Nonregular representations of CCR algebras and algebraic fermion bosonization

,Vol.

33 (1993)

REPORTS

ON

MATHEMATICAL

NONREGULAR REPRESENTATIONS AND ALGEBRAIC FERMION F. ACERB1

G. hIORCXI0

and

”

No. I/2

PHYSICS

OF CCR ALGEBRAS BOSONIZATION a

F. STRO~~HI ’

a International School for Advanced Studies and GNFILI, via Beirut 2-4, I-34013. ‘Trieste, Italy ” Dipartimento di Fisica dell’UniversitL and INFN, Piazza Torricelli 2, I-56126, Pisa, Italy ’ International School for Advanced Studies and INFN, via Beirut 2-4, I-34013, Trieste, Italy (Received

December

17, 1992)

General properties of nonregular representations of CCR algebras are discussed. The relevance of such representations for the theory of superselection sectors and for the mechanism of confinement is outlined. They also play a crucial rGle in providing an algebraic construction of local Fermi fields as ultrastrong limits of bosonic

variables.

Motivations

1.

and results

Nonregular representations models (in QFT, Many Body and sometimes

solved in terms

Such variables Hilbert space. -

Quantum

~

Bloch

which display a singular

particle

infrared

behaviour.

as operators

on a

on a circle

electrons

~

Infinite

Free Bose gas in d < 2

--

Massless

quantum

~~ Schwinger

harmonic

lattice

in d < 2

scalar field in 1+1 dimensions model

~~ Stiickelberg-Kibble U(1)

of variables

formally satisfy the CCR, but cannot be represented Examples of models of this type are given by:

~

~

of CCR algebras naturally show up in the analysis of Theory and Quantum Statistical Mechanics) formulated

current

algebra

model on the circle.

Our strategy in discussing the above models is to keep the canonical structure (in Weyl form) and allow nonregular representations to describe the infrared singular fields. The use of nonregular representations appears to be helpful also for the quantization of systems with constraints, see [1.2]. Buchholz and Fredenhagen have argued that

VI

8

P. ACERBI.

nonregular

representations

plicitly implement

G. RIOHCHIO

naturally

and F. S~L‘ROCY’HI

arise in the construction

of charged

states

For the use of nonregular theories see [4:5]. We focus our analysis ~$1 whose representation

representations

in the canonical

formulation

[6] on the case in which t,he CCR rr, is quasi-free.

algebra

A has a suhalgehra

Then

on A” are given so that ~$2 is uniquel\- determined

ii. Simple conditions alent representations iv. Irreducible

of the (maximal

nonregular

The above framework ~

of A decompose

representations

into sectors

regularj

representations

nonregular) hy K,.

corresponding

subalgebra

to inequiv-

A,,.

are characterized.

can be used. for inst,ance.

clarify t,he H-angle structure

field [3].

of Chern-Simons

i. Given the represent’ation X, of do. one can always define a (possibly representation 7ril of A extending TV. iii. Nonregular

and ex-

their idea in the case of the algebra of the free electromagnetic

for Bloch

to [7]

electrons

and its connection

with H-vacua in

QCD, ~~ determine

the mechanism

of confinement

as the instability

of the charged

sectors

under time evolution. ~

offer a concrete

canonical

strat,egy

for the construction

la DHR [S] in terms of int,ertwiners. algebra on the circle.

of superselection

As an application.

,4s a final point, nonregular representations have heen the essential of the problem of fermion hosonization in If1 dimensions [9]: 1. we show how to construct

fermionic

degrees of freedom

2. local Fermi fields are then given as dtrastrong 2.

Nonregular A CCR

algebra

representations

of CCR

sectors

a

we discuss the U( 1) current

tool in the solution

in terms of CCR

algebras:

limit,s of Bose operators.

algebras

A(V. CT)is the *-algebra generated by elements 1,1’(F), with F bespace (V. CT).the product and the involution in the algebra

longing to a real symplectic being defined bJ

W(F)W(G)

W(F)’

=

II’(F

=

W-F).

