DIRAC-BRACKET QUANTIZATION OF CHIRAL SCALAR TWO-DIMENSIONAL QED

PHYSICAL

REVIE&

0

VOLUME 39,NUMBER 6 15MARCH 1989

Dirac-bracket

quantization

of

chiral

scalar

two-dimensional

_QED

E.

Abdalla

Instituto deFssica, Universidade deSaoPaulo, C.P.2051'6, 01498 SaoPaulo, SaoPaulo, Brazil M.

C.

B.

Abdalla,

F.

P.

Devecchi, and

A.

Zadra

Instituto de Fssica Teorica, Universidade Estadual Paulista, Rua Pamplona, 145,01405 SaoPaulo, SaoPaulo, Brazil (Received 23March 1988;revised _manuscript received 11 July 1988)

%"ediscuss the interaction oftwo-dimensional chiral bosons with an electromagnetic field,

quan-tizing the system via the Dirac procedure. Aquantum theory isobtained; however, due to the mass

generation the system isconsistent only iffurther constraints are imposed. In such a case, the elec-tromagnetic field is trivial. The role ofthe gauge Jackiw-Rajaraman mass parameter is also

dis-cussed.

I.

INTRODUCTION motion are also discussed, as well as the particular value a

=0.

In Sec. IVwe draw conclusions.

Frequently, symmetries do not survive the quantization procedure. A major example

of

such an anomalous con-servation law is given by chiral gauge theories, ' where gauge invariance is broken at the quantum level. In the case

of

two-dimensional spinor

QED

and QCD, there have been several proposals ' _{forcanceling anomalies}

in-troducing Wess-Zumino (WZ) terms. Consistent quanti-zation

of

the chiral Schwinger model has been achieved by Jackiw and Rajaraman, who showed the existence

of

a one-parameter family

of

chiral Schwinger models. Girotti, Rothe, and Rothe quantized the bosonized theory with the Wess-Zumino term via the Dirac pro-cedure for constrained systems. Thus two-dimensional

QED

(QED2) has been shown to depend on the value

of

the Jackiw-Rajaraman (JR) parameter, and may be preserved as two types

of

theories, with either two orfour constraints.

The non-Abelian case was considered in Refs. 8and

9.

In the former reference, the author based his discussion on Witten's' bosonization prescription using Coleman's arguments.

"

In the latter work, a "first-principles" derivation

of

the effective action involving the WZ term

was given, and thereafter the Dirac quantization pro-cedure was applied. As in the Abelian case there were two cases, involving either two or four constraints, the latter in a distinguished value

of

the

JR

a parameter.

In this paper we discuss the coupling

of

chiral bosons toan Abelian gauge field.

Chiral bosons have been recently studied by a number

of

authors. '

'

The theory has been consistently quan-tized' using a nonlocal Lagrangian and alocal commuta-tion rule, or equivalently a local Lagrangian but a nonlo-calcommutation rule.

It

was also shown that the system may be quantized via the Dirac procedure. '

In

Sec.

II

we consider the coupling to a gauge field,

writing the (classically) gauge-invariant action, and the constraints. In Sec.

III

we derive the constraint algebra and quantize the theory via Dirac brackets. Equations

of

II.

CHIRAL SCALAR QED2

A pure left-moving bosonic field _P(xo

—

x

&)may be

de-scribed by the Lagrangian

x

=-,

'a~pa„y

(2.1)

L2(x)

=7r(x)

—

P'(x)

=0

. (2.2)

Some authors proposed to substitute the above con-straints by the first-class constraint'

L

(x)

=

Q(x)

(2.3)

which is dealt by the action'

x=

—,

'& —

gg

PO

yaP

with the gravity field truncated as

(2.4)

0 1

1 (2.&)

which leaves us with the expression

X=d+Pr)

P —,'A,

(8

P)—

(2.6)

where the doubly self-dual field realizes the first-class constraint. A description

of

the model by means

of

the second-class constraint (2.2) is feasible using the Dirac method. The constraint satisfies

I

Q(x),

Q(y)I

= —

25'(x

—

y),

(2.7)

with the constraint

a

y=o,

which may be canonically quantized, as far aswe are able to deal with the second-class constraint

39

BRIEF

REPORTS 1785

which determines a second-class system (although the number

of

constraints seems to be odd, going to a discre-tized version

of

space one sees that an antisymmetric ma-trix emerges' ).

Thus one derives the commutator'

[P(x),

P(y)]=

—

—

~(x

—

y)

.

(2.8) The interaction

of

the above model with a gauge field

was proposed in

Ref.

12. We write the Lagrange density

X=8

PB

P+

A _{8 P}

—

A d _P

,

'F„—

+

——,

'aA"

A„,

(2.9)

with the nonvanishing Dirac brackets

IP(x),

~(y)

j»=&(x

—

y),

[A

'(x

),rr,(y)

]»

=

5(x

—

y),

and the identities

m0=0,

Ao=

—

1

(~I

—

0')

.

