1 ; ðxio x Þ: Rescaling the exponential field x, pffiffiffi ~o 1 2 pl 1 ioþ x; x ioþ x ¼ x 2 ~ m
ð2:67Þ ð2:68Þ
ð2:69Þ
we obtain from (2.66) after an integration by parts, ( Z ( Z ~ 4 1 ~ 2 1 1 m 1 m 2 1 2 1 ioþ xÞ DxDg exp i d z ½o ð xio Þ þ ðg ioþ g x x 8p l 8p l ~2o ~2o )) ~ 2 1 1 m 1 1 1 x ioþ xÞð Þ xio x Þ : ðgio g xio ðx ð2:70Þ 8p l ~2o Defining a new gauge invariant field h, : 1 ioþ x; h1 ioþ h ¼ g1 ioþ g x
ð2:71Þ
: x 1 ; hio h1 ¼ gio g1 xio
ð2:72Þ
the field g can be factorized as, g ¼ hx:
ð2:73Þ
The total effective action is now written in terms of the gauge invariant fields h, x, the gauge non-invariant field h and the free Fermi field v. To obtain the complete bosonized action, we introduce the bosonized form for the action of the free Fermi field [16], Z 1 ð0Þ d2 zol uol u; S F ¼ C½fu ¼ ð2:74Þ 2 with fu ¼ e2i
pffiffi pu
;
ð2:75Þ
and the Mandelstam representation for the bosonized free massive Fermi field, Z 1 n p o
j 12 pffiffiffi 5 o 5 1 0 1 ~ ðxÞ þ ~ ðx ;z Þ :; vðxÞ ¼ exp i c : exp i p c u dz o0 u ð2:76Þ 4 2p x1 where jo is an arbitrary finite mass scale. The total local action is given in the bosonized form by, ~2 m h ¼ C½fu C½h þ C½x þ 2o C½h S½f ; g; h ¼ S½f ; h; x; ~o l 4 Z ~ 1 m xÞ 2; ð2:77Þ d2 z½o1 ðxio 2 8p l ~o
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as, where the field h is quantized with negative metric. Parametrizing the fields h and x pffiffi ð2:78Þ hðxÞ ¼ e2i p~gðxÞ ; pffiffiffi l ~ ~ ; ð2:79Þ xðxÞ ¼ exp 2i p o2 RðxÞ ~ m performing the field scaling, l 0 ~ g! o~ g; ~ m ~ !m ~ 0; ~R R
ð2:80Þ ð2:81Þ
and streamlining the notation by dropping primes everywhere, we obtain for the e gauge invariant field /, ~o e ¼l ~ ð~ g þ RÞ; / ~ m pffiffiffi l ~o ~ ¼ exp 2i p ð~ g þ RÞ : g ¼ hx ~ m
ð2:82Þ ð2:83Þ
The effective bosonized total Lagrangian density is then given by, ~2 ~2 1 ~ 2o 1 1 1m 2 2 ~ 2m ~ Þ2 þ ðol RÞ R ðol ~ L ¼ ðol u ðol fÞ : gÞ þ 2 2 2 2 l 2 ~2o
ð2:84Þ
The field ~ g is quantized with negative metric. The gauge non-invariant field f is a non-canonical free massless decoupled field, ½fðxÞ; fðyÞ ¼
~2o l Dðx y; 0Þ: ~ 2o m
ð2:85Þ
For massless Fermi fields, although the total partition function factorizes into free field partition functions, Z ¼ Zf Zh Zx Zh ;
ð2:86Þ
the generating functional does not factorizes.