+ G)P0(F.Gj’2:

The above equations imply II-(O) = I, WT(F)P’ = II’(F)*. In general, a representation r of A(V. (T) 15 ‘: said to be regular are strongly

continuous

in X, YF

E V.

representation 7riT,is regular. Quasi-free states over CCR algehras quadratic forms 4(.) on (V>0) satisfying Io(F. Gj]”

I

A state

[lO,ll]

;c, is regular

are determined

if ?i(ll’(XF)),

if the

associated

by nondegenerate

X E Iw GNS Hilbert

q(0) = 0 and

q(F) q(G) >

vF.G

E V.

(1)

NONREGULARREPRESENTATIONSOFCCR The corresponding

quasi-free

9

ALGEBRAS

state ~1~ is t,hen given by

w,(W(F))

:=

exp(-i

q(r))

YF E V.

:

To define

nonregular quasi-free states we consider the case in which q(.) is allowed to be degenerate and to take also the value +ccj? still satisfying (1) whenever q(F) and q(G) are finite. The properties of a Hilbert quadratic form (non-negativity, homogeneity. triangle inequality,

parallelogram

law.

) still hold, suitably

extended

this more general case, q will be called a generalized point is PROPOSITION A(V.o)

1. Let q(.)

be a g.q.f.

over

to the case q(F)

quadratic form (g.q.f.).

(V, a);

then

= +a.

In

The relevant

the linear function,al

over

defined by

if q(F) < +c= otherwise is positive nonregular

and hence

defin,es a state (generalized

if there are F E V such that q(F)

The proof of the above proposition

of regular

The subalgebra of d(V,0) generated will be called the regular subalgebra. applications

(g.q.s.)).

This state is

be obtained by generalizing the argument = 0 (YF such that q(F) < +w) and g nondegener-

ate, or by showing that wq is a w*-limit removal of an infrared cutoff).

For the physical

state

can either

in [3] which deals with the case q(F)

states

quasi-free

= SW.

states

by the elements

(with a strategy W(F)

of a regular

to the

q(F)

< +KI

such that

and also as a step towards the classification

it is useful to ask when the knowledge

similar

of nonregular

state w over a CCR

subalgebra

A(Vb,ao), Vo c V, oI~,,~v,, = 00, uniquely determines its extension to d(V.u).To this end we introduce the following notion: a regular state w defined on A(Vo, a~) has A(Vo? 00) as maximal

domain

of regularity in A(V. c) if there

is no regular

extension

of

u: to a larger CCR subalgebra A(V,, o) > d(V0,OO), Vo g VI c V. A state w on A(V, a) is minimally nonregular if its regular subalgebra is its maximal domain of regularity. Lie have then PROPOSITION 2. Let A(&, a~) c d(V,g) be CCR algebras, wq a regular quasi-free state over A(&, o(j) defined by a quadratic form q(.) on (Vo, aa); then the following are equivalent: i. wq has A(Ve, oo) as maximal

domain

ii. ‘Us has no regular and quasi-free

of regularity in A(V, a);

extension

to larger CCR subalgebras of A(V,a);

iii. for any G E V\Vo, a(G, .) is not a bounded linear functional product induced by q;

on VO in the inner

10

F. ACERRI, G. IUORCHIO and F. STROCCHI

iv.

drI

admits a mique

ezten.sio,n

RQ

to A(V. a),

dqfined by the g.q.f. Q:

In particu~lar~ if w4 is pure so is also 0~2. With

the help of this

CCR algebra representations

proposition

we can analyse

the quasi-free

representations

of a

A(V, o) determined by minimally nonregular states wrl in terms of the of the corresponding regular subalgebra d(Vtl, (TO). To this end we define

TF := 7rcro p,P: F E V. where ?r[) := 7rtiv]A(\;,,~,)) and

Hence the following

result holds:

PROPOSITION 3. The represen,tation alent representatiow

rti4 of a, CCR algebra A(V, a) given by a miniinto the direct sum, of kequiu-

quasi-free state wp decomposes

,mally nonregular generalized

TF of the re,qular subalgebra A(L$, o(j), labelled by the equivalen,ce

classes F E V/v/;,.

