Quantum Logic and Geometric Quantization

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1. Introduction

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 [1] . About the quantization of Kähler manifolds and the Berezin-Toeplitz quantization we suggest the following literature [2] [3] [4] [5] [6] .

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 [7] [8] [9] . In order to emphasize the im- portance of these studies we shall notice that these are used in quantum physics to describe the probability aspects of a quantum system. A quantum state is generally described by a density operator and the result used to introduce a notion of probability in the Hilbert space is a celebrated theorem due to Gleason in [10] . We will see how recent developments in POVM theory (positive operator-valued measure) suggest to see the classical methods of quantization as special cases of the POVM formalism. Regarding these developments on POVMs see [11] [12] [13] .

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.

2. Preliminaries

2.1. Quantum Logic, Hilbert Lattice and Quantum Probability

In the usual meaning of classical logic, “propositions” can be interpreted as sets and implications as the subset relation Ì. Let a family of subsets of the phase space M. These subsets are associated to “experimental propositions” in the sense of [7] . Assume that is a partially ordered system respect the inclusion Ì. Assume in addition that there are two relations “meet” and “joint” with a relation of complementation of sets ^. We shall take as an orthocomplemented lattice. Now we shall focus on a crucial point that differentiates the logic associated to a classical system respect the logic associated to a quantum system. The main issue is the validity of the following distributive law:

(1)

for every experimental propositions. An orthocomplemented lattice is said Boolean if (1) holds.

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 obeys to (1). The answer is negative and further developments on this problem are due to [7] [8] [9] , let us clarify the issue. We will consider orthocomplemented lattices such that:

(2)

with experimental propositions of. The identity (2) is called the orthomodular law and the associated lattice orthomodular. What happens is that orthomodular lattices are models for a quantum logic.

We shall take as quantum space an Hilbert space and as the collection of all closed linear subspaces of. The Hilbert space generally is an infinite complete function space possessing the structure of an inner product, a typical example is the set of square integrable functions. We notice that is an orthomodular lattice and we call it the Hilbert lattice. A way to describe is by the one to one correspondence between closed subspaces and projectors P such that, where is the adjoint operator. The link between observables and projectors is guaranteed by the spectral theorem:

(3)

where A is a self-adjoint operator, the associated spectral resolution of the identity with and is the Stieltjes measure associated to the distributional function. Much information about the spectral theorem can be found in [14] .

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 for all. In this case there is a trace class associated:

(4)

where the series (4) converges and is an orthonormal basis for.

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. An answer to this problem was given by [15] that introduced a probability function. The function p is σ-additive and can be understood in the sense of [16] with as probability space. We shall observe that it is a non-Kol- mogorovian measure because the lattice is interpreted as a non-Boolean σ-algebra.

A fundamental result concerned the probability measure is due to [10] , this called the Gleason theorem. Let us recall the statement of this theorem.

Theorem 2.1 (Gleason). Let be a separable Hilbert space over (or) with. There exists a positive semi-definite self-adjoint operator T of the trace class such that for all projector in

(5)

The operator T is called the von Neumann density operator.

2.2. Geometric Quantization, Berezin-Toeplitz Quantization and POVM

In this section we will examine the quantization procedures usefull to pass from a phase space, generally a symplectic manifold, to an Hilbert space. Let be a complex projective compact manifold and a Kähler form. Let be an hermitian line bundle on M with associated hermitian product h. Let the curvature of the unique Levi-Civita connection compatible with L. We shall assume the prequantization condition. Let us denote with X the S^{1}-bundle of L and with the Hardy space where stands for the Cauchy-Riemann operator.

We shall follow the scheme used in [17] under the action of a d_{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 over irreducible representations of G and T:

(6)

The finite dimensionality of is guaranteed under assumptions on the moment maps associated to the actions (details are in [18] and [19] ).

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, we define the covariant Berezin symbol by the map:

(7)

where is the coherent vector associated to such that:

for every section s, where is the scalar product on the space of sections. The material regarding this topic can be found in [2] and [20] .

