DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 593–596

NONEXISTENCE OF FINITE-DIMENSIONAL QUANTIZATIONS OF A NONCOMPACT SYMPLECTIC MANIFOLD

MARK J. GOTAY AND HENDRIK B. GRUNDLING

Abstract. We prove that there is no faithful representation by skew-hermitian matrices of a “basic algebra of observables” 𝔟 on a noncompact symplectic manifold M. Consequently there exists no finite-dimensional quantization of any Lie subalgebra of the Poisson algebra C^{∞}(M) containing 𝔟.

1. Introduction

Let M be a connected noncompact symplectic manifold. On physical grounds one expects a quantization of M, if it exists, to be infinite-dimensional. This is what we rigorously prove here, in the framework of the paper [Go1]. Our precise hypotheses are spelled out below.

A key ingredient in the quantization process is the choice of a basic algebra of observables in the Poisson algebra C^{∞}(M). This is a Lie subalgebra 𝔟 of C^{∞}(M) such that

(B1) 𝔟 is finitely generated,

(B2) the Hamiltonian vector fields X_{f}, f ∈ 𝔟, are complete,

(B3) 𝔟 is transitive and separating, and

(B4) 𝔟 is minimal with respect to these conditions.

A Lie subalgebra 𝔟 ⊂ C^{∞}(M) is “transitive” if {X_{f}(m) | f ∈ 𝔟} spans T_{m}M at every point. 𝔟 is “separating” provided its elements globally separate points of M. Throughout this paper, we assume that 𝔟 is finite-dimensional.

Now fix a basic algebra 𝔟, and let O be any Lie subalgebra of C^{∞}(M) containing 1 and 𝔟. Then by a finite-dimensional quantization of the pair (O, 𝔟) we mean a Lie representation Q of O by skew-hermitian matrices on some ℂ^{k} such that (Q1) Q(1) = I,

1991 Mathematics Subject Classification. Primary 81S99; Secondary 58F06.

Key words and phrases. Quantization, Poisson algebra.

This paper is in final form and no version of it will be submitted for publication elsewhere.

(Q2) Q↾𝔟 is irreducible, and

(Q3) Q↾𝔟 is faithful.

We refer the reader to [Go1] for a detailed discussion of these matters. We remark that in the infinite-dimensional case there are additional conditions which must be imposed upon a quantization. We also elaborate briefly on (Q3). Although faithfulness is not usually assumed in the definition of a quantization, it seems to us a reasonable requirement in that a classical observable can hardly be regarded as “basic” in a physical sense if it is in the kernel of a quantization map. In this case, it cannot be obtained in any classical limit from the quantum theory.

2. The Obstruction

Given the definitions above, we state our main result:

Theorem 1. Let 𝔟 be a finite-dimensional basic algebra on a noncompact symplectic manifold M. Then 𝔟 has no faithful representations by skew-hermitian matrices.

Proof. We argue by contradiction. Suppose there exists such a representation Q of 𝔟 on some ℂ^{k}. As Q(𝔟) consists of skew-hermitian matrices, Q is completely reducible. Since Q is faithful, one deduces from [Va, Thm. 3.16.3] that 𝔟 is reductive, i.e. 𝔟 = 𝔷 ⊕ 𝔰 where 𝔷 is the center of 𝔟 and 𝔰 is semisimple. We show that 𝔷 = {0}. Indeed, by the transitivity condition (B3), the elements of 𝔷 must be constant but, if these are nonzero, then 𝔰 alone would serve as a basic algebra, contradicting the minimality condition (B4). Thus 𝔷 = {0} and 𝔟 = 𝔰 is semisimple.

Let B be the connected, simply connected Lie group with Lie algebra 𝔟. We claim that B is noncompact. Now the map f ↦ X_{f} can be thought of as an action of 𝔟 on M. By (B2) the vector fields X_{f} are complete, so by a theorem of Palais [Va, Thm. 2.16.13] this action of 𝔟 can be integrated to an action of the group B on M. Condition (B3) implies that this action is locally transitive and thus globally transitive as M is connected. Thus the noncompact manifold M is a homogeneous space for B, and so B must be noncompact as well.