‘Ft44

--

%F.

@

7i,,,

=

As we will see later, “observable” algebra

the tiF’s have the meaning A( Vo ~‘TO).

To characterize the generalized introduce the following

@

Ft

F~Vll’r,

of the charged

disjoint

sectors

of the

states

which are pure;

it is convenient

DEFINITION 4. A g.q.f. q is said to be minimal

on a symplectic

space

o not necessarily q’(F)

I q(F)

nondegenerate,

quasi-free

TF

V/V,>

if there

exists

no g.q.f.

q’ # q on (V. o)

to

(V. a), with such that

YF E V.

This is the generalization to g.q.s. of the standard characterization states (also degeneracy of (T is allowed!). Then we have

PROPOSITION 5. Given a symplectic

[ll] of pure quasi-free

space (V. o), with o possibly degenerate,

a g.q.s.

wq on A( VT;a) is pure ilff’q is minim,al on (V. o). 3.

Applications

3.a. Bloch electrons The ground state

and O-angle structure

of an electron

in a periodic

potential

provides

an example

of non-

regular representation of a canonical algebra. Within the above framework. it is possible (without relying on semiclassical approximations) to discuss in a mathematically rigorous way the phenomenon of §ors for Bloch electrons. The connection with a similar can now be phenomenon in &CD, advocated [la] on the basis of WKB approximations,

NONREGIJLAR

REPRESENTATIONS

11

OF CCR ALGEBRAS

made precise, with a discussion of the analog of the large gauge t,ransformations, the chiral transformations and their breaking. Bloch’s theorem says that the eigenfunctions of the energy operator for an electron in a periodic

V(x) = V(x + a) (Bloch

potential

wave functions)

have the form

Each $F (x) is non normalizable and gives rise to a nonregular representation of the CCR algebra A := A(IW x I;R,a). In particular, the ground state $ (i.e. n = k = 0) is given by (we set a = 1) n;(W(N.q)

=

The Weyl operators

IV(a.

0), generating

represented

The

regular

by a(:.

o#2nrr,

if

0.

the boosts

subalgebra

on

nEZ.

d(R x R,CT), are

du is generated

nonregularly

by the Weyl

operators

IIV(O: ,/3) = expl;$p. The (unbroken) gauge group is generated by the lattice translations T,, = expinp, 71 E W; they correspond to the large gauge transformations of &CD. The observable algebra Ao~,sr defined as the algebra left pointwise fixed by the action of the gauge group, is generated

by expipp

with ,!3 E Ps, m E Z. space X0,, o of the algebra A decomposes

and by exp2imnx,

It is easily seen that the GNS representation into disjoint

irreducible

One notices

alSO

that

sectors

&bs

by an angle e E [0,27r)

has a nontrivial

state ni is pure on &bs. The analog of the chiral automorphisms

labelled

transformations

of Aohs generat,ed

center

generated

in QCD

by exp iax,

by the T,L’s and that

the

is given by the group of charged

cv E [0,27r).

The following

relations

hold,

with \I’0 := W(6), O)eq:

Since the algebra invariant irreducible 3.b.

of the observables

by the above

“chiral”

representations

Extended

CCR

of

has a nontrivial

transformations,

dohsgiven

algebras

center

which is not left pointwise

it follows that

by the states

and superselection

they are broken

in the

qo. structures

The natural occurrence of superselection sectors in the above model suggests that the use of nonregular representations can be helpful for the construction of charged sectors starting from a regular purpose we introduce

representation

of a canonical

(observable)

algebra

do

[8]. To this

12

F. ACERBI, C:. MORCHIO

and

F. STROCCHI

DEFINITION 6. Let A(&, go) be a CCR algebra and (V, o) a symplectic space such that V > V,, and alr/bx~, = o(); then the CCR algebra A( V, o) is called a canonical extension of A(&, a”). One may then show that

a canonical

group G and can be interpreted the space v, Each

4 E V” defines

:=

extension

d(V,a)

of d(Vo,oo)

as a charged field algebra. (4 E Vrkal :

an automorphism

4(F) = 0

identifies

To this purpose

a gauge

we consider

YF E VO}.