3. Field operators and Hilbert space The Bose fields (U, V) are given by, pffiffiffi pffiffiffi l ~ ~ ~ h½f; U ¼ expf2i p/gh½f ¼ exp 2i p o ð~ g þ RÞ ~ m pffiffiffi pffiffiffi l ~o 1 ~ ~ V ¼ expf2i p/gh ½f ¼ exp 2i p ð~ g þ RÞ h1 ½f: ~ m
ð3:1Þ ð3:2Þ
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The bosonized form of the Fermi field operator is given in terms of the free Fermi field as,
j pffiffiffi l ~o 5 p 5 o ~ wðxÞ ¼ expfi c g : exp 2i p c f~gðxÞ þ RðxÞg : ~ 4 2p m Z 1 pffiffiffi 5 0 1 1 ~ ~ o0 uðx ; z Þ dz g : h1 ½f; ð3:3Þ : exp i pfc uðxÞ þ x1
and the vector field is given by, Al ¼
1~ ~ 1 ol ð R þ ~ gÞ þ hiol h1 : ~ m e
ð3:4Þ
The divergent part h [f] has the form of a gauge term. The bosonized expressions for fields (3.3) and (3.4) correspond to those of the Schwinger model gauged by a diver~ o ! 0. gent operator-gauge transformation as long as m The vector current is computed with the standard point-splitting limit procedure, . .. 1 J l ðxÞ ¼ ..wðxÞc l wðxÞ. ¼ Z ðÞ
Z xþ l lo Al ðzÞ dz wðxÞ V :E:V : : wðx þ Þcl exp ia~ ð3:5Þ x
In terms of the GI and GNI field combinations (u ± v), the vector current is given by, ~ ~ e e l l J l ¼ |l o ða þ 2Þ~ ol ðu þ vÞ o ða 2Þol ðu vÞ; 2 2 2 2
ð3:6Þ
where |l is the conserved current associated with the free Fermi field v, . . 2 ~: ol u |l ¼ .. vcl v.. ¼ pffiffiffi ~ p
ð3:7Þ
The bosonized form of the vector current is given by, ~ m2o Þ ~ ~ e ðm ~þ ~ o ðm ~ 2o m2o Þol f: J l ¼ 2~ ol ð R þ ~ ol u lo ~ gÞ l ~ 2 m
ð3:8Þ
The conserved current that acts as the source of Flm is given by ~2 ~ ~2 e e m m Kl ¼ m2o Al J l ¼ |l þ ol ðu þ vÞ þ o ol ðu vÞ; 2 2 2~ lo 2~ lo which can be written as, ~ þ Ll ; ~~ Kl ¼ m ol R
ð3:9Þ
ð3:10Þ
where Ll is a longitudinal current of zero norm given by, Ll ¼ ol L;
ð3:11Þ
with the potential L given by, ~ þ L ¼ 2~ lo u þ mg
~ 2o m f: ~o l
ð3:12Þ
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The longitudinal current Ll commutes with itself and thus generates zero-norm states from the vacuum, hWo jLl ðxÞLm ðyÞjWo i ¼ 0:
ð3:13Þ
Due to the presence of the longitudinal current Ll, the Proca equation is not satisfied as an operator identity, e ð3:14Þ om Fml þ m2o Al J l ¼ Ll : 2 w; Al g generate a local field algebra I. These field operThe fundamental fields fw; ators constitute the intrinsic mathematical description of the model and whose Wightman functions define the model. The field algebra I is represented in the indefinitemetric Hilbert space of states H, H ¼ IjWo i: ð3:15Þ In a gauge invariant model, the field f is not a dynamical degree of freedom. In the indefinite metric formulation, the GaussÕ law is satisfied in weak form, om Fml eJ l ¼ Ll ; ð3:16Þ where Ll is the zero-norm longitudinal piece of the vector current that acts as the source in the GaussÕ law. The local gauge transformations of the intrinsic fields w; Al g are implemented by the longitudinal current Ll. The physical gauge fw; invariant field algebra Iphys , is a subalgebra of the intrinsic field algebra I that commutes with the longitudinal current, Iphys I;
ð3:17Þ
½Iphys ; Ll ¼ 0:
ð3:18Þ
The physical Hilbert space Hphys ¼ Iphys jWo i; is a subspace of the Hilbert space H ¼ IjWo i, Hphys H:
ð3:19Þ ð3:20Þ
