Foundations of Physics

, Volume 48, Issue 8, pp 982–1006 | Cite as

On Defining the Hamiltonian Beyond Quantum Theory

  • Dominic Branford
  • Oscar C. O. DahlstenEmail author
  • Andrew J. P. Garner
Part of the following topical collections:
  1. Foundational Aspects of Quantum Information


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.


Hamiltonian Generalized probabilistic theories Energy Time evolution 



We thank George Knee and Benjamin Yadin for useful comments. We are grateful for financial support from the UK Engineering and Physical Sciences Research Council, the John Templeton Foundation, the Foundational Questions Institute, EU Collaborative Project TherMiQ (Grant Agreement 618074), the London Institute for Mathematical Sciences and Wolfson College, University of Oxford.


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Authors and Affiliations

  1. 1.Department of PhysicsUniversity of WarwickCoventryUK
  2. 2.Atomic and Laser Physics, Clarendon LaboratoryUniversity of OxfordOxfordUK
  3. 3.Southern University of Science and Technology (SUSTech)ShenzhenChina
  4. 4.London Institute for Mathematical SciencesLondonUK
  5. 5.Institute for Quantum Optics and Quantum Information, Austrian Academy of SciencesViennaAustria
  6. 6.Centre for Quantum TechnologiesNational University of SingaporeSingaporeSingapore

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