Observables and States p-Mechanics

TL;DR

This paper surveys p-mechanics, a unified framework that describes classical and quantum mechanics simultaneously using convolution operators on the Heisenberg group, connecting various quantisation methods and interpretations.

Contribution

It provides a comprehensive overview of p-mechanics, detailing its construction, states, observables, and dynamical equations, with applications to harmonic oscillators and links to quantum interpretations.

Findings

p-mechanics unifies classical and quantum descriptions

States and observables are represented as functions/distributions on the Heisenberg group

Dynamical equations are applicable to systems like harmonic oscillators

Abstract

This is an up-to-date survey of the p-mechanical construction (see funct-an/9405002, quant-ph/9610016, math-ph/0007030, quant-ph/0212101, quant-ph/0303142), which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics. Observables in p-mechanics are defined to be convolution operators on the Heisenberg group H^n. Under irreducible representations of H^n the p-observables generate corresponding observables in classical and quantum mechanics. p-States are defined as positive linear functionals on p-observables. It is shown that both states and observables can be realised as certain sets of functions/distributions on the Heisenberg group. The dynamical equations for both p-observables and p-states are provided. The construction is illustrated by the forced and unforced harmonic oscillators. Connections with the contextual interpretation of quantum mechanics are discussed. Keywords: Classical mechanics, quantum mechanics, Moyal brackets, Poisson brackets, commutator, Heisenberg group, orbit method, deformation quantisation, symplectic group, representation theory, metaplectic representation, Berezin quantisation, Weyl quantisation, Segal--Bargmann--Fock space, coherent states, wavelet transform, Liouville equation, contextual interpretation, interaction picture, forced harmonic oscillator.