On Defining the Hamiltonian Beyond Quantum Theory

Foundations of Physics volume 48, pages982–1006(2018)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

US$ 39.95

Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

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.

References

  1. 1.

    Hankins, T.L.: Sir William Rowan Hamilton, p. 64. Johns Hopkins University Press, Baltimore (1980)

    Google Scholar 

  2. 2.

    Coopersmith, J.: Energy, the Subtle Concept: The Discovery of Feynman’s Blocks from Leibniz to Einstein. Oxford University Press, Oxford (2010)

    Google Scholar 

  3. 3.

    Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, New York (1989)

  4. 4.

    Cohen-Tannoudji, C., Diu, B., Laloe, F.: Quantum Mechanics. Wiley, New York (1977)

    Google Scholar 

  5. 5.

    Hardy, L.: Quantum theory from five reasonable axioms (2001). arXiv:quant-ph/0101012

  6. 6.

    Barrett, J.: Information processing in generalized probabilistic theories. Phys. Rev. A 75(3), 032304 (2007). https://doi.org/10.1103/PhysRevA.75.032304

    ADS  Article  Google Scholar 

  7. 7.

    Masanes, L., Müller, M.P.: A derivation of quantum theory from physical requirements. New J. Phys. 13(6), 063001 (2011). https://doi.org/10.1088/1367-2630/13/6/063001

    ADS  Article  Google Scholar 

  8. 8.

    Janotta, P., Hinrichsen, H.: Generalized probability theories: what determines the structure of quantum theory? J. Phys. A 47(32), 323001 (2014). https://doi.org/10.1088/1751-8113/47/32/323001

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Popescu, S., Rohrlich, D.: Quantum nonlocality as an axiom. Found. Phys. 24(3), 379–385 (1994). https://doi.org/10.1007/BF02058098

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Hardy, L.: Reformulating and reconstructing quantum theory (2011). arXiv:1104.2066 [gr-qc, physics:hep-th, physics:quant-ph]

  11. 11.

    Dakić, B., Brukner, Č.: Quantum theory and beyond: is entanglement special? In: Hans, H. (ed.) Deep Beauty: Understanding the Quantum World Through Mathematical, pp. 365–392. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  12. 12.

    Chiribella, G.: DAriano, G.M., Perinotti, P.: Informational derivation of quantum theory. Phys. Rev. A 84(1), 012311 (2011). https://doi.org/10.1103/PhysRevA.84.012311

    ADS  Article  Google Scholar 

  13. 13.

    Barnum, H., Müller, M.P., Ududec, C.: Higher-order interference and single-system postulates characterizing quantum theory. N. J. Phys. 16(12), 123029 (2014). https://doi.org/10.1088/1367-2630/16/12/123029

    Article  Google Scholar 

  14. 14.

    Barnum, H., Barrett, J., Leifer, M., Wilce, A.: Cloning and broadcasting in generic probabilistic theories (2006). arXiv:quant-ph/0611295

  15. 15.

    Chiribella, G., Scandolo, C.M.: Entanglement as an axiomatic foundation for statistical mechanics (2016). arXiv:1608.04459 [quant-ph]

  16. 16.

    Barrett, J., Hardy, L., Kent, A.: No signaling and quantum key distribution. Phys. Rev. Lett. 95(1), 010503 (2005). https://doi.org/10.1103/PhysRevLett.95.010503

    ADS  Article  Google Scholar 

  17. 17.

    Acín, A., Gisin, N., Masanes, L.: From Bell’s theorem to secure quantum key distribution. Phys. Rev. Lett. 97(12), 120405 (2006). https://doi.org/10.1103/PhysRevLett.97.120405

    ADS  Article  MATH  Google Scholar 

  18. 18.

    Acín, A., Brunner, N., Gisin, N., Massar, S., Pironio, S., Scarani, V.: Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98(23), 230501 (2007). https://doi.org/10.1103/PhysRevLett.98.230501

    ADS  Article  Google Scholar 

  19. 19.