Q$ of d(V.0)

by cu$(W(F))

:=

ei’P(F)W(F).

VF E V. The gauge group G identified by the structure (V> 0) > (Vo, au) is then defined a ++,4 E Vu; notice that A(&. a,)) is left pointwise fixed as the group of automorphisms by the action of s. 1’e then let $ be a g.q.s. on d(V. a), whose restriction We denote

by V, the subspace

PROPOSITION invariant

00) is a regular state. Hence: it results

that

The subgroup GQ, of th,e gauge group G which leaves the state 62,

7.

is characterized

Furthermore,

to A(&,

of V on which the g.q.f. 4 is finite.

by CQ E &I, iff

Gn, is implemented,

in the representation

ra2,: by strongly continuous

uni-

tary operators. Their generators define th,e gauge charges: they lenve 0, invariant. If !&I, is nontrivial, th,en f14 gives rise to n nonregular representation of the field algebra d( V, g). Further 12,:

information

PROPOSITION 0,

is minimally i. &

is obtained

when d(V 0. o(j) is the maximal

domain

Let A(&,

00) be the regular svbalgebra of

d(V,cr)

8.

nonregular,

The

Then, if

one has th,at:

of A(&,

a”),

~0~ decomposes

into the direct sum of inequivalen,t representa-

lnbelled by the gauge charges

iii. the charged fields M’(F), F E V/Vu, act as intertwiners representations o,f d( Vu. 00). 3.c.

in TQ,.

for

= G;

ii. th,e” representation tions

of regularity

Schwinger

Nonregular

between the inequivalent

model

representations

play also a crucial rGle in the discussion

of the Schwinger

model in the Coulomb gauge and provide a characterization of the mechanism of confinement as the instability of the charged sectors under time evolution. This model may

NONREGULAR

be defined starting (Weyl)

algebra s

REPRESENTATIONS

from the time zero U(1)

over the real symplectic :=

and for F = (fi, fz), G = (gl,pJ)

of motion

(where, for simplicity, the CCR algebra A(S

{fES:

in 8s

dz (fd+z(4

read

the variables

realized

as the CCR

forsome

gES}

(2)

- fz(+dd)

.

(3)

at 03 [13] have been frozen to zero) and lead to as the observable algebra Aobs.

x S, a), which can be considered

the strategy

explained

of Aobs obtained by adding the algebra A(V, a), where

V :=

algebra

x S

This algebra is stable under time evolution massive scalar field with m2 = e2/n. Following

f=ag

s

o(F, G) := The equations

current

x U(1)

space V, := (~545 x S, a), where

ELS :=

&X%1@) ?

13

OF CCR ALGEBRAS

in Section

bosonized

{F = (fl,fz)

and the corresponding 3.b, we consider

fermion

E S x iYIS:

-

is that of a

the canonical

extension

at t = 0: in this way one gets

variables

fz(+m)

dynamics

fz(-co)

= n&,

n E Z}

and d-lS

:=

The gauge group is then the U(1)

The ground

state,

{f

E C”,

af

E S}

group ,L?’ defined by

which is a regular

quasi-free

state

on Aohs, is minimally

nonregular

on A(V: 0). It follows that

the gauge group ,@’ is unbroken. Furthermore, the GNS represenof the ground tation of the field algebra A(V, a) given by the (unique) extension 0, state decomposes into a direct sum of irreducible inequivalent representations of &bs labelled by the gauge charge q = R E Z. One verifies that the charged sectors are not stable under time evolution, but are mapped into a one-parameter family of inequivalent representations.

