Operational Theories in Phase Space: Toy Model for the Harmonic Oscillator

TL;DR

This paper introduces a novel operational probabilistic theory framework in phase space, exemplified by a toy model of the harmonic oscillator that exhibits unique quantum-like features.

Contribution

It develops a general probabilistic theory incorporating energy observables in phase space, providing a new approach distinct from classical and quantum theories.

Findings

The toy model has a discrete energy spectrum.

It features a state with non-positive Wigner function.

The model demonstrates tunneling properties.

Abstract

We show how to construct general probabilistic theories that contain an energy observable dependent on position and momentum. The construction is in accordance with classical and quantum theory and allows for physical predictions, such as the probability distribution for position, momentum and energy. We demonstrate the construction by formulating a toy model for the harmonic oscillator that is neither classical nor quantum. The model features a discrete energy spectrum, a ground state with sharp position and momentum, an eigenstate with non-positive Wigner function as well as a state that has tunneling properties. The toy model demonstrates that operational theories can be a viable alternative approach for formulating physical theories.