In a model with gauge symmetry breakdown, the field f is a dynamical degree of freedom. The intrinsic field algebra commutes with the longitudinal current, ½I; Ll ¼ 0: ð3:21Þ w; Al g are singlet under gauge transformations genThis implies that the fields fw; erated by the longitudinal current and thus are physical operators. The intrinsic field algebra is by itself the physical field algebra, I Iphys :
ð3:22Þ
3.1. The QED2 limit It is very instructive to make some comments about the gauge invariant limit ~ o ! 0. From the Lagrangian (2.84), written in terms of the non-canonical field f, m we obtain in the zero mass limit,
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Lm~ o !0 ! LQED :
ð3:23Þ
In this indefinite-metric Hilbert space formulation, the GI limit can be performed in the operator field algebra written in terms of the non-canonical free field f. The operator solution of the QED2, as given by Lowenstein–Swieca [6], can be formally obtained from (3.3), (3.4), (3.10), and (3.11). In the gauge invariant limit, the field f decouples from the quantum action corresponding to the Lagrangian (2.84) and thus it is not a dynamical degree of freedom. The field f becomes a c-number, f (x) fi K (x), and we get, pffiffiffi pffiffiffi ~ wðxÞ ¼: expfi pc5 f~ gðxÞ þ RðxÞgg : vðxÞ expfi pKðxÞg; ð3:24Þ pffiffiffi pffiffiffi p~ ~ p ol ð R þ ~ ol KðxÞ; Al ¼ gÞ þ e e
ð3:25Þ
1 ~ þ Ll ; ol R J l ¼ pffiffiffi ~ p
ð3:26Þ
where Ll is the longitudinal current of zero norm, e Ll ¼ pffiffiffi ol ðu þ gÞ: p Taking this into account, we obtain for the generating functional, #; |l Z½#; ~ o !0 ! Z½#; #; |l QED : m
ð3:27Þ
ð3:28Þ
It must be stressed that this gauge invariant limit can be formally performed only in this level. It cannot be performed in the general Wightman functions due to the singular commutation relation for the field f. This limit is well defined only for the subset of Wightman functions that are independent of the field f. By considering the quantum action and the field operators written in terms of the non-canonical field f, to take the limit mo fi 0 is equivalent to start from the beginning with a zero bare mass for the vector meson. As a matter of fact, in this indefinite-metric Hilbert space formulation, the GI limit can be formally performed as above, since the Fermi field operator is written in terms of the free Fermi field and not in terms of the charge-carrying Fermi field operator of the Thirring model. The intrinsic field algebra I is the physical field algebra, which is singlet under gauge transformations generated by the longitudinal current.
4. The positive-definite-metric formulation The bosonization in the indefinite-metric formulation introduces a larger Bose field algebra IB that contains more degrees of freedom than those needed for the description of the model, I IB . To extract the redundant degrees of freedom, as well as, to obtain the solution of the equations of motion as operator identities in a positive definite-metric Hilbert space of states, we introduce two free Bose fields ~ NÞ, ~ by the following canonical transformation: ðU;
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pffiffiffi l ~ b ~ pffiffiffi ~ þ2 p o~ U ¼ pu g; ~ 2 m
ð4:1Þ
pffiffiffi l pffiffiffi ~ b~ ~ þ p~ N¼ 2 p ou g: ~ 2 m
ð4:2Þ
~ and N ~ are The negative metric quantization for the field ~g ensures that the fields U independent degrees of freedom, ~ ~ ½UðxÞ; NðyÞ ¼0