In fact, the charged OF(.)

is mapped

:=

into a one-parameter

Ft

state

. W(F)),

R,(W(-F) family

of states

=

(ff,fi)

f:(k)

=

cos(w,t)fi(k)

j;(k)

=

-w;l

F E V\Vo

a>

:= flF* on Aobs with

E d-‘&T x d-IS, + w, sin(w,t)fi(lc)

sin(w,,t)fi(lc)

+

,

cos(w,t)fi(lc) ,

14

I;. ACERBI,

where di (k) := JG;”+ m2.

Hence,

G. LIORCHIO

the extended

and F. STROCCHI

ground

state

defines a Hilbert

space in

which the time translations are implemented by unitary operators which are not strongly continuous. except on the vacuum sector (confinement as breaking of time tradations). The

3.d.

U( 1) current

Representations

of the U(1)

role in connection rasoro

algebras

the U(1) strategy.

algebra

on the

current

algebra

with the classification [13.15;16].

circle on the circle have played an important

of the representations

The classification

of Kac-Moody

of the positive

energy

and Vi-

representations

of

current algebra on the circle has been given in [17] by implementing the DHR Here, we offer a canonical approach which provides the charged morphisms as

Weyl operators which intertwine between the vacuum representation and the charged sectors. A virtue of our approach is that it works equally well in the case in which the U(l)

current

algebra

lives on the (not compactified)

In two dimensional conserved current).

chiral current

line rather

than

algebra

.

cJ(u) :=

u(z)

.I

A. := sl(S(Sl).

0): tl re symplectic

o”(W(u)) models,

:=

W(Ut) )

whose common

~~(2)

E S(S’)

form being given by

do has a nontrivial center generated by J( 1) = Q. The conformal to the following group of automorphisms of Au:

The various CQFT

the circle.

on

the U(1)

Weyl operators

:= expiJ(u)

the CCR

QFT,

current algebra is generated by the living on a circle (we use here one of the two copies of the total

The associated

l&‘(u) generate

conformal

Hamiltonian

:= u(e-‘tz).

germ is the current

gives rise

(4)

algebra

on

the circle, can

then be recovered on the basis of the classification of all positive energy representations of Au: and in [17] this is done, in the spirit of DHR approach, by using localized morphisms of do.

We show that these representations can also be obtained by using a canonical extension of Au: one recovers a charged field algebra A, a gauge group G and the decomposition of the quasi-free representations of A into inequivalent positive energy representations of AtI. To this end one int,roduces the canonical extension of A0 given by A := A(V, 0); where I’ and X is characterized O(X.U)

by g(X,

:=

{S(S)

.i

~__ 27ri

x E R}

X) = 0 and 1

dz u(z)

:=

& XX :

2

=

G

ILK J’ ()

7469

dOI

vu

E

S(S1).

NONREGULAR

This extension S(Sl)).

REPRESENTATIONS

of o to V is nondegenerate

Notice

(remember

a constant.

that

o is degenerate

on S(S1)

x

also that e”QW(XX)e-“Q

and so W(X)

15

OF CCR ALGEBRAS

has unit “charge”. Hence,

W(X)

77 is a phase factor. The new element

X is uniquely

is uniquely

X is essential

the local automorphisms

e”‘W(AX) characterized

determined

by the above formulas

in A up to q(X)

in the construction

exp (iX(X)Q),

up to where

of the charged fields implementing

(is&P(L)u(I)) W(u),

yP is implemented

by IY(F,,)

carried

zp(z) E S(P).

with

Fp = a,X + pl where up is the charge

(5)

of Ae given by

7,(W(u)) := exp It is easily seen that

=

E

V.

by y,,, up =

s

and p1 is defined by the decomposition

d := i_-pl(Z)

p(z)

dz

UP + -, z

Hence, X implements the singular part of p, namely up/Z. Notice that p1 is determined up to an additive constant, corresponding to the multiplication of W(F,) by an element of the center

of

do.

The canonical extension of the current algebra also allows for a simple To this end we consider of the fusion rules for localized automorphisms. automorphisms

y,, with p localized

the point 8 = 7r. Equation of p and gives locally

in subintervals

of the circle, which do not include e.g.

