[Paper Review] Nonexistence of Finite-dimensional Quantizations of a Noncompact Symplectic Manifold

Mark J. Gotay, Hendrik Grundling|ArXiv.org|Oct 22, 1997

Advanced Topics in Algebra7 references3 citations

TL;DR

This paper proves that no faithful finite-dimensional representation by skew-Hermitian matrices exists for the algebra of observables on a noncompact symplectic manifold, implying that finite-dimensional quantization of any Lie subalgebra of the smooth Poisson algebra containing this observable algebra is impossible. The result establishes a fundamental obstruction to finite-dimensional quantization in noncompact settings using standard representation-theoretic methods.

ABSTRACT

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.

Motivation & Objective

• To investigate the possibility of finite-dimensional quantization for Lie subalgebras of the smooth Poisson algebra on noncompact symplectic manifolds.
• To determine whether a faithful finite-dimensional representation by skew-Hermitian matrices can exist for the basic algebra of observables on such manifolds.
• To establish a fundamental obstruction to finite-dimensional quantization in noncompact geometric settings.
• To clarify the limitations of standard finite-dimensional representation methods in geometric quantization of noncompact phase spaces.

Proposed method

• The analysis focuses on the algebraic structure of the basic algebra of observables B on a noncompact symplectic manifold M.
• It employs representation theory to examine the existence of faithful finite-dimensional representations by skew-Hermitian matrices.
• The proof relies on properties of Lie algebras and the structure of the Poisson algebra C∞(M) on noncompact manifolds.
• It uses the fact that skew-Hermitian matrices generate compact Lie groups, which cannot faithfully represent noncompact symplectic structures.
• The argument proceeds by contradiction, assuming such a finite-dimensional representation exists and deriving a topological inconsistency.
• The core technique involves showing that the image of the representation would have to be a compact Lie group, contradicting the noncompactness of the underlying manifold.

Experimental results

Research questions

• RQ1 Can a faithful finite-dimensional representation by skew-Hermitian matrices exist for the basic algebra of observables on a noncompact symplectic manifold?
• RQ2 Is finite-dimensional quantization possible for any Lie subalgebra of the smooth Poisson algebra containing this observable algebra?
• RQ3 What topological or algebraic obstruction prevents finite-dimensional quantization in noncompact symplectic geometry?
• RQ4 Why do standard finite-dimensional representation methods fail in the context of noncompact phase spaces?

Key findings

• There exists no faithful finite-dimensional representation by skew-Hermitian matrices of the basic algebra of observables on a noncompact symplectic manifold.
• Consequently, no finite-dimensional quantization exists for any Lie subalgebra of the smooth Poisson algebra containing this observable algebra.
• The obstruction arises from the incompatibility between the compactness of unitary groups generated by skew-Hermitian matrices and the noncompactness of the symplectic manifold.
• The result implies that finite-dimensional quantization schemes cannot capture the full observable algebra in noncompact settings.
• The proof establishes a fundamental limitation of finite-dimensional approaches in geometric quantization for noncompact phase spaces.
• The nonexistence result holds regardless of the specific choice of observable algebra, as long as it contains the basic algebra of observables.