8ðx; yÞ:
ð4:3Þ
Imposing canonical commutation relations for the fields U and N, we get 2 ~o b2 m ¼ > 0; ð4:4Þ 4p ~2 m ~ is quantized with negative metric. Defining the canonical-free field, and the field N n¼
~o m f; ~o l
ð4:5Þ
the Fermi field and the vector field operators (3.3) and (3.4) can be re-written as ‘‘gauge transformed fields,’’ ^ w ¼ wq;
ð4:6Þ
1 Al ¼ A^l þ qiol q1 : e
ð4:7Þ
Here, the operator q is a pure gauge excitation given by, pffiffiffi l ~o q ¼: exp 2i p ðN nÞ :; ~o m and
~ ~ l ^ ¼: exp 2ic5 pffiffiffi w : WU~ ; p oR ~ m ~o ~ 1 ~ 2l ol R U ; A^l ¼ ~ ~ ~o m m
ð4:8Þ
ð4:9Þ
ð4:10Þ
Here W is the charge-carrying Fermi field operator of the Thirring model, which is given by the Mandelstam representation, Z
l 12 p b~ 2p þ1 ~ 0 1 1 i WðxÞ ¼ o expfi c5 g : exp ic5 UðxÞ o0 Uðx ; z Þ dz : : 4 2 b x1 2p ð4:11Þ The vector current Jl, given by (3.10), can be re-written as ~ 2 m2o Þ ~ ~ 2~ e ðm l m2 ~ Jl ¼ ol R þ o o ~ ol U: ~ ~ om ~ 2 m m
ð4:12Þ
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and the current Kl is given by, ~ þ Ll ; ~~ Kl ¼ m ol R
ð4:13Þ
where the longitudinal current is now given by, ~o m ~ o ol ðN nÞ ¼ pffiffiffi qol q1 : ‘ l ¼ ol L ¼ m ~o 2 pl
ð4:14Þ
The longitudinal current carries no fermion selection rule since it is independent of the vector current of the Thirring model, J Th l ¼
2~ lo m2o ~ ~ ol U: ~ om ~ m
ð4:15Þ
For the GI regularization a = 2, we obtain, b2 m2 ¼ e2 o ; 4p p þ m2o
ð4:16Þ
b ~ ~ J Th ð4:17Þ l ¼ pffiffiffi ol U; p and the field operators given by (4.6) and (4.7) correspond to those operators obtained by Lowenstein–Rothe–Swieca [6,7] gauged by a singular operator gauge transformation. The operator q has zero scale dimension, commutes with itself and thus generates infinitely delocalized states that leads to constant Wightman functions hWo jq ðx1 Þ q ðxn Þqðy 1 Þ qðy n ÞjWo i ¼ 1:
ð4:18Þ
The spurious operator q does not carry any fermionic charge selection rule, and since it commutes with all operators belonging to the field algebra I, it is reduced to the identity operator in H. The position independence of the state q (x)Wo can be seen by computing the general Wightman functions involving the operator q and all operators belonging to the local field algebra I. Thus, for any operator R w; Al g, the poOðfz Þ ¼ OðzÞf ðzÞ d2 z 2 I, of polynomials in the smeared fields fw; sition independence of the operator q can be expressed in the weak form as hWo ; q ðx1 Þ q ðx‘ Þqðy 1 Þ qðy m ÞOðfz1 ; . . . ; fzn ÞWo i ¼ Wðz1 ; . . . ; zn Þ hWo ; Oðfz1 ; . . . ; fzn ÞWo i 8Oðfz Þ 2 I;
ð4:19Þ
where Wðz1 ; . . . ; zn Þ is a distribution independent of the space-time coordinates (xi, yj). Within the functional integral approach, the Hilbert space of states is constructed from the generating functional (2.3), which can be rewritten as, Z ^ þ wq ^ # þ |l ð A #; |l ¼ exp i d2 z #wq ^ l þ 1 q1 iol qÞ ; Z½#; ð4:20Þ e where the average is taken with respect to the functional integral measure, Z Z ð0Þ ~ ~ Re ~ iS½R~ U dl ¼ DNDneiS ½N;n DUD ;
ð4:21Þ
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and the actions appearing in (4.21) are given in terms of the corresponding bosonized Lagrangian densities, 2
2
Lð0Þ ¼ 12ðol NÞ þ 12ðol nÞ ;
ð4:22Þ
~2 ~2 1 ~ 2 þ 1 ðol RÞ ~ 2m R: L ¼ ðol UÞ 2 2 2
ð4:23Þ
