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Quantum mechanics from stochastic processes

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Abstract

We construct an explicit one-to-one correspondence between non-relativistic stochastic processes and solutions of the Schrödinger equation and between relativistic stochastic processes and solutions of the Klein–Gordon equation. The existence of this equivalence suggests that the Lorentzian path integral can be defined as an Itô integral, similar to the definition of the Euclidean path integral in terms of the Wiener integral. Moreover, the result implies a stochastic interpretation of quantum theories.

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Notes

  1. For \(\alpha \in (0,\infty )\) one obtains the heat equation and for \(\alpha \in (-\infty ,0)\) the time-reversed heat equation.

  2. For \(\alpha \in \textrm{i}\times (0,\infty )\) one obtains the Schrödinger equation and for \(\alpha \in \textrm{i}\times (-\infty ,0)\) the time-reversed Schrödinger equation.

  3. Starting in Sect. 9, we will generalize this to pseudo-Riemannian manifolds \((\mathcal {M},g)\).

  4. The statistical theory is often referred to as the Euclidean theory as the Wick rotation does not only change the value of the diffusion constant, but also the signature of the spacetime.

  5. The presence of this divergent term resolves Wallstrom’s criticism of stochastic mechanics [23, 24].

  6. We stress that the complex process studied here is different from the processes that were previously studied in stochastic mechanics [13,14,15,16,17], including the complex formulation due to Pavon [18, 19]. The advantage of this reformulation is twofold: (i) the complex process unifies quantum mechanics (\(\alpha =\textrm{i}\)) and Brownian motion (\(\alpha =1\)) in a single framework; (ii) the complex process correctly reproduces all aspects of quantum mechanics, whereas previous formulations failed to recover the correct multi-time correlations [20].

  7. In non-relativistic theories the evolution parameter is the time t, while in relativistic theories the time t is promoted to a coordinate \(x^0=c\,t\). Instead, the evolution of relativistic dynamics is defined with respect to an arbitrary affine parameter \(\lambda\). If \(m>0\), this parameter reduces to the proper time \(\tau\) of the particle after gauge fixing \(\varepsilon\).

  8. This is not the first work to point this out, as relativistic quantum theories for a single particle have been developed earlier in the literature, cf. e.g. Ref. [28].

  9. \(\Psi\) is the solution of a complex diffusion equation on the space \(\mathcal {M}\times \mathcal {T}\), where the dynamics is measured with respect to the affine parameter \(\lambda\). Due to the reparameterization invariance of the relativistic theory, this diffusion equation can be solved by separation of variables, yielding Eq. (78), where \(\Phi\) satisfies the Klein-Gordon equation (79).

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Acknowledgements

This research was carried out in the frame of Programme STAR Plus, financially supported by UniNA and Compagnia di San Paolo.

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  1. INFN, Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via Cintia Edificio 6, 80126, Napoli, Italy

    Folkert Kuipers

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Correspondence to Folkert Kuipers.

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Kuipers, F. Quantum mechanics from stochastic processes. Eur. Phys. J. Plus 138, 542 (2023). https://doi.org/10.1140/epjp/s13360-023-04184-x

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