Geometric Quantization of Generalized Oscillator

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arXiv:math-ph/9902024v1

18Feb1999

Geometric quantization of generalizedoscillator

Sergey V. Zuev1

Department of Theoretical Physics, Penza State Pedagogical University, 440039

Penza, Russia

Abstract

Using geometric quantization procedure, the quantization of algebra of ob-

servables for physical system with Ricci-flat phase space is obtained. In the

classical case the appointed physical system is reduced to harmonic oscillator

when the one real parameter is vanished.

Keywords: Generalized oscillator, Geometric quantization, Ricci-flat Kahler manifold.

1. Introduction

There exist a number of procedures and methods to quantizea physical systems, but it is geometric quantization that takesinto account the geometrical background (i. e. geometry of phasespace) of the physical system. The procedure of geometric quan-tization was discovered by B. Kostant [2] and J.M. Souriau [4]in 1970. During last 25 years it was highly developed by mul-tiply authors. J.M. Tuynman in [5] has compared some known

methods of quantization (in particular, geometric quantization)using 2-dimensional generalized harmonic oscillator, i. e. hamil-tonian system with algebra of observables su(1,1). For physical

1Tel./Fax +7.8412.441604, E-mail: [email protected]

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applications it would be helpful to quantize the generalized har-monic oscillator for an arbitrary dimension. Theorem 3 of thepresent paper gives such quantization for even dimensions.

Recently a Ricci-flat Kahler metric for any real dimension 4n

was constructed by the author [6]. The metric has the followingform

g = uzz + u =

am

r2(rm am)

1m

m zz +(rm am)

1

m

r,

(1)

whereu

du

dr=

(rm am)1/m

r, (2)

u du/dr, r m=1

zz, m 2n, R a = const. As it

was shown in [6] (this proposition is almost evident) the groupof complex isometries of the metric (1) is SU(m). It is notablethat Ricci-flat Kahler phase space M with abovementioned met-ric has an subalgebra F of Poisson algebra ofC(M)-functions

on M and F is appropriate to quantize by geometric quanti-zation procedure. Let us call the physical system with phasespace M as generalized oscillator and the algebra F as algebraof observables of generalized oscillator.

2. Antiholomorphic polarization

Let us consider a Kahler polarization (see [2, 4] for detailes)F T M R C which is determined by the condition

Fp

= {X T Mp

C |X = |p

ad (z)p

, |p

C }, (3)

where |p are C(M)-functions of a point p M. The de-

fined polarization is called antiholomorphic polarization. Func-tion f C(M) preserves polarization F if it obeys to the

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equationLad (f)X F for all X F. (4)

It is equal to the next condition

[ad (f), ad (z)] =(f)a ad (z

). (5)

Theorem 1 [1] Let M be a Kahler manifold and F T M RC be an antiholomorphic polarization. Function f C(M)preserves polarization F if and only if in every complex chart(U, z, z), = 1, . . . , m, on M the equation

f = (z) + (z), (6)

holds. Here (z), (z) are an arbitrary holomorphic functions.

On the manifold (M, g) where g is defined by (1) the formula(6) takes the following form

f =

uz(z) + (z), (7)

where u is defined by (2).Let us consider the next functions

N = uzz, , = 1, . . . , m . (8)

It is easy to show that the fuctions N preserve polarization F.From here and from theorem 1 we find

N = uz , (9)

where = z .

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The Hamiltonian vector fields of functions N

are definedby the following equalities

V ad (N) = i(z z).

The Poisson brackets of functions N and the commutators ofits Hamiltonian vector fields have the next form

{N, N} = i(N N

)

[V, V] = V V,

Since V are Hamiltonian and holomorphic that the corre-sponding transformations preserve complex structure on M andfundamental 2-form (X, Y) g(JX,Y). Hence they preservemetric g and they are Killing vector fields on M which preservecomplex structure. So the vector fields V form the algebrasu(m) of infinitesimal holomorphic isometries of the metric (1).

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3. Geometric quantization of the algebra of observables

The transformations from group SU(m) preserves complexstructure and metric g as well as fundamental form . Thismeans that the action of SU(m) on M is symplectic [3]. Thecohomology group H2(su(m), C ) is trivial and as it was shownin [3] the action of SU(m) on M is Poisson action.

Let us consider an algebra (with respect to Poisson brack-

ets) F(m) of linear functions on N

, , = 1, . . . , m . As itwas mentioned in Introduction, this algebra is called as alge-bra of observables of 2m-dimensional generalized oscillator. Itis evident that F(m) coincides with an algebra of functions pre-serving antiholomorphic polarization on M. For such a functionsthe following theorem exists.

Theorem 2 [1] Let M be a Kahler manifold, F T M R Cbe an antiholomorphic polarization and FF(M) C

(M) be an

algebra of functions on M which preserve polarization F. Thenin every chart (U, z, z), = 1, . . . , m, the quantization Q ofFF(M) is defined by the formulae

Q(f) 0 = ( + h( +1

2)) 0, (10)

where f = + FF, , is holomorphic functions onU and 0 is non-vanished at every point of U section of theHermitian vector bundle L over M.

One can construct the quantization of algebra F(m) usingtheorem 2. As far as the map Q is linear, it is sufficient todefine the action of Q on functions N.

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Using (9) we find from (10)

(Q(N)) 0 = h

z

z+

1

2

0. (11)

The last formula defines quantization of algebra F(m) of ob-servables of m-dimensional generalized oscillator.

By summation (11) with = from 1 to m we have

(Q(H)) 0 = h(z

z+

m

2) 0,

Therefore, the eigenvalues of operator QH are defined by thenext relation

l = h

l +m

2

, l = 0, 1, 2, . . .

and coincide with energy levels of harmonic oscillator in flatspace.

4. Corollary

The main result of the paper can be formulated as the fol-lowing

Theorem 3 Let (M, g) be Kahler Ricci-flat space, dimR M =2m = 4n with metricg, defined by(1). LetF be an antiholomor-phic polarization defined by (3) and F C(M) be an algebraof functions on M which are linear functions on variables N

defined by (8). Then in every chart (U, z, z), = 1, . . . , m ,

the quantization Q of F is defined dy operators (11) where isholomorphic function on U and 0 is non-vanished on U sectionof Hermitian vector bundle L over M.

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Acknowledgement

I would like to thank Prof. A.V. Aminova and Dr. D.A. Kalininfor their helpful discussions.

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References

[1] Aminova A.V., Kalinin D.A. Quantization of Kahler mani-folds admitting H-projective mappings. // Tensor. 1995. V.56. P.1-11.

[2] Kostant B. Quantizations and unitary representations. //Lecture Notes Math. 1970. V.170. P.87-208.

[3] Sniatycki J. Geometric quantization and quantum mecha-

nics. Berlin: Springer, 1980.

[4] Souriau J. M. Structures des systems dinamiques. Paris:Dunod, 1970.

[5] J.M. Tuynman. Quantization. Towards a comparison betwe-en methods. // J. Math. Phys.. 1987. V.28. P.2829-2840.

[6] Zuev S.V. A 4n-dimensional Kahler Ricci-flat metric. //

Proc. Int. Conf. Geometrization of Physics III. Kazan,October 1-5, 1997. P.191.

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Geometric Quantization of Generalized Oscillator