    Short, A.J., Wehner, S.: Entropy in general physical theories. N. J. Phys. 12(3), 033023 (2010). https://doi.org/10.1088/1367-2630/12/3/033023

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Barnum, H., Barrett, J., Clark, L.O., Leifer, M., Spekkens, R., Stepanik, N., Wilce, A., Wilke, R.: Entropy and information causality in general probabilistic theories. N. J. Phys. 12(3), 033024 (2010). https://doi.org/10.1088/1367-2630/12/3/033024

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Kimura, G., Nuida, K., Imai, H.: Distinguishability measures and entropies for general probabilistic theories. Rep. Math. Phys. 66(2), 175–206 (2010). https://doi.org/10.1016/S0034-4877(10)00025-X

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Müller, M.P., Dahlsten, O.C.O., Vedral, V.: Unifying typical entanglement and coin tossing: on randomization in probabilistic theories. Commun. Math. Phys. 316(2), 441–487 (2012). https://doi.org/10.1007/s00220-012-1605-x

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Chiribella, G., Scandolo, C.M.: Entanglement and thermodynamics in general probabilistic theories. N. J. Phys. 17(10), 103027 (2015). https://doi.org/10.1088/1367-2630/17/10/103027

    Article  Google Scholar 

  24. 24.

    Barnum, H., Barrett, J., Krumm, M., Müller, M.P.: Entropy, majorization and thermodynamics in general probabilistic theories. Electron. Proc. Theor. Comput. Sci. 195, 43–58 (2015). https://doi.org/10.4204/EPTCS.195.4

    MathSciNet  Article  Google Scholar 

  25. 25.

    Chiribella, G., Scandolo, C.M.: Operational axioms for diagonalizing states. Electron. Proc. Theor. Comput. Sci. 195, 96–115 (2015). https://doi.org/10.4204/EPTCS.195.8

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kimura, G., Ishiguro, J., Fukui, M.: Entropies in general probabilistic theories and their application to the Holevo bound. Phys. Rev. A 94(4), 042113 (2016). https://doi.org/10.1103/PhysRevA.94.042113

    ADS  Article  Google Scholar 

  27. 27.

    Krumm, M., Barnum, H., Barrett, J., Müller, M.P.: Thermodynamics and the structure of quantum theory. N. J. Phys. 19(4), 043025 (2017). https://doi.org/10.1088/1367-2630/aa68ef

    Article  Google Scholar 

  28. 28.

    Chiribella, G., Scandolo, C.M.: Microcanonical thermodynamics in general physical theories. N. J. Phys. 19(12), 123043 (2017). https://doi.org/10.1088/1367-2630/aa91c7

    Article  Google Scholar 

  29. 29.

    Dahlsten, O.C.O., Garner, A.J.P., Thompson, J., Gu, M., Vedral, V.: Particle exchange in post-quantum theories (2013). arXiv:1307.2529 [quant-ph]

  30. 30.

    Sorkin, R.D.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 09(33), 3119–3127 (1994). https://doi.org/10.1142/S021773239400294X

    ADS  MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Ududec, C., Barnum, H., Emerson, J.: Three slit experiments and the structure of quantum theory. Found. Phys. 41(3), 396–405 (2011). https://doi.org/10.1007/s10701-010-9429-z

    ADS  MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Garner, A.J.P., Dahlsten, O.C.O., Nakata, Y., Murao, M., Vedral, V.: A framework for phase and interference in generalized probabilistic theories. N. J. Phys. 15(9), 093044 (2013). https://doi.org/10.1088/1367-2630/15/9/093044

    MathSciNet  Article  Google Scholar 

  33. 33.

    Dahlsten, O.C.O., Garner, A.J.P., Vedral, V.: The uncertainty principle enables non-classical dynamics in an interferometer. Nat. Commun. 5, 4592 (2014). https://doi.org/10.1038/ncomms5592

    ADS  Article  Google Scholar 

  34. 34.

    Garner, A.J.P., Müller, M.P., Dahlsten, O.C.O.: The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer. Proc. R. Soc. A 473, 20170596 (2017). https://doi.org/10.1098/rspa.2017.0596

    ADS  MathSciNet  Article  Google Scholar 

  35. 35.

