PHYSICAL REVIEW D VOLUME 45, NUMBER 4 15 FEBRUARY 1992

Chiral-boson quantum mechanics

M. M. Horta Barreira and C. Wotzasek* Instituto de Fisica, Uniuersidade Federal do Rio de Janeiro, Caixa Postal 68528, 21945 Rio de Janeiro, Brazil

(Received 5 September 199 1)

We analyze the chiral quantum oscillator: a quantum-mechanical massless particle, restricted to move on a two-dimensional plane, under a constant, perpendicular magnetic field. The oscillator, which is constrained to wind around the origin in one sense only, mimics the behavior of the two-dimensional self-dual field. This model is used to study certain aspects of the chiral-boson theory.

PACS number(s): 03.65. - w, 11.30.Rd

I. INTRODUCTION

Chiral bosons in two-dimensional space-time and ( 2 + 1 )-dimensional Chern-Simons (CS) gauge theories are related problems which have been attracting much atten- tion. These problems are important for the string pro- gram and for the development of the quantum Hall effect (QHE) [1,2]. The inclusion of a gauge field controlled by a Chern-Simons Langrangian as the kinetic term pro- duces the phenomenon of boson-fermion transmutation [3]. The resulting excitations, called anyons, carry frac- tional spin and statistics and are very important at the present theoretical status of the Q H E [4]. A quantum- mechanical version of CS field theory has been studied by Jackiw and collaborators [5,6]. This model mimics the behavior of its field theory counterpart and makes it easier to obtain information. The model proposed in Ref. [5] is, basically, a quantum-mechanical particle con- strained to move on a two-dimensional plane, interacting minimally with a constant magnetic field, perpendicular to the plane, whose Lagrange function is

This model is the analogue to the three-dimensional topo- logically massive electrodynamics in the Weyl gauge A,=O, whose Lagrange density is

The behavior of the model (1) in the vanishing-mass limit has been thoroughly studied in Refs. [5] and [6] and was shown to be the analogue of a purely CS gauge theory.

In Ref. [6] Bazeia has made the interesting observation that the model described by Eq. (I) , with m =O, is able to mimic the behavior of two-dimensional chiral bosons. This can be seen in the following way. The spectrum of the full theory ( m f O), which has been completely solved in Ref. [5], is characterized by two simultaneously com- mutating operators, the angular momentum M and the

*~lectronic address: USERVOTZ @ LNCC.

Hamiltonian H, whose eigenvalue equations are

where f l = d ~ ~ / 4 r n ~ + ~ / m . The integer n labels the particle angular momentum and Ni t s energy.

Observe that states of both chiralities, i.e., positive and negative n, are present in the massive theory. Classically, this corresponds to a particle winding around the origin both in the clockwise and anticlockwise senses. In the limit m -0, all excited states with N > 0, as well as those with N = O and n < 0, get separated from those states with N=O and n 2 0 by an infinite energy gap and decouple from the spectrum. The remaining states, with only one chirality, get aligned with the external magnetic field. The limiting process "kills" one of the chiralities, result- ing in a massless particle playing the role of the chiral bo- son in two-dimensional (2D) field theory.

It is our intention here to explore the analogy proposed by Bazeia, extending his idea to other formulations of the chiral boson and examining its consequences. To obtain the massless formulation mentioned above we use the fol- lowing alternative route. Observe first, following the spirit of the above discussion, that the two-dimensional harmonic oscillator can be seen as the analogue of a free, massless scalar particle in two space-time dimensions:

This analogy can be achieved making use of the correspondence

2D field theory tt quantum mechanics,

This correspondence maps the Klein-Gordon Lagrangian into a two-dimensional harmonic oscillator, as explained above. To transform the harmonic oscillator into a chiral harmonic oscillator, we use Harada's chiral bosonization technique [7]. The effective action is obtained by impos-

1410 @ 1992 The American Physical Society

45 CHIRAL-BOSON QUANTUM MECHANICS 141 1

ing, in the phase-space path integral, the chiral con- straint, which here is, using the correspondence (51,

Since the Faddeev-Popov determinant for this set of con- straints is independent of the dynamical variables, the re- sult of evaluating the phase-space path integral is equivalent to imposing the constraint directly into the first-order Langrangian [a]. The resulting effective action is

This effective theory is our chiral oscillator, which is the quantum-mechanical analogue of the two-dimensional chiral boson as proposed by Floreanini and Jackiw, and is easily seen to be the vanishing-mass limit of Eq. ( I ) , when w = K / B .

