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Epistemic Uncertainty from an Averaged Hamilton–Jacobi Formalism

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Abstract

In recent years, the non-relativistic quantum dynamics derived from three assumptions; (i) probability current conservation, (ii) average energy conservation, and (iii) an epistemic momentum uncertainty (Budiyono and Rohrlich in Nat Commun 8:1306, 2017). Here we show that, these assumptions can be derived from a natural extension of classical statistical mechanics.

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Notes

  1. Perhaps, the meaning of “average” is more cleared, if we rewrite this axiom as \(\int f(x,p) [\partial _t S+H(x,p)]dx dp=0\)

  2. This variational principle is a direct generalization of the variational formulation of classical H–J formalism, i.e. \(\delta \int \rho [\partial _t S+H(x,\nabla S)]dx dt=0\), which leads to classical H–J and continuity equations [26]. Similar generalized actions, also used in some other methods for derivation of Schroedinger equation [8, 9, 27,28,29]. Moreover, one can use of an “average Hamilton’s equations” instead of this variational principle; i.e. \(\partial _t \rho =\delta \langle H\rangle /\delta S\) and \(\partial _t S=-\delta \langle H\rangle /\delta \rho\), where \(\rho\) and \(S\) are considered as canonical conjugate variables [19, 20]. In addition, it is easy to see that, the equation-of-motions which finally we find from these axioms, (4) and (5), have clear physical meanings in our interpretation: probability and average classical energy conservation, \(d\langle H\rangle /dt=0\), which together ensure conservation of the total classical energy of ensemble

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  1. Department of Physics, Faculty of Science, University of Qom, Qom, Iran

    M. J. Kazemi & S. Y. Rokni

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Kazemi, M.J., Rokni, S.Y. Epistemic Uncertainty from an Averaged Hamilton–Jacobi Formalism. Found Phys 52, 54 (2022). https://doi.org/10.1007/s10701-022-00571-z

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