Now consider a unitary representation U of B on ℂ^{k}. Decompose B into a product B_{1} × ⋯ × B_{N} of connected simple groups. Then (at least) one of these, say B_{1}, must be noncompact. But it is well-known that a connected, simple, noncompact Lie group has no nontrivial finite-dimensional unitary representations [BR, Thm. 8.1.2]. Thus U(b) = I for all b ∈ B_{1}. Since every representation Q of 𝔟 by skew-hermitian matrices is a derived representation of some finite-dimensional unitary representation U of B, it follows that Q ↾ 𝔟_{1} = 0, and so Q cannot be faithful. □

From this we immediately have

Corollary 2. Let M be a noncompact symplectic manifold, 𝔟 a finite-dimensional basic algebra on M, and O any Lie subalgebra of C^{∞}(M) containing 1 and 𝔟. Then there is no finite-dimensional quantization of (O, 𝔟).

As the proof above shows, we do not need conditions (Q1) or (Q2) to obtain this result. Moreover, the subalgebra O is irrelevant since the proof depends only on the Lie theoretical properties of the basic algebra 𝔟 and its action on M.

3. Discussion

Corollary 2 is complementary to the recent result of [GGG] which states that there are no nontrivial quantizations (finite- or infinite-dimensional) of (P (𝔟), 𝔟) on a compact symplectic manifold M, where P (𝔟) is the Poisson algebra of polynomials generated by the basic algebra 𝔟. The proof of that result leaned heavily on the algebraic structure of P (𝔟); indeed, when M is compact, it turns out that 𝔟 must be compact semisimple, and such algebras do have faithful representations by skew-hermitian matrices. Thus in the compact case, the obstruction to the existence of a quantization is Poisson, rather than Lie theoretical. Combining [GGG] with Corollary 2, we can now assert, roughly speaking, that no symplectic manifold with a (finite-dimensional) basic algebra has a finite-dimensional quantization.

Neither Theorem 1 nor Corollary 2 are valid when the representation is infinitedimensional. For instance, in [GGra] an explicit quantization of the polynomial algebra generated by the “‘affine” basic algebra span{pq, q^{2}} on T^{∗}ℝ_{+} = {(q, p) ∈ ℝ^{2} | q > 0} is constructed on the Hilbert space L^{2}(ℝ_{+}, dq/q). On the other hand, there do exist obstructions to obtaining infinite-dimensional quantizations of other noncompact symplectic manifolds, such as ℝ^{2n} [Go2] and T^{∗}S^{1} [GGru]. We hope to determine the circumstances under which such obstructions occur in future work. This appears to be a difficult problem; still, some results are already known in this direction [Go1].

Acknowledgments

M.J.G. thanks the University of New South Wales and the U.S. National Science Foundation (through grant DMS 96-23083) for their support while this research was underway.

H.G would like to thank the Australian Research Council for their support through a research grant.

References

[BR] Barut, A. O. & Ra̧czka, R., Theory of Group Representations and Applications, Second Ed., World Scientific, Singapore 1986.

[Go1] Gotay, M. J., Obstructions to quantization, in: The Juan Simo Memorial Volume, Marsden, J. E. & Wiggins, S., Eds., Springer, New York 1999, in press.

[Go2] Gotay, M. J., On the Groenewold-Van Hove problem for ℝ^{2n}. J. Math. Phys. 40 (1999), 2107–2116.

[GGra] Gotay, M. J. & Grabowski, J., On quantizing nilpotent and solvable basic algebras, (1999), preparint math-ph/9902012.

[GGG] Gotay, M. J., Grabowski, J., & Grundling, H. B., An obstruction to quantizing compact symplectic manifolds, Proc. Amer. Math. Soc. (1999), in press.

[GGru] Gotay, M. J. & Grundling, H. B., On quantizing T^{∗}S^{1}, Rep. Math. Phys. 40 (1997), 107–123.

[Va] Varadarajan, V. S., Lie Groups, Lie Algebras and Their Representations, Springer-Verlag, New York 1984.

Mark J. Gotay Department of Mathematics, University of Hawai‘i 2565 The Mall, Honolulu, HI 96822 USA E-mail: gotay@math.hawaii.edu Home Page: www.math.hawaii.edu/˜ gotay

Hendrik B. Grundling Department of Pure Mathematics, University of New South Wales P.O. Box 1, Kensington, NSW 2033 AUSTRALIA E-mail: hendrik@maths.unsw.edu.au