Fractional Fourier Transform and Geometric Quantization

TL;DR

This paper introduces a coordinate-independent generalized Fourier transform based on symplectic geometry, enabling exact quantum dynamics for linear systems and revealing new symmetries of the Schrödinger equation.

Contribution

It develops a novel, geometry-based construction of quantum transformations, extending the fractional Fourier transform concept without relying on linear structures.

Findings

Quantum dynamics of linear systems can be derived from classical dynamics via this transform.

New symmetries of the Schrödinger equation are identified.

A natural connection in the quantum state bundle is defined, explaining the non-existence of a full quantum observable algebra representation.

Abstract

Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of the phase-space: no linear structure is necessary. It is shown that the "fractional Fourier transform" provides a simple example of this construction. As an application of this techniques we show that for any linear Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of the corresponding classical dynamics by means of the above transformation. Moreover, it can be deduced from the free quantum evolution. This way new, unknown symmetries of the Schr\"odinger equation can be constructed. It is also argued that the above construction defines in a natural way a connection in the bundle of quantum states, with the base space describing all their possible representations. The non-flatness of this connection would be responsible for the non-existence of a quantum representation of the complete algebra of classical observables.