In this paper we shall study the chiral-boson quantum mechanics, or chiral oscillator, following the formula- tions for the quantum chiral oscillator analogue to those proposed by Siegel [9], by Floreanini and Jackiw [lo], and by Srivastava [ l l ] . In Sec. I1 we quantize the Floreanini-Jackiw formulation, given by Eq. (7), accord- ing to the simplectic method proposed by Faddeev and Jackiw [8,12,13]. In Dirac's language [14], this is a second-class constrained system. To covariantly quantize the model we follow Faddeev's ideas and convert the second-class set of constraints into a first-class one by the introduction of Wess-Zumino terms. This conversion of constraints, done in Sec. 11, follows an algorithm developed by one of us in the framework of the chiral Schwinger model [15,16,17]. Siegel formulated the origi- nal chiral-boson theory introducing the chiral constraint squared to obtain a first-class constraint, which presents a sort of gauge symmetry. Its quantum-mechanical coun- terpart is studied in Sec. 111. We see that an analogue of Siegel's symmetry is also present in this model. Finally, in Sec. IV, we analyze the implementation of the chiral constraint in a linear way. For completeness, we review in the Appendix, the Faddeev-Jackiw simplectic quanti- zation for constrained systems.

11. FLOREANINI-JACKIW CHIRAL HARMONIC OSCILLATOR

The Floreanini-Jackiw (FJ) formulation for the chiral quantum-mechanical oscillator is given in Eq. (7). This model is an example of a constrained theory in Dirac's Hamiltonian language, but it has a nonsingular simplectic matrix [a], which can be inverted immediately to provide the generalized bracket as

These brackets coincide with those calculated by Dirac's procedure, as Costa and Girotti have shown [la]. The equation of motion is easily obtained from (7) and (8):

whose solution reads

where q i+ '=q , +iq2. The particle is constrained to move only in the counterclockwise direction. This is analogous to the chiral restriction in the 2 0 self-dual model. The effective Lagrangian (7) is invariant under a global rotation

where h is a time-independent infinitesimal parameter and the symmetry generator is the angular momentum operator M = q X p = w q 2 , which in this case is propor- tional to the energy operator H =w2q '.

As remarked by Floreanini and Jackiw, their chiral bo- son also admits a nonlocal formulation in terms of local fields. Translating to the case under study here, this transformation gives us back the action (7). In fact the transformation analogue to Eq. (19) in Ref. [lo] here is

However, because of the identity cijcjk =6ik, we end up with exactly the Floreanini-Jackiw chiral oscillator we started with.

Covariant quantization for the chiral oscillator

In the language of constrained systems introduced by Dirac, the FJ chiral oscillator presents only second-class constraints. In order to covariantly quantize the model above we can follow the suggestion given by Faddeev and Shatashvili and introduce a Wess-Zumino term, along with some auxiliary variables, turning the second-class constraints into first-class ones. In this section we show how this program can be implemented. Changing slight- ly the notation to accommodate the new (Wess-Zumino) variables and the iterative change of the Lagrangian, we rewrite the original model as

L = w Q ! O ' E . , q ~ O ' - 2 10) ( 0 ) o i 1 , w q i q i . (1 3)

This theory has two primary constraints:

These constraints are second class as is easily seen by computing the Poisson brackets among them:

(fiiO),ny)] = -&.. 11 ' (15)

In order to transform these second-class constraints into gauge-symmetric first-class ones, we introduce some counterterms, changing LO+Ll as follows:

The new theory has four primary constraints:

1412 M. M. HORTA BARREIRA AND C. WOTZASEK

One can easily check that the Poisson brackets of these constraints among themselves are nonvanishing. Never- theless the determinant of the matrix of these constraints is zero, indicating the existence of zero modes. This is signaling the presence of a set of first-class constraints, which can be singled out by a simple redefinition of the constraints as [14]

The original set of constraints is now first class, while the new constraints a:'' remain second class. Ob- serve that the chiral restriction is now applied only over the auxiliary variable q!" that has been introduced along with the Wess-Zumino term. But this is exactly the situa- tion we started with, so we have to repeat the above pro- cedure. Repeating the same algorithm N times we arrive at the following situation:

The model now has N + 1 sets of constraints, N of them being first class and one second class. The conclusion we arrive at is that one has no right to stop the constraint- conversion process at any finite step in order to obtain a purely first-class constraint set. The final theory, ob- tained after an infinite repetition of the constraint- conversion process, has the form

The final model, made of the original chiral oscillator plus its infinite set of auxiliary chiral oscillators, has a gauge-invariant set of constraints.