Since the free massless fields N and n decouples in the quantum action, the partition function factorizes, ð0Þ
ð0Þ
ð0Þ
ð0Þ
Z ¼ Zn ZN ZU ZR :
ð4:24Þ
In the computation of general correlation functions from the generating functional (4.20), due to the opposite metric quantization, the space-time contributions of the functional integration over the field n cancel those contributions coming from the integration over the field N. In this way, the field q becomes the identity with respect to the functional integration and we get the identities, ^ w w;
Al A^l :
ð4:25Þ
c is builded from the generating funcThe positive-definite metric Hilbert space H tional, Z
^ 2 l l l^ ^ c Z½#; #; | Z½#; #; | ¼ exp i d z #w þ w# þ | Al ; ð4:26Þ where the average is taken with respect to the functional integral measure Z ~ c ¼ DUD ~ Re ~ iS½R~ U : dl
ð4:27Þ
c corresponds to the quotient space The Hilbert space H c¼ H; H H0
ð4:28Þ
where H0 is the zero-norm space. The fields (4.6) and (4.7) provide the operator solution for the coupled Dirac–Proca equations icl ol wðxÞ þ ecl : Al ðxÞwðxÞ :¼ 0;
ð4:29Þ
e om Fml þ m2o Al J l ¼ 0: 2
ð4:30Þ
For the gauge invariant regularization a = 2 and mo „ 0, we recover from (4.25) the operator solution for the TW model obtained by Lowenstein–Rothe–Swieca [6,7]. For mo = 0 and the gauge non-invariant regularization a „ 2, we obtain the operator solution of QED2 with broken gauge symmetry. In the positive-metric Hilbert space formulation, the QED2 limit does not exist for the Fermi field and vector field themselves. In this case, the equations of motion are satisfied as operator identities and the Fermi field is a charge-carrying operator,
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½J 0TW ðxÞ;
WðzÞx0 ¼z0 ¼ e
m2o dðx1 z1 ÞWðzÞ: ~ 2o m
ð4:31Þ
As stressed in [7], if this limit were to exist, we would obtain for the QED2 a local charge-carrying Fermi field operator, which is incompatible with the MaxwellÕs equation being satisfied in the strong form. Nevertheless, this limit is well defined for the gauge invariant field subalgebra, as for instance, 1 ~ ol R; J l ! pffiffiffi ~ p
ð4:32Þ
e ~ Flm ! lm pffiffiffi R: p
ð4:33Þ
4.1. Massive fermions The introduction of a mass term for the Fermi field gives a contribution to the action, ¼ M o fvy v ðUV Þ þ vy v ðUV Þ1 g ¼ M o vy v g2 þ vy v g2 : M ¼ M o ww 1 2
2 1
1 2
2 1
ð4:34Þ Using the decomposition for the GI field g and the bosonized form for the free Fermi field and the bosonized chiral density of the free Fermi field,
j pffiffi o v 1 v2 ¼ ð4:35Þ : e2i pu :; 2p we get,
pffiffi M o jo 2ipffiffipu~ 2 þ e2i pu~ ½xh 2 Þ ðe ½xh 2p pffiffiffi pffiffiffi l ~ M o jo ~ : ~ þ 4 p o ð~ cos 2 pu g þ RÞ ¼ ~ p m
M¼
ð4:36Þ
Performing the canonical transformation (4.1) and (4.2), the mass term can be written as pffiffiffi l ~o ~ M o jo ~ M¼ cos 2bU þ 4 p R : ð4:37Þ ~ p m ~ Notice that, even for a massive fermion field, the field f (n) is a free field. The fields U ~ and R are coupled by the sine-Gordon interaction and the total Lagrangian density is now given by 1 1 1 2 2 ~ 2 þ 1 ðol RÞ ~ 2 L ¼ ðol NÞ þ ðol nÞ þ ðol UÞ 2 2 2 2 pffiffiffi ~o ~ 2 ~ 2 M o jo m ~ þ4 pl ~ : cos 2bU R R ~ 2 p m
ð4:38Þ
For a = 2 we recover from (4.38) the bosonized Lagrangian density for the massive TW model, as obtained by Rothe–Swieca [7].
