0941.17017
Gotay,
Mark J.; Grundling,
Hendrik B.
Nonexistence of finite-dimensional quantizations of a noncompact symplectic
manifold. (English)
[CA] Kolár, Ivan (ed.) et al., Differential geometry and applications.
Proceedings of the 7th international conference, DGA 98, and satellite conference
of ICM in Berlin, Brno, Czech Republic, August 10-14, 1998. Brno: Masaryk
University. 593-596 (1999). [ISBN 80-210-2097-0/pbk]
A basic algebra of observables $\frak b$ on a symplectic manifold $M$ is
a Lie subalgebra of $C^\infty(M)$ which is finitely generated, transitive
and separating, the Hamiltonian vector fields $X_f$ of elements $f\in\frak
b$ are complete, and which is minimal with respect to the previous requirements.
Note that $\frak b$ is separating means that elements of $\frak b$ separate
globally points of $M$, and transitivity means that the Hamiltonian vector
fields $\{ X_f:f\in{\frak b}\}$ span the tangent bundle $TM$. \par The main
result states that finite-dimensional basic algebras on noncompact symplectic
manifolds do not admit faithful representations by skew-hermitian matrices.
This can be understood as a rigorous proof that quantizations of a noncompact
symplectic manifold, if exist, have to be infinite-dimensional.
[ Janusz
Grabowski (Warszawa) ]
- MSC 2000:
- *17B63
Poisson algebras
53D17
Poisson manifolds
37J05
Relations with symplectic geometry and topology
81S99
General quantum mechanics and problems of quantization
Keywords: quantization; Poisson algebra