{'fraction_non_alphanumeric': 0.04080746689819839, 'fraction_numerical': 0.005209463859344476, 'mean_word_length': 4.242320819112628, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Nonexistence of Finite-dimensional Quantizations of a Noncompact Symplectic Manifold

We prove that there is no faithful finite-dimensional 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^\infty(M) containing B.


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 [GGT]. Our precise hypotheses are spelled out below.
A key ingredient in the quantization process is the choice of a basic set of observables in the Poisson algebra C ∞ (M ) . This is a finite-dimensional linear subspace B of (B3) (Minimality) B is minimal with respect to these conditions.
In addition to these conditions we assume in this paper that B forms a Lie algebra under the Poisson bracket. We then refer to B as a basic algebra. (Note also that unlike in [GGT], we do not require here that 1 ∈ B. We refer the reader to [GGT] 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 result: Theorem. Let M be a noncompact symplectic manifold, B a basic algebra on M , and O any Lie subalgebra of C ∞ (M ) containing B . Then there is no finite-dimensional quantization of (O, B) .
As the proof will show, we do not need conditions (Q1) or (Q2) to obtain the theorem. 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 . Let B be the connected, simply connected Lie group with Lie algebra B . We show that B is noncompact. Let g be the Lie algebra {X f f ∈ B} . By (B1) the vector fields in g are complete and so by [V,Thm. 2.16.13] this infinitesimal action of g can be integrated to an action of the connected, simply connected Lie group G with Lie algebra g . Condition (B2) 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 G , and so G must be noncompact as well. Now B is isomorphic either to g or to a central extension of g by constants. Since B is semisimple, the latter alternative is impossible. Hence B is isomorphic to G and so is noncompact. Now consider a unitary representation π of B on C n . Decompose B into a product B 1 × · · · × B K of simple groups. Then (at least) one of these, say B 1 , must be -3-noncompact. But it is well-known that a connected, simple, noncompact Lie group has no nontrivial unitary representations [BR,Thm. 8.1.2]. Thus π|B 1 is trivial, i.e. π(b) = I for all b ∈ B . Since every finite-dimensional quantization Q of B is a derived representation of some unitary representation π of B , it follows that Q|B 1 = 0 , and so Q cannot be faithful.

Discussion
This theorem is complementary to a recent result of [GGG] which states that there are no nontrivial quantizations (finite-dimensional or otherwise) 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 finite-dimensional 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 the present theorem, we can now assert, roughly speaking, that no symplectic manifold with a basic algebra has a finite-dimensional quantization.
We hope to address the issue of whether there are obstructions in general to infinitedimensional quantizations of noncompact symplectic manifolds in future work. Certainly such obstructions exist in specific examples, such as R 2n [GGT] and T * S 1 [GG]. This appears to be a difficult problem, however.