Physical Applications Related to Differentiable Deformations of Poisson Brackets: Quantum Mechanics

Abstract

Quantum mechanics is usually formulated in terms of linear operators on a Hilbert space of physical states, while classical mechanics relies on the algebra N of functions over phase-space. In one of the most naive presentations, it is said that the passage from classical to quantum mechanics is realized through the correspondence principle of Bohr, according to which one replaces the position variable q by the operator of multiplication by q, momentum p by iℏ∂/∂q where ℏ is the Planck constant, ‘and the Poisson bracket by the commutator’; the classical limit of a quantum theory being obtained by letting ℏ→0. This smells of deformation theory - but how can one ‘deform’ a function into an operator? Due to the fundamental difference in the nature of the observables in classical and quantum theories, it may seem at first sight hopeless to interpret quantum mechanics as a deformation. We shall nevertheless show in this chapter that one can give an autonomous phase-space formulation of quantum mechanics, in the framework of which computations can be made, and for which quantum mechanics will appear naturally as a differentiable deformation of classical mechanics. Quantization will manifest itself in a deformation of the algebra of observables rather than in a radical change of their nature. The link between this formulation of quantum mechanics and the usual one is provided by the Weyl application, that we shall explain in Section 1.