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Relevance of stochasticity for the emergence of quantization

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Abstract

The theories of stochastic quantum mechanics and stochastic electrodynamics bring to light important aspects of the quantum dynamics that are concealed in the standard formalism. Here, we take further previous work regarding the connection between the two theories, to exhibit the role of stochasticity and diffusion in the process leading from the originally classical + zpf regime to the quantum regime. Quantumlike phenomena present in other instances in which a mechanical system is subject to an appropriate oscillating background that introduces stochasticity, may point to a more general appearance of quantization under such circumstances.

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Notes

  1. There are of course a variety of theories containing a stochastic element, aimed to explain or reproduce quantum mechanics. These may widely differ from one another in their method, purpose or philosophy; see, e. g., [2,3,4,5,6,7,8,9,10,11,12,13,14]. It is not our intention to review such theories, but to draw on the specific connections between sqm and sed for a better understanding of the role of stochasticity in the emergence of quantization.

  2. This approximation is justified a posteriori [16], as it turns out that the wavelength of the relevant modes of the zpf, i. e., those with which the particle interacts resonantly in the quantum regime (see below), is indeed much larger than the displacements of the electron around its mean position.

  3. More detailed derivations are presented in [16] and references therein. Furthermore, the radiative terms neglected here have been explored to the lowest order of approximation and shown to reproduce the predictions of nonrelativistic quantum electrodynamics [24,25,26].

  4. This is not a unique situation in theoretical physics; there are several (although related) examples in which stochasticity brings about a qualitative change in the dynamics. The title of section 2.4 of the forerunner paper by Chandrasekhar [27] reads “The Fokker–Planck Equation. The Generalization of Liouville’s Theorem.” The generalization at issue is just an extension of the Hamiltonian dynamics of Liouville’s theorem, so as to embed fluctuations and dissipation into the scheme. A more recent example, closer to our case, is the discovery by Nelson [3] of the need of two velocities for an appropriate description of the dynamics of a stochastic system, just as discussed above. However. Nelson continued to call his theory Newtonian.

  5. In a separate work [28], it been shown that under conditions of ergodicity, the dynamical variables describing the statistical properties of an ensemble in a given (pure) state are expressed by the corresponding quantum operators; this represents a complementary derivation of quantum mechanics á la Heisenberg.

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Acknowledgements

The authors acknowledge partial support from DGAPA-UNAM through project PAPIIT IN113720.

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  1. Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico

    A. M. Cetto, L. de la Peña & A. Valdés-Hernández

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  1. A. M. Cetto
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  2. L. de la Peña
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Correspondence to A. M. Cetto.

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Cetto, A.M., de la Peña, L. & Valdés-Hernández, A. Relevance of stochasticity for the emergence of quantization. Eur. Phys. J. Spec. Top. 230, 923–929 (2021). https://doi.org/10.1140/epjs/s11734-021-00066-4

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