(6) above can be integrated pi(z)

=

in the complement

of log z at 0 = r, continuity

where c,’ = cP as 0 \ can impose that

= cP as 19/ r.

+ = -i %Cp

of p1 implies c,’ - c;

= 27r,

Since p1 is defined up to a constant,

we

.I

&Pww

This uniquely fixes the implementers W( Fp), and the Weyl relations rule for y,,, yU localized in disjoint intervals:

W(F,)W(F,)

of the support

a,(i log 2 + cP) .

If we fix the cut in the definition r and c;

derivation as in [17]

= W(Fp+g)e*Zsa~ao ,

in

A

give the fusion

F. ACERBI,

16

G. MORCHIO

and F. STROCCHI

where the * signs correspond to the cases in which p is localized sense of increasing 0. in the punctured circle.

to A as a canonical

We now turn to the gauge group G associated It is clear from the definition t,hat G =

{ck&V(F))

= eLP3(%-(F)

after/before

(7, in the

extension

of

A().

: F E V. c$E I/;La,, 4(u)= 0 'duE S(S1)}

and that every ob E 6 is of the form a,; , ( where 4 is fixed and X E LR. Since c$(7~ + XX) X4(X)

for every IL E S(Sl). one can choose 4 such that s’ 1s . a group of inner automorphisms

=

J(X) = 1 and this amounts to of A: the automorphism ax$!

fix (;j(.) = a(., 1).

X E R, is implemented by the central element W(X) (= exp(iXQ) in every representation in which the center of Au is regularly represented). As a further point, we know from the analysis in [17] how to characterize the sets of (a’. B)-KhIS

and ground states

The factor

for the dynamics

(4) above.

(a”, S)-KIWIS stat,es on .& are defined by

:= exp{?g&

w$(W(u))

where g E IR and 21,, := (27ri)-’ The pure ground states

on

~ a 2

ncoth

z

I&/“}

:

vu E S(S1)

s ZL(Z)Z”~~ dz.

A0 are

defined by

In the following we will call ~~ the representation determined by tii. We are now faced with the problem of extending the dynamics in (4) to the algebra

A: we require

to this end that A is stable under the action of the extended dynamics (which we call 6’). Using the fact that A is the smallest *-algebra containing Au and

elements

which are Q-charged, $(@Q)

equation =

,t(,jxQ)

(5) above and the fact that =

?iW ,

YiXER:

one obtains that the most general extension of the dynamics following time evolution for r/tr(X) (77 is a phase factor): &‘(W(gX))

=

r/(9\t)ezA(“.f)QW(.9X)

,

to

A

is determined

by the

VgeIW.

The following results are stated for the choice v(g,t) E 1, X(g,t) E 0: but they hold for all values of these parameters. It is easily shown that both LL(;Iand ~‘0 (we set here g = 0) have do as a maximal domain of regularity in A. It then follows that the unique extensions 00 and Ro of WL~ and wg to A are a factor (G’, O)-KMS state and a pure ground state, respectively. We can also recover the decomposition into sectors given in [17] ( we use for the sake of simplicity only the pure states

here):

NONREGULAR REPRESENTATIONS PROPOSITION decomposes

9.

The gauge group G is unbroken

into a direct sum of inequivalent

17

OF CCR ALGEBRAS The representation

in x0,,.

irreducible

representations

of A,

7rnC,

labelled

by the value of the gauge charge:

TO” = @%. SEW The charged fields W(gX),

g E II%,act as intertwiners

between the inequiwalen ,t represen-

tations of do. Algebraic

4.

fermion

bosonization

It is well known [18,19,20]

that fermion

bosonization

in l+l

dimensions

involves two

steps: i. An infrared problem which essentially consists in smearing bosonic fields with test functions which do not vanish at infinity (infrared singular fields). To this purpose the use of nonregular representations greatly simplifies the discussion. ii. An ultraviolet problem involving the construction of local Fermi fields. By exploiting nonregular representations we can reach them as ultrastrong limits of bosonic variables in all the representations which are locally normal with respect to the vacuum representation of the massless scalar field in l+l dimensions. This result was regarded as unlikely in the literature [19,20], where the best result was a strong limit on a dense set of states

in specific

bosonic

models

[20].