Let us now add the chiral constraint

Q,

(x)=sr+

A,

—

P'=0

.

(3.7b)

(3.8a)

(3.8b)

(39) where the last term has an arbitrary parameter a related

to

the gauge symmetry breaking, and should be put in, to take into account the renormalization arbitrariness.

If

this is the only constraint to be imposed, we get, us-ing the Dirac procedure, the equation

of

motion

III.

DIRAC QUANTIZATION OF THE MODEL

P=P'+

—,'

f

dy vr'(y)e(x

—

y),

(3.10)

—

_A1,

_(3.1a)

(3.1b)

1 0 1 10 (3.1c)

Canonical quantization

of

Lagrangian (2.9) may be achieved defining the momenta

which from the point

of

view

of

taking P as being a right-mover field is erroneous, unless we impose the fur-ther constraint m.

_{&=0. For}

_{this reason we impose}

_(3.

_{9) as}

an external constraint to the equations

of

motion in order to achieve a consistent system. In order that

0,

be con-stant in time, with the Hamiltonian (3.6),we get the new

constraint

with the canonical Hamiltonian given by

Qp=~+(1+a)

A,

=0

. (3.11)

M,

=

f

dx'[ ,

'vr,

+

_,

'(m—+A—,)

+

—,

'P'

+

Ao(~')

—

Q')

On the other hand, the equations

of

motion for P,

~,

and

_A],

—

—,

'a(AO

—

A ₎

)],

with the primary constraint Q, =pro(x)

=0

.

(3.2)

(3.

3)

P=~+

A(,

~=/"

+

—

1(

~,

+p)",

—

(3.12a) (3.12b)

The chirality constraint in the canonical formalism is

given by

A,

=

—

m._,

+

—

(m._,

—

P"),

~ ₁

imply

(3.12c)

Q(x)

=~(x)

—

P'(x)+

A,

(x)

=0

. (3.4)

Qq(x)=

—

~',

+P'+aAO=0

.

(3.5)

The system constrained by

01

and

₀₂

is second class, with atotal Hamiltonian given by

HT=

f

dx'

',

(sr+

A,

)

+

,

'm—_,

+

—,

'P'

+

1

——

(

mI+P')

However, we will refrain from imposing (3.4)at the be-ginning, since quantizing the system with (3.3) and

(3.

4) leads toa further constraint

~,

=0.

This condition is ob-tained a posteriori, considering the bosonic equation

of

motion in terms

of

Dirac brackets, after a long, albeit straightforward calculation using the Dirac procedure.

In order to clarify the issue, we first consider the sys-tem using (3.3).

Time conservation

of Q,

leads to the Gauss law

P+~, =0

.

Since Pis a right-mover field, we obtain

(3.13)

03=+1=0

.

(3.14)

The system Q1,Qz,Q3 is consistent and does not gen-erate any further constraint. An even better set

of

con-straints isgiven by

1+a

Q,

(x) =rr

—

P'=0,

a

Q~(x)

=

7r,

=0,

—

4

Q3(x)

=sr+(1+a)

A,

=0

.

(3.15a) (3.15b) (3.15c) The Dirac matrix isreadily computed, and we have the nonvanishing commutators:

+

—

a A 1 (3.6)

[P(x),

P(y)]

=

iA e—

(x

—y),

[P(x),

m(y)]=

iA

5(x

—

y),

(3.

16a)

1786 BRIEFREPORTS 39

1+a

[sr(x), sr(y)]=i@

5'(x

—

_y)

.

Za (3.16c)

Further commutators are obtained using (3.15). We may

write the Hamiltonian using now the constraints strongly

a=

jdx'

'+'y'

_a (3.17)

From (3.16a) and (3.17)we get

(3.18)'

The system is well defined only for a

)

0 or a

(

—

1.

The casea

=0

must betreated separately, since further constraints arise. As it turns out, in that case the theory

is trivial, since we eventually obtain

/=0.

This is the same result as the one obtained from

(3.

16a), making a

=0.

IV. CONCLUSIONS

'The _interaction

_of

_{chiral bosons with a quantized} elec-tromagnetic field yields a theory where all gauge field de-grees

of

freedom disappear due to constraints. The whole theory reduces to a single chiral boson, proportional to the Floreanini-Jackiw solution, the constant

of

propor-tionality being a function

of

the gauge-fie1d mass parame-ter

a.

This isthe only trace

of

the gauge-field interaction. A similar effect appears in the pure chiral boson with a Mess-Zumino term related to Siegel symmetry as it turns out, the efFect

of

the extra WZ term is trivial, due to constraints.

ACKNOWLEDGMENTS

The work

of

E.

A.and M.

C.

B.

A.

was partially support-ed by Conselho Nacional de Desenvolvimento Cientifico

e Tecnologico (Brazil). The work

of

F.

P.D.

and

A.

z.

was

supported by Fundacao de Amparo e Pesquisa do Estado de SaoPaulo (Brazil).

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Abdalla,

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