On defining the Hamiltonian beyond quantum theory

TL;DR

This paper explores how to define the Hamiltonian in general probabilistic theories, extending the concept beyond quantum mechanics, and provides a concrete definition for 3-dimensional systems with discussion on higher dimensions.

Contribution

It introduces a set of desiderata for defining the Hamiltonian in general probabilistic theories and offers a complete recipe for 3D systems, advancing the understanding of energy in generalized frameworks.

Findings

Provided a fully-defined Hamiltonian for 3D systems

Discussed the freedom of choice in higher-dimensional cases

Applied the definition to toy theories and analyzed time evolution

Abstract

Energy is a crucial concept within classical and quantum physics. An essential tool to quantify energy is the Hamiltonian. Here, we consider how to define a Hamiltonian in general probabilistic theories, a framework in which quantum theory is a special case. We list desiderata which the definition should meet. For 3-dimensional systems, we provide a fully-defined recipe which satisfies these desiderata. We discuss the higher dimensional case where some freedom of choice is left remaining. We apply the definition to example toy theories, and discuss how the quantum notion of time evolution as a phase between energy eigenstates generalises to other theories.