On Defining the Hamiltonian Beyond Quantum Theory

The central conception of all modern physics is the “Hamiltonian”

Erwin Schrodinger

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.

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Notes

  1. 1.

    By n-dimensional theories, we mean those where normalized states are determined by n real degrees of freedom.

  2. 2.

    Specifically, from the Robertson–Schrödinger uncertain relation [45, 46] as applied to the three Pauli operators—see, for instance, Appendix A of [44]

  3. 3.

    Geometrically, the set of normalized states correspond to a hyperplane intersection with the cone of all states.

  4. 4.

    This definition follows similar definitions in [15, 54]. In general, an observable can be treated as a functional that maps states onto a number, for example representing an expectation value of a measurement, as per [13].

  5. 5.

    Without time-independence, this map is not necessarily homomorphic. E.g. composition of evolutions for 3 seconds and for 5 seconds is equivalent to evolving the system for 8 seconds only if “5 seconds of evolution” corresponds to the same group element whether the evolution begins at time \(t=0\) or at time \(t=3\;{\mathrm{seconds}}\).

  6. 6.

    For time-dependent A, one integrates Eq. (19) into a time-ordered exponential: \(M(\tau ) = \mathcal {T}\left\{ \exp \left[ \int _0^\tau A(t)\, \mathrm{d}t \right] \right\} \), where \(\mathcal {T}(\cdot )\) denotes that every term in the expansion of the exponent only appears in increasing time order. This explicit ordering is necessary since in general A(t) and \(A(t')\) might not commute at different times.

  7. 7.

    [13] use an inverse statement of GEN, and consider the “observability of energy” as a postulate for quantum theory. Namely, they specify that the generator of dynamics should be able to uniquely determine an observable. On top of a set of axioms that restricts theories to Jordan algebras, this uniquely singles out quantum theory.

  8. 8.

    Allowed means that the transformation maps all states to states and satisfies any other constraints that are part of the theory.

  9. 9.

    As a pathological example, consider a three level system where \(E_1 = 0\), and \(E_3 = 2 E_2\); a transformation from a state \(\vec {\rho }\) with well-defined energy in \(E_2\) () to \(\rho '\) where would satisfy INV but not INV*.

  10. 10.

    This does not mean that there are no phase dynamics—take for instance the Aharonov-Bohm effect in quantum theory, whereby some global operation induces phases between branches. However, in quantum theory, a transformation with statistically identical action on the states can also be induced by putting pieces of glass on each individual branch; in box-world, a global transformation akin to the Aharonov–Bohm effect is permissible, whereas the analogous local construction that induces the same phase transformation is not possible.

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Acknowledgements

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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Branford, D., Dahlsten, O.C.O. & Garner, A.J.P. On Defining the Hamiltonian Beyond Quantum Theory. Found Phys 48, 982–1006 (2018). https://doi.org/10.1007/s10701-018-0205-9

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Keywords

  • Hamiltonian
  • Generalized probabilistic theories
  • Energy
  • Time evolution