LANDAU LEVELS AND GEOMETRIC

International Journal of Modern Physics A, Vol. 4, No. 15 (1989) 3939 - 3949

© World Scientific Publishing Company

LANDAU LEVELS AND GEOMETRIC QUANTIZATION' J. R. KLAUDER Departmenrs oj Physics and Malhematics, University oj Florida, Gainesville, FL32611, USA

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and

E. ONOFRI Dipartimento di Fisica del/'Unillersitd di Trenw, 38050 Povo, Ilaly and IN FN, Gruppo Collegato di Parma, 43100 Parma, lIaly

Received 23 December 1988

The geometrical approach to phase-space quantization introduced by Klauder [KQ] is inler· preted in terms of a universal magnetic field aeting on a free particle moving in a higher dimensional configuration space; quantization corresponds to freezing the particle to its first Landau level. The Geometric Quantization [GQ] scheme appears as the natural technique to define the interaction with the magnetic field for a particle on a general Riemannian manifold. The freedom of redefining the operators' ordering makes it possible to select that particular definition of the Hamiltonian which is adapted to a specifie polarization; in this way the first Landau level acquires the expected degeneracy. This unification with GQ makes it clear how algebraic relations between classical observables are or are not preserved under quantization. From this point of view all quantum systems appear as the low energy seetor of a generalized theory in whieh all classical observables have a uniquely assigned quantum counterpart such that Poisson bracket relations are isomorphic to the commutation relations.

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Introduction

In a series of recent papers,'-3 a new approach to quantization has heen formulated in terms of a regularized form of Feynman path integral on phase space. In a number of explicitly worked out examples, it has been shown that the introduction of a Riemannian metric g in the classical phase space allows one to build a genuine path integral which in a suitable limit converges to the quantum propagator expressed in a coherent state basis. The Riemannian metric, which is not a natural classical object, is the key ingredient to the construction. Many questions can be raised about this quantization scheme, which we will refer to as KQ, for short. First of all, one may ask whether there is any physical meaning in this Riemannian metric, or, more specifically, whether the quantum system which comes out ofKQ depends for some ofits properties on the choice of g. Secondly, since the theory is formulated in phase space, it is natural to compare it with another quantization scheme which also lives in phase space, namely Kirillov-Kostant-Souriau's Geometric Quantization 4 - 6 (hereafter GQ). The applica­ .. Work supported in part by the Italian Ministero Pubblica Istruzione.

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3940 J. R. Klauder &- E. pnofri

Landau Levels and Geometric Quantization

tions of GQ have been rather limited in scope, mainly to systems whose Hamiltonian belongs to some Lie algebra which acts transitively in phase space, and from a general viewpoint such examples are mathematically equivalent to the theory of induced representations, modulo technical details. However, GQ's firm geometrical foundation has led to a revived interest in it recently'"" This fact and its natural connection with coherent states suggests that it would be advisable to understand the connection between GQ and the new quantization scheme KQ. Our main thesis will be the following:

(P2,q2Iexp( -iIH)lpI,qI) =

feXPi{f~

p(r)dq(r) -

I h(p(r),q(r»)dr}~q~p

3941

(2.1)

(it is understood that p(O) = PI' p(t) = p" etc.). In a general set of coordinates (not necessarily canonical) ~ = (~" ~', ... , ~"), the same formal path integral would be written as

f

(~,Iexp( -itH)I~I) ~ eXPi{f~ B(r) - f>(~(r»)dr} II w'

(2.2)

KQ GIVES THE RIGHT PHYSICAL IDEA

AND TIlE GENERAL MATHEMA TICAL FRAMEWORK, WHILE

GQ PROVIDES THE RIGHT MATHEMATICAL TOOLS

OF ONE AND THE SAME QUANTIZATION SCHEME

=

which can be rephrased by saying that KQ is physically very appealing, as we shall try to convince the reader, and it leads quite naturally to a well-defined path integral; to give a precise mathematical meaning to its formal definitions in general contexts, however, it needs input from GQ with its line bundles and its concept of prequantiza­ tion. On the other hand, while GQ is well established as a mathematical discipline, it has always lacked physical intuition-the main idea of associating physical states with sections of a line bundle over phase space has always been justified only a posteriori by its effectiveness. From the viewpoint of KQ, it follows that GQ's line bundle is nothing other than the customary object which must be considered in order to quantize a free, nonrelativistic particle in a magnetic field. The plan of the paper is the following: In Sec. 2, we will show that the phase-space path integral of KQ is mathematically equivalent to a configuration space path integral of a particle moving under the action of a magnetic field B given by the symplectic two-form w of classical Hamiltonian mechanics and in addition, we shall identify the space of physical states with the degenerate subspace known as the first Landau level. In Sec. 3, the connection with GQ is established; we shall discuss the concept of polarization in the complex (Kaehler) case, which leads directly to spaces of holomor­ phic sections, that is to Hilbert spaces with a regular reproducing kernel (generalized coherent states). The case of real polarizations is still not entirely clear; we shall discuss only the flat case in detail and give a proposal for more general cases. In Sec. 4 we discuss the algebraic properties of KQ, showing how the commutation properties, which are isomorphic to the classical Poisson brackets, are in general broken in the limit which freezes the system to its first Landau level. Sec. 5 contains our conclusions. 2.

The Classical Symplectic Structure as a Universal Magnetic Field

Let us start with the usual (formal) coherent state path integral in phase space for a Hamiltonian h(p, q), (p, q denoting canonical coordinates in R"; h ~ 1):

_______________...-jl

where B denotes the one form P' dq and w the two-form dp !'. dq when expressed in terms of ~. Equation (2.2) can be taken as the formal path integral on a general symplectic manifold -It, provided we give a meaning to B, which has only a local status in general. Now we adopt the regularization introduced in Ref. 1 by choosing a Riemannian metric g on the phase space -It, which we will denote by do' = gijd~i d~j; the regularized path integral is then (~2Iexp( -iIH)I~l)

=

~~~ e'"

f

exp {

I

-~ V-I gi;(~)~i~j dr + i

I

B(r) - i

I h(~(r»)dr} II

w'

(2.3)

where K is a suitable constant which makes the limit finite. As already noticed in Ref. 2, the path integral corresponding to h(p, q) '" 0 (which defines the projection X" onto the physical states) is equivalent, for finite V, to the Euclidean path integral 9 for a particle with kinetic energy defined by g coupled to a magnetic field B with potential 8; that is the factor exp(i JB) is interpreted as the usual phase factor in electrodynamics exp (i JAi d~i), which retains its imaginary prefactor also in the Euclidean regime. Notice that in the identification B = dB", w we are actually forced to consider general magnetic fields which still satisfy div B = 0, but will in general develop a nonzero flux

cI>(~) =

ff

B' do .. 0

through closed surfaces ~ which are not boundaries (in geometrical words, this happens when the classical phase space has nontrivial second cohomology class). Following Dirac, I 0 this magnetic flux must be quantized, in order that exp (i fA' dx) be well-defined; this condition will impose some quantization rule on the parameters which characterize the classical phase space, e.g. spin quantization arises in this way. The reproducing kernel X" which defines the physical Hilbert space is given by

%(~1; ~2) =

lim e KV1 fe-1/2V-1 fbgij(.:')~i~jdt'+ifb8{t") "'

- i[' , .] of Poisson brackets into commutators. Now, our point of view is that GQ's constructs­ line bundle, horizontal vector fields, global sections-are precisely the standard tools that one needs to adopt in order to define in a general setting the coupling of a mass particle to a magnetic field B = co which is just the curvature of the connection form

{t/J(Z)lf l t/J 12 e-f w'