It is important to stress that, in our approach, the Boson-Fermion correspondence will emerge at the canonical level. Our starting point is the U(1) x U(1) current algebra 0 ne then introduces the canonical (2) and (3) above). ( see equations d(BS x S,a) extension

A(S

d-IS,

x

0) (see above).

The first point is to show that the field algebra A(SxdPIS, variables.

For this purpose

Given a bounded localized in I if

we introduce

interval

DEFINITION

W(F)

10.

:=

supp(fl)

K of localized

elements

i. is invariant ii. is invariant iii. elements

under the adjoint

intervals

is said to describe

subset, if the corresponding

operation; in disjoint

is

in I.

under space translations;

which are localized

E S x d_lS

c 1.

of S x a_lS

fermionic degrees of freedom, or briefly is called a fermionic set .?=K of elements W(F), F E K, is such that:

anticommuting

of localization:

u supp(af~)

will also be said to be localized

A subset

C) contains

notion

1 of the real line, we say that F = (fl, fi)

sump The corresponding

the following

anticommute.

18

F. ACERBI.

A straightforward

analysis

by two real parameters, by

shows that

:=

./’

qz(F)

:=

fz(fO0)

=

EC +

where c is a (fixed) The *-algebra

and F. STROCCHI

maximal

q (the “charge”)

a(F)

fz(-30)

G. LlORCHIO

fermionic

and c. Their

subsets

elements

F=*l,

.fl(X)dX = cq, - fz(-IX)

real number.

by 3C_q. It is also finitely

=

t (2,;

I)“.

are characterized

qEW,

n E Z!

y,

generated

exist and are labelled

W(F)

m,EZ, Such subsets

by the W(F),

generated

will be denoted by KC,,. (f ermionic subalgebra)

F E K,.,

i. all elements W(F) with F localized, ql(F) = tq, q2(F) = &r/q, (fermionic operators with charge eq/& and chiral charge *e&?/q); ii. the operator

is denoted

by

WX := IV(&),

F,

f2(-3cj)

= cr,

:= (0,27r/q) E S x d_lS.

It is easily shown that 11. The *-algebra ZFc,, has a nontrivial center generated the Fcc.q are isomorphic for different values of c.

PROPOSITION

thermore,

It follows that

one can set c = 0.

the sense that any extension algebra 3Cc.rl. The following p(z) E D(R)

construction

Set p’(x)

subset

the algebras 3CJ.q are maximal in Kc,, g ives rise to the same fermionic

allows one to construct c [-l/2,1/2]

approximate

Fermi

fields:

let

and

&G/qO)

P(X) 2 0,

fur-

One has also that

of the fermionic

with supp (p(z))

by W,:

:=

.I’

p(z)&

=

fi.

:= 1,‘~ ,D(z/E), E > 0, and define F:(x)

:=

(QE(x),

(0 * p’)(x))

E S x d-lS.

We call tiO the state on A(dS

x S, a) defined by the vacuum

in 1+1 dimensions.

fermion

Algebraic

bosonization

of the massless

is provided

scalar field

by the following

THEOREM 12. Let w be any state on the current algebra A(AS x S,a), which is locally normal with respect to WC),and let Q denote an extension of w to the field algebra A(S x d_lS, o); then for any g E D(Iw) the operators

$4(g)

:=

&

.i

rn (4W%)))g(y)

do 1

with C a suitable constant, converge as E \ 0 in the ultrastrong operator topology a(&). The limits define right/left handed fermions which satisfy the CAR.

on

NONREGULAR

REPRESENTATIONS

19

OF CCR ALGEBRAS

Acknowledgement One of us (F.A.) would atica) of CNR for financial

like to thank support.

the GNFh’I

(Gruppo

Nazionale

di Fisica

Matem-

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