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

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.

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Cited by in corpus (14)

• General probabilistic theories: An introduction
• Detecting quantumness in uniform precessions
• Measurement incompatibility is strictly stronger than disturbance
• Unification of energy concepts in generalised phase space theories
• From reasonable postulates to generalised Hamiltonian systems
• Generalized dynamical theories in phase space and the hydrogen atom
• Quantum harvester enables energy transfer without randomness transfer or dissipation
• All three-angle variants of Tsirelson's precession protocol, and improved bounds for wedge integrals of Wigner functions
• Tsirelson's Inequality for the Precession Protocol is Maximally Violated by Quantum Theory
• Newton's laws of motion generating gravity-mediated entanglement
• Probing the nonclassical dynamics of a quantum particle in a gravitational field
• Symmetries and Wigner representations of operational theories
• Existing experiments suffice to indirectly verify the quantum essence of gravity
• Information-theoretic derivation of energy, speed bounds, and quantum theory