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5. A glance into the non-Abelian model The classical Lagrangian density defining the non-Abelian TW model is given by icl ol ecl Al w þ 1m2 trAl Al ; ð5:1Þ L ¼ LYM þ w 2 o where the Yang–Mills Lagrangian is given by, LYM ¼ 14trFlm Flm :
ð5:2Þ
Let us apply the Wess–Zumino–Witten theory to obtain the effective bosonic action. To this end, we shall consider the change of variables (2.7), (2.8), (2.12), and (2.13), in terms of the Lie-algebra-valued Bose fields (U, V). The partition function can be factorized as [13,14], ð0Þ
ð0Þ
Z ¼ ZF Zgh Z;
ð5:3Þ
ð0Þ
where ZF is the partition function of free fermions, Z R ð0Þ i d2 z vi v ZF ¼ DvD ve ;
ð5:4Þ
ð0Þ
Zgh is the partition function of free ghosts associated with the change of variables (2.7) and (2.8), Z Z Z ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ Dbð0Þ Zgh ¼ Dbþ Dcþ exp i d2 z tr bð0Þ io c þ þ Dc Z 2 ð0Þ ð0Þ exp i d z tr b ioþ c ; ð5:5Þ and ¼ Z
DU DV expfiS YM ½UV g exp ifC V C½UV þ C½U 1 þ C½V g Z 1 l2o a þ m2o g d2 z tr½ðU1 oþ UÞðVo V1 Þ ; 2 f~ 2e Z
with the Yang–Mills action given by, Z 1 1 2 S YM ½UV ¼ 2 d2 z tr ½oþ ðGio GÞ : 4e 2 In the non-Abelian case, the WZW functional is given by [11,16], C½g ¼ S P rM ½g þ S WZ ½g; where SPrM[g] is the principal sigma model action, Z 1 d2 x tr ½ðol gÞðol g1 Þ; S P rM ½g ¼ 8p and the functional SWZ[g] is the Wess–Zumino action, Z 1 S WZ ½g ¼ d3 xeijk tr½ð^ g1 oi g^Þð^ g1 oj g^Þð^ g1 ok g^Þ: 12p S B
ð5:6Þ
ð5:7Þ ð5:8Þ
ð5:9Þ
ð5:10Þ
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Using the PW identity, the total partition function can be rewritten as, Z ð0Þ ð0Þ DU DV expfiS YM ½UV g Z ¼ ZF Zgh ( 1 1 exp i 2 f~ l2o ða þ 2C V Þ þ m2o gC½UV þ i 2 f~ l2 ða 2Þ þ m2o gC½V 2~ lo 2~ lo o ) 1 2 1 2 iC½U þ i 2 f~ l a þ mo gC½U : ð5:11Þ 2~ lo o From the partition function (5.11), we read off the equivalence of the QCD2 provided with a GNI regularization b „ 2 and the non-Abelian TW model presenting a fixed ~2o ðb aÞ for the vector field. Similar to the Abelian case, this isobare mass m2o ¼ l morphism also holds in the non-Abelian model with massive Fermi fields.
6. Conclusion We have re-analyzed the TW model from the functional integral approach using the Wess–Zumino–Witten theory. The present approach give us easiness to read off the equivalence of the QED2 (QCD2) with gauge symmetry breakdown and the TW model (non-Abelian TW model). w; Al g are singlet under gauge In the indefinite-metric formulation, the fields fw; transformations generated by the longitudinal current. This implies that the field algebra is by itself the physical field algebra. The GI limit can be performed in the field operators written in terms of the non-canonical free field f, leading to the Lowenstein–Rothe–Swieca solution for the QED2. The positive-definite formulation is obtained by performing a canonical transformation that maps the redundant Bose degrees of freedom into a zero-norm gauge excitation. The fermion field operator is now written in terms of the charge-carrying fermion field of the Thirring model. In this case, the QED2 limit is defined only for the GI field subalgebra. The ‘‘anomalous’’ vector Schwinger model considered in [9] is nothing but the Thirring–Wess model. As a matter of fact, the structural physical aspect underneath the conclusion given in [9], according with the parameter a apparently controls the screening and confinement properties in the ‘‘anomalous’’ vector Schwinger model, is that the Hilbert space of states of the TW model exhibits charge sectors and thus there is no confinement at all.
Acknowledgment One of us (L.V.B.) wishes to thank R.L.P.G. Amaral for some very clarifying conversations.
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