Remarks on the naturality of quantization

We explain that when quantising phase spaces with varying symplectic structures, the bundle of quantum Hilbert spaces over the parameter space has a natural unitary connection. We then focus on symplectic vector spaces and their fermionic counterparts. After reviewing how the quantum Hilbert space depends on physical parameters such as the Hamiltonian and unphysical parameters such as choices of polarisations, we study the connection, curvature and phases of the Hilbert space...

Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. More precisely, is it possible to capture certain quantum idiosyncrasies within the symplectic framework of classical mechanics? Beginning with Dirac, the idea of quantizing a classical system involved associating the phase space variables with Hermitian operators which act on some Hilbert space, as well as associating the Poisson bracke...

These notes give an introduction to the quantization procedure called geometric quantization. It gives a definition of the mathematical background for its understanding and introductions to classical and quantum mechanics, to differentiable manifolds, symplectic manifolds and the geometry of line bundles and connections. Moreover, these notes are endowed with several exercises and examples.

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle H over the space J of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle H -> J is a rescaled orthogonal proj...

Geometric quantization is an attempt at using the differential-geometric ingredients of classical phase spaces regarded as symplectic manifolds in order to define a corresponding quantum theory. Generally, the process of geometric quantization is applicable to other symplectic manifolds, not only cotangent spaces. The resulting formalism provides a way of looking at quantum theory that is distinct from conventional approaches to the subject, e.g., the Dirac bra-ket formalism....

The classical limit $\hbar$->0 of quantum mechanics is known to be delicate, in particular there seems to be no simple derivation of the classical Hamilton equation, starting from the Schr\"odinger equation. In this paper I elaborate on an idea of M. Reuter to represent wave functions by parallel sections of a flat vector bundle over phase space, using the connection of Fedosov's construction of deformation quantization. This generalizes the ordinary Schr\"odinger representat...

Quantum mechanics is among the most important and successful mathematical model for describing our physical reality. The traditional formulation of quantum mechanics is linear and algebraic. In contrast classical mechanics is a geometrical and non-linear theory that is defined on a symplectic manifold. However, after invention of general relativity, we are convinced that geometry is physical and effect us in all scale. Hence the geometric formulation of quantum mechanics soug...

A comparison on some facts concerning the geometric quantization of symplectic manifolds is presented here. Criticism, facts and improvements on the sophisticated theory of geometric quantization are presented touching briefly, all the "salient points of the theory". The unfamiliar reader can consider this as a "soft" introduction to the topic.

Classical mechanics can be formulated using a symplectic structure on classical phase space, while quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This letter is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space.

We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert space as a linear real manifold equipped with its canonical symplectic form and restricting only to the expectation-value functions of Hermitian operators. Then in this framework, we reformulate the structure of quantum mechanics in the lan...