    Lee, C.M., Selby, J.H.: Higher-order interference in extensions of quantum theory. Found. Phys. 47(1), 89–112 (2017). https://doi.org/10.1007/s10701-016-0045-4

    ADS  MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    Barnum, H., Lee, C.M., Scandolo, C.M., Selby, J.H.: Ruling out higher-order interference from purity principles. Entropy 19(6), 253 (2017). https://doi.org/10.3390/e19060253

    ADS  Article  Google Scholar 

  37. 37.

    Lin, Y.L., Dahlsten, O.C.O.: Tunnelling necessitates negative Wigner function. arXiv:1607.01764 [quant-ph] (2016)

  38. 38.

    Lee, C.M., Barrett, J.: Computation in generalised probabilisitic theories. N. J. Phys. 17(8), 083001 (2015). https://doi.org/10.1088/1367-2630/17/8/083001

    Article  Google Scholar 

  39. 39.

    Lee, C.M., Selby, J.H.: Generalised phase kick-back: the structure of computational algorithms from physical principles. N. J. Phys. 18(3), 033023 (2016). https://doi.org/10.1088/1367-2630/18/3/033023

    Article  Google Scholar 

  40. 40.

    Lee, C.M., Hoban, M.J.: Bounds on the power of proofs and advice in general physical theories. Proc. R. Soc. A 472(2190), 20160076 (2016). https://doi.org/10.1098/rspa.2016.0076

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Lee, C.M., Selby, J.H.: Deriving Grover’s lower bound from simple physical principles. N. J. Phys. 18(9), 093047 (2016). https://doi.org/10.1088/1367-2630/18/9/093047

    Article  Google Scholar 

  42. 42.

    Barrett, J., de Beaudrap, N., Hoban, M.J., Lee, C.M.: The computational landscape of general physical theories (2017). arXiv:1702.08483 [quant-ph]

  43. 43.

    Lee, C.M., Selby, J.H., Barnum, H..: Oracles and query lower bounds in generalised probabilistic theories (2017). arXiv:1704.05043 [quant-ph]

  44. 44.

    Garner, A.J.P.: Interferometric computation beyond quantum theory. Found. Phys. (2018). https://doi.org/10.1007/s10701-018-0142-7

    Google Scholar 

  45. 45.

    Robertson, H.P.: The uncertainty principle. Phys. Rev. 34(1), 163–164 (1929). https://doi.org/10.1103/PhysRev.34.163

    ADS  Article  Google Scholar 

  46. 46.

    Schrödinger, E.: Zum Heisenbergschen Unschrfeprinzip. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse 14, 296–303 (1930)

    MATH  Google Scholar 

  47. 47.

    Hioe, F.T., Eberly, J.H.: $N$-level coherence vector and higher conservation laws in quantum optics and quantum mechanics. Phys. Rev. Lett. 47(12), 838–841 (1981). https://doi.org/10.1103/PhysRevLett.47.838

    ADS  MathSciNet  Article  Google Scholar 

  48. 48.

    Kimura, G.: The bloch vector for N-level systems. J. Phys. Soc. Jpn. 72(Suppl.C):185–188 (2003). https://doi.org/10.1143/JPSJS.72SC.185

  49. 49.

    Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  50. 50.

    Bertlmann, R.A., Krammer, P.: Bloch vectors for qudits. J. Phys. A 41(23), 235303 (2008). https://doi.org/10.1088/1751-8113/41/23/235303

    ADS  MathSciNet  Article  MATH  Google Scholar 

  51. 51.

    Jakóbczyk, L., Siennicki, M.: Geometry of Bloch vectors in two-qubit system. Phys. Lett. A 286(6), 383–390 (2001). https://doi.org/10.1016/S0375-9601(01)00455-8

    ADS  MathSciNet  Article  MATH  Google Scholar 

  52. 52.