111. SIEGEL'S FORMULATION

Siegel was the first one to propose an action for the chiral-boson problem. In order to avoid second-class constraints he introduced the chiral constraint squared, maintaining in this way a (gauge) symmetry that would help to eliminate the Lagrange multiplier field used to im- pose the chiral restriction. Following the general scheme presented in the first section, we can write Siegel's version for the quantum-mechanical chiral boson as [6]

To examine the existence of a sort of Siegel symmetry in this model we rewrite the model as

where we have defined a,qi= l / ~ ~ ( ~ i + w ~ i J q j ) . As in its field-theoretical counterpart, we can write (21) as

with the metric gpv given by

The equations of motion are derived by making varia- tions with respect to h and q,:

6h: (a-q, 12=0 ,

sqi: a-(a, +ha- )qj = O

One can check, after some algebra, that under the follow- ing transformations, the Lagrange function changes only by a total derivative:

where r] is an infinitesimal, time-dependent parameter. This is the quantum-mechanical analogue of Siegel's sym- metry. Introducing the creation and destruction opera-

t tors a; and a i , respectively, as

we can write the (squared) chiral constraint as

Observe that no ordering ambiguities appear in the ex- pression above. I t turns out, as expected, that this sym- metry is free from local anomalies but a global one can still manifest itself, as remarked by Reuter [19].

To see the equivalence of the model with the FJ chiral oscillator of Sec. I, we write the Lagrangian L in its first- order form ( L =piqi - V)

This is a truly constrained system in the simplectic ap- proach of Faddeev and Jackiw (see appendix), which means that the simplectic two-form is degenerated and cannot be inverted directly to provide the generalized brackets. This can be clearly seen writing the simplectic matrix for this problem in terms of the variables < = ( q i , ~ i , h l I ,

which is seen to be singular. Its (left) zero mode is 4 '0 '=(0 ,0 , 1 ) and selects the constraints

The implementation of this constraint into Eq. (29) shows the equivalence between Siegel's action, Eq. (211, and the FJ action, Eq. (7).

45 - CHIRAL-BOSON QUANTUM MECHANICS

IV. SRIVASTAVA MODEL multipliers vi. The first-order Lagrangian becomes

Srivastava considered the problem of implementing the chiral constraint linearly. H e showed that the Lagrange multiplier field does not become dynamical, as was sup- posed at the time, but his solution has been criticized by Harada as violating positivity. We have proposed a modification of Srivastava's model in order to solve the positivity problem of its original formulation [20,21]. Let us consider the chiral oscillator with a linear constraint in the form

where p is a free parameter. To examine the simplectic structure we must write this theory in its first-order La- grangian form, i.e., L =pigi - V, where

The simplectic matrix is singular and its zero mode selects the Lagrangian constraint

The potential V becomes, after strongly implementing the constraint,

Introducing the "Kac-Moody" currents J,"'=pi koei ,q, , which satisfy the algebra

we can write v a s

where v is defined in Eq. (35) above. The simplectic ma- trix is, in terms of the simplectic variables {i=(qi,Pi,hi,17iI, given by

which is not singular and can be inverted by the usual means if p# 1. Observe that when P- m the simplectic matrix becomes block diagonal, with the Lagrange multi- pliers sector [ hi, g i ] decoupled from the harmonic- oscillator variables { q, ,pi ] . The matrix will become singular for p+ 1. This case will be treated separately below. The inverse of (41) is

For arbitrary values of /3 we have a mixture of right- going and left-going harmonic oscillators. Observe that the limit p+ m gives back the free, unconstrained har- monic oscillator. The limit p-+ 1, on the other hand, is singular and produces the chiral oscillator (after a con- venient redefinition of the currents to absorb the infinity)

This can also be seen directly from the Lagrange function (32), which gives, after integrating out in the path in- tegral the Lagrange multiplier fields hi, the Floreanini- Jackiw chiral oscillator

Let us go back and consider the simplectic structure f,,. In the simplectic point of view, the constraints (34) must be incorporated into the theory using the Lagrange

from which we obtain the generalized brackets directly by inspection:

Observe the behavior of the simplectic matrix for the different B regimes. For f l tending to infinity we have

( 0 6, 0 O 1

This result corroborates the conclusion attained before for this limit. In this limit the chiral constraint disap- pears resulting in the (free) harmonic oscillator. The lim- it p+ 1, on the other hand, renders the (inverse) simplec- tic matrix singular, as expected. This is signaling the ap- pearance of new constraints for this specific value of 8. Either (carefully) taking the limit of (41) or working out the /3= 1 case directly, one finds the simplectic matrix, given in terms of the variables li = [ qi ,pi hi ) , to be

M. M. HORTA BARREIRA AND C. WOTZASEK - 45

whose inverse is

and gives the generalized brackets

which gives a justification for the brackets posed before in Eq. (8).

It is clear that for any value of B the harmonic oscilla- tor with a linear chiral constraint produces a well-defined and positive theory, whose quantization can be effected directly from the mapping of the generalized brackets coming from (43) and (47) into quantum commutators.

ACKNOWLEDGMENTS

This work was partially supported by CNPq (Conselho Nacional de Desenvolvimento Cientifico e Tecnologico), Brasil.

APPENDIX

The Hamiltonian formalism for constrained systems was developed a long time ago by Dirac [14]. A more geometrically motivated approach, based on the simplec- tic structure of the phase space, has been offered recently by Faddeev and Jackiw [a]. This description of the sys- tem can be thought of as a Hamiltonian with geometrical structure, originally given by the Poisson brackets being inferred from the Euler-Lagrange equations of motion. The generalized brackets obtained in this way are related to the so-called simplectic two-form, and the classification of the system as constrained or uncon- strained depends on the singular behavior of this func- tion.

In a recent note [13] one of us proposed an extension of the technique developed in Ref. [8] for first-order Lagrange functions for the case of singular simplectic two-forms. By incorporating the constraints into the canonical part of the first-order Lagrangian, instead of enlarging the Hamiltonian as is done in the Dirac method, we succeeded in obtaining a nonsingular sim- plectic two-form, from which the generalized brackets can be inferred. In this Appendix we shall review the for-

malism developed in Refs. [8] and [13]. In order to analyze the canonical structure of a physi-

cal theory, one is bound to consider the concepts of sim- plectic geometry [22], which is the geometry of a simplec- tic manifold. The conventional phase space can be re- garded as a simplectic manifold if we identify gi =(q i ,p i ) as its local coordinates. For a first-order system the La- grangian one-form is

where a "'( { ) =a,'O'({)d{' is the canonical one-form. The geometric structure is induced by the closed simplectic two-form

with

Correspondingly, the Euler-Lagrange equations of motion are

When (f(P')-' exists, one can directly map its elements into generalized brackets that will render the equations of motion into a Hamiltonian form, as shown by Faddeev and Jackiw. Let us then consider the more interesting case when the two-form f"' is singular. In this case the matrix fhO' has, say, m (left) zero modes dlO',a = 1, . . . ,m, which can be used to single out the (primary) Lagrangian constraints of the theory. From the equations of motion it becomes clear that, in this situ- ation, V({) imposes specific restrictions over the field variables. These primary constraints then become, in matrix notation.

In order for these conditions to provide a consistent description of the system, the constrained manifold must be stable under time evolution, which provides us with secondary (Lagrangian) constraints to be satisfied:

These secondary constraints must be incorporated into the original action with appropriate Lagrange multiplier fields rlbo). As usual, these multipliers will be incorporat- ed into the set of simplectic variables. Consequently, the dimensions of the simplectic matrices will be modified after every iteration. Being linear on the velocities, these constraints will produce a shift on the original general- ized momenta and, consequently, produce a deformation on the corresponding simplectic two-form, which now reads

CHIRAL-BOSON QUANTUM MECHANICS 1415

with

If det fLyi.0, then one has succeeded in eliminating the constraints and obtaining the basic geometric struc-

ture, which is given by the tensor field inverse to the sim- plectic metric. If, on the other hand, the simplectic ma- trix remains singular, one has to repeat the above pro- cedure as many times as necessary, say N, to render the matrix fhN) invertible. In the case of gauge theories, it will be necessary to add gauge-fixing terms to the sim- plectic potential V ( g ) in order to render the simplectic matrix invertible. From the resulting nonsingular object one can retrieve the generalized brackets that will be mapped to quantum commutators (up to ordering prob- lems) by simple inspection.

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