Quantum Logic and Geometric Quantization

Quantum Logic and Geometric Quantization 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). 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 kind of objects are involved in this procedure. This mathematical approach is very classic and basic results are in [1]. About the quantization of K¨ahler manifolds and the Berezin–Toeplitz quantization we suggest the following literature [2], [3],[4], [5] and [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. Fundamental works in this field are [7], [8] and [9]. In order to emphasize the importance 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 AMS Subject Classification (2010). Primary 03G12; Secondary 03G10, 81P10, 53D50. The first part of these notes was inspired by a series of conversations when the author was in Wien at E.S.I. in occasion of GEOQUANT2013. 1 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] and [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.