Quantum Logic and Geometric Quantization

Simone Camosso

doi:10.4236/jqis.2017.71003

Abstract

We assume that M is a phase space and H an Hilbert space yielded by a quantization scheme. In this paper we consider the set of all "experimental propositions" of M and we look for a model of quantum logic in relation to the quantization of the base manifold M. In particular we give a new interpretation about previous results of the author in order to build an "asymptotic quantum probability space" for the Hilbert lattice L(H).

Highlights

Geometric quantization is a scheme involving the construction of Hilbert spaces by a phase space, usually a symplectic or Poisson manifold
We assume that M is a phase space and an Hilbert space yielded by a quantization scheme
In this paper we consider the set of all “experimental propositions” of M and we look for a model of quantum logic in relation to the quantization of the base manifold M

Summary

Introduction

Geometric quantization is a scheme involving the construction of Hilbert spaces by a phase space, usually a symplectic or Poisson manifold. The principal idea that inspires this work is to consider the special case of the geometric quantization as a “machine” of Hilbert lattices and try to find a possible measurable probability space. Let us denote with ⋅,⋅ the inner product on the Hilbert space and recall that a self-adjoint operator A is said to be positive if As, s ≥ 0 for all s ∈. In this case there is a trace class T associated:. Let A be a self-adjoint operator on the space of sections ( ) H 0 M , L⊗k , we define the covariant Berezin symbol σ ( A) by the map: x∈ X σ.

Examples

Scaling Limits for the Probability Measure

Conclusion