Observation 1. In order to compare the two schemes we take in consideration the remarkable relation between, the well know operator of geometric quantization and given by

where is the Laplace-Beltrami operator with respect the Kähler metric. This suggest we have the same semi-classical behaviour as (the result is due to Tuynman in [21] ). This semi-classical behaviour is understood if we put

where is the Plank constant and we imagine to send.

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 [11] and [13] ).

More precisely, if we equip the symplectic manifold M with a Borel σ-algebra there exists a sequence of -valued POVM on M such that the Toeplitz operator associated to is

(8)

where.

On the previous upshot we refer to proposition 1.4.8 of Chapter II in [13] and the same theme is treated in [12] .

3. From the Geometric Quantization to QL

3.1. Realization of the Hilbert Lattice

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:

(9)

where and are irreducible representations of a Lie group G and a torus T as explained in the previous section.

Theorem 3.1. The family with is an orthoalgebra.

Proof. The family with satisfies the properties for poset (partially ordered set). It is an orthocomplemented lattice with meet, joint and the complementation. The orthogonal space is defined as

where is the hermitian product and an ortho- normal basis. We observe that the decomposition of by the Peter-Weyl theorem provides isotypes that are pairwise orthogonal.

The lattice is orthomodular and we have that the joint is in fact the direct sum. ,

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. In this case the previous argumentation is almost trivial.

3.2. Examples

Example 3.2. Let us consider. Let us take in account the standard circle action induced by the representation on given by . It is holomorphic and Hamiltonian with moment map. The equivariant decomposition:

provides the Hilbert lattice.

Example 3.3. Let us consider now the action of a torus on induced by the representation on given by. Also in this case it is a holomorphic Hamiltonian action with moment map given by:

(10)

Let us assume that is a regular value of and let, then

For every we have

In this case.

Example 3.4. In this last example let us start with and the action of. The group acts linearly on, and it’s action descends to an action on. We may equivariantly identify with. Let us assume

that has radius. This is an holomorphic, Hamiltonian action with

moment map that corresponds to the inclusion, where here. Let us consider the line bundle and the space of holomorphic sections. For every the irreducible representations of are given by the symmetric polynomials so let an irreducible representation for G we have that:

Here corresponds to the atomic elements of the equivariant de- composition.

3.3. Scaling Limits for the Probability Measure

In the same setting of [17] , we have the action of the product group on the symplectic manifold M. We shall interpret the von Neumann density operator as the equivariant Szegö projector. Now we spend few words on the Szegö projector.

Given a pair of irreducible weights and for G and T, respectively, we shall denote by the orthogonal projector. We refer

to its Schwartz kernel in terms of an orthonormal basis of

as:

(11)

In the paper [17] the main subject studied is a local asymptotics of the equivariant Szegö kernels, where the irreducible representation of T tends to infinity along a ray, and the irreducible representation of G is held fixed. The Szegö kernel is usually expressed in Heisenberg local coordinate centered at and for our purpose we shall need the scaling limits of on the diagonal of. We shall observe that is an orthogonal projector, self-adjoint (with microsupport see [22] ), positive and it is a trace class. Looking at these key features, we shall force the interpretation of the equivariant kernel as a “fundamental state of the system” in the sense of quantum physics.

Let us assume that the dimension of is, then there exists a von Neumann density operator such that:

(12)

where is the dimension of the product group, the probability function, , is the canonical projection from the circle bundle to M, is the dimension of the torus, the dimension of the group G, is a quantity associated to the metric and are respectively the moment map of the group G and the torus T. Here we were under the assumptions that is a regular value for and (for more datails see [17] ).

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

a Toeplitz operator, where f is,

is the Szegö kernel and denotes multiplication by f. We shall consider fixed, and. Then is a self- adjoint endomorphisms of. We shall reinterpret a result of [17] obtaining an asymptotic of the principal term of (the mean value operator) for. We shall have:

(13)

with the following principal term in the asymptotic expansion:

(14)

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.

4. Conclusion

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.

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