_{1}

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 “asymptotics quantum probability space” for the Hilbert lattice L(H).

Geometric quantization is a scheme involving the construction of Hilbert spaces by a phase space, usually a symplectic or Poisson manifold. In this paper, we will see how this complex machinery works and what kinds of objects are involved in this procedure. This mathematical approach is very classic and basic results are in [

From another point of view we have the quantum logic. This is a list of rules to use for a correct reasoning about propositions of the quantum world. Fun- damental works in this field are [

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.

In the usual meaning of classical logic, “propositions” can be interpreted as sets and implications as the subset relation Ì. Let

for every experimental propositions

We shall regard the classical phase space M as a Boolean algebra through the lattice

It is then natural to ask if also a quantum space

with

We shall take as quantum space

where A is a self-adjoint operator,

Let us denote with

where the series (4) converges and

Now we have a model for a quantum logic and we are able to describe it in terms of quantum observables. What we need to complete the description of the quantum picture is a notion of probability on

A fundamental result concerned the probability measure is due to [

Theorem 2.1 (Gleason). Let

The operator T is called the von Neumann density operator.

In this section we will examine the quantization procedures usefull to pass from a phase space, generally a symplectic manifold, to an Hilbert space^{1}-bundle of L and with

We shall follow the scheme used in [_{G}-dimensional compact Lie group G and a d_{T}-dimensional torus T. We assume that these actions are Hamiltonian and holomorphic and that commute togheter. By virtue of the Peter-Weyl theorem we may unitarily and equivariantly decompose

The finite dimensionality of

Another scheme of quantization is called the Berezin-Toeplitz quantization. In this picture the main rule is played by the notion of covariant Berezin symbol σ and coherent vector. Let A be a self-adjoint operator on the space of sections

where

for every section s, where

Observation 1. In order to compare the two schemes we take in consideration the remarkable relation between

where

A last mathematical formalism permits to express the Berezin-Toeplitz quan- tization in the modern language of POVM (that stands for Positive Operator Valued Measure, details on definitions are in [

More precisely, if we equip the symplectic manifold M with a Borel σ-algebra

where

On the previous upshot we refer to proposition 1.4.8 of Chapter II in [

The goal of this paper is a reinterpretation of main ideas of geometric quantization in the framework of quantum logic. The key strategy is to use the quantization of geometrical objects (manifolds) in order to have a quantization of “experimental propositions” that are the principal subjects of a logic formalism. We shall try in this section to develop these ideas. We shall start observing that from the quantization machinery we have a collection of finite dimensional Hilbert spaces given by the equivariant Hardy spaces:

where

Theorem 3.1. The family

Proof. The family

where

The lattice is orthomodular and we have that the joint

We shall use the geometric quantization to produce orthomodular lattices and obviously, it is not distributive because contains the diamon:

Observation 2. We are primarily interested in the equivariant case because it is more general, nothing change if we have only the standard action of

Example 3.2. Let us consider

provides the Hilbert lattice

Example 3.3. Let us consider now the action of a torus

Let us assume that

For every

In this case

Example 3.4. In this last example let us start with

that

moment map

Here

In the same setting of [

Given a pair of irreducible weights

to its Schwartz kernel in terms of an orthonormal basis

In the paper [

Let us assume that the dimension of

where

Let us consider now the setting of Berezin-Toeplitz quantization and let

with the following principal term in the asymptotic expansion:

where

The previous formulas (12) and (14) are respectively corollaries of more general asymptotic expansions of the equivariant Szegö and Toeplitz kernels near to the diagonal of

The case of geometric quantization presented here is a very special case that works because it requires some restrictions on the space M, for example one of those is that M must be simply connected. We have seen how this procedure fits well with the pourpose of quantum logic to find a general “formal” procedure to quantize “experimental propositions”. This suggests a chain of inclusions between differents methods of quantization described as follow:

where GQ is the geometric quantization; BQ is the Berezin Toeplitz quantization and QL is the quantum logic.

Camosso, S. (2017) Quantum Logic and Geometric Quantization. Journal of Quantum Information Sci- ence, 7, 35-42. https://doi.org/10.4236/jqis.2017.71003