    Chiribella, G., D’Ariano, G.M., Perinotti, P.: Probabilistic theories with purification. Phys. Rev. A 81(6), 062348 (2010). https://doi.org/10.1103/PhysRevA.81.062348

    ADS  Article  Google Scholar 

  53. 53.

    Janotta, P., Lal, R.: Generalized probabilistic theories without the no-restriction hypothesis. Phys. Rev. A 87(5), 052131 (2013). https://doi.org/10.1103/PhysRevA.87.052131

    ADS  Article  Google Scholar 

  54. 54.

    Garner, A.J.P.: Phase and interference phenomena in generalised probabilistic theories. DPhil thesis, University of Oxford (2015)

  55. 55.

    Spekkens, R.W.: Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A 75(3), 032110 (2007). https://doi.org/10.1103/PhysRevA.75.032110

    ADS  Article  Google Scholar 

  56. 56.

    Wilce, A.: A royal road to quantum theory (or thereabouts) (2016). http://arxiv.org/abs/1606.09306

  57. 57.

    Simon, B.: Representations of Finite and Compact Groups. American Mathematical Society, Providence (1996)

    Google Scholar 

  58. 58.

    Gottesman, D.: Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54(3), 1862–1868 (1996). https://doi.org/10.1103/PhysRevA.54.1862

    ADS  MathSciNet  Article  Google Scholar 

  59. 59.

    van Enk, S.J.: A toy model for quantum mechanics. Found. Phys. 37(10), 1447–1460 (2007). https://doi.org/10.1007/s10701-007-9171-3

    ADS  MathSciNet  Article  Google Scholar 

  60. 60.

    Pusey, M.F.: Stabilizer notation for Spekkens’s Toy theory. Found. Phys. 42(5), 688–708 (2012). https://doi.org/10.1007/s10701-012-9639-7

    ADS  MathSciNet  Article  MATH  Google Scholar 

  61. 61.

    Tuckerman, M.: Operators and numerical propagation methods (2002). URL http://www.nyu.edu/classes/tuckerman/mol.dyn/lectures/lecture_9/node1.html

  62. 62.

    Graydon, M.A.: Quaternions and quantum theory. PhD thesis, University of Waterloo (2011). URL http://hdl.handle.net/10012/6168

  63. 63.

    Goold, J., Huber, M., Riera, A., del Rio, L., Skrzypczyk, P.: The role of quantum information in thermodynamicsa topical review. J. Phys. A 49(14), 143001 (2016). https://doi.org/10.1088/1751-8113/49/14/143001

    ADS  MathSciNet  Article  MATH  Google Scholar 

  64. 64.

    Skrzypczyk, P., Short, A.J., Popescu, S.: Work extraction and thermodynamics for individual quantum systems. Nat. Commun. 5, 4185 (2014). https://doi.org/10.1038/ncomms5185

    ADS  Article  Google Scholar 

  65. 65.

    Horodecki, M., Oppenheim, J.: Fundamental limitations for quantum and nanoscale thermodynamics. Nat. Commun. 4, 2059 (2013). https://doi.org/10.1038/ncomms3059

    ADS  Article  Google Scholar 

Download references

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.

Author information

Affiliations

  1. Department of Physics, University of Warwick, Coventry, CV4 7AL, UK

    Dominic Branford

  2. Atomic and Laser Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK

    Dominic Branford, Oscar C. O. Dahlsten & Andrew J. P. Garner

  3. Southern University of Science and Technology (SUSTech), Nanshan District, Shenzhen, China

    Oscar C. O. Dahlsten

  4. London Institute for Mathematical Sciences, 35a South Street, Mayfair, London, W1K 2XF, UK

    Oscar C. O. Dahlsten

  5. Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, 1090, Vienna, Austria

    Andrew J. P. Garner

  6. Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore, 117543, Singapore

    Andrew J. P. Garner

Authors
  1. Dominic Branford
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oscar C. O. Dahlsten
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew J. P. Garner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Oscar C. O. Dahlsten.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Hamiltonian
  • Generalized probabilistic theories
  • Energy
  • Time evolution