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arXiv:2512.10497 · New Journal of Physics

The relativistic reason for quantum probability amplitudes

Karol Sajnok, Kacper Dębski, Andrzej Dragan

quant-phgr-qcSubmitted Dec 11, 2025• Announced Dec 12, 2025• 0 citations• Revised Oct 17, 2026 (v2)
v1 · Dec 11, 2025v2 · Oct 17, 2026
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Abstract

We show that the quantum-mechanical probability distribution involving complex probability amplitudes can be derived from three natural conditions imposed on a relativistically invariant probability function describing the motion of a particle that can take multiple paths simultaneously. The conditions are: (i) pairwise Kolmogorov additivity, (ii) time symmetry, and (iii) Bayes' rule. The resulting solution, parameterized by a single constant, is the squared modulus of a sum of complex exponentials of the relativistic action, thereby recovering the Feynman path-integral formulation of quantum mechanics.

Journal reference
New Journal of Physics, 28(3), 034511 (2026)
DOI
10.1088/1367-2630/ae4d79
Comments
6 pages, 2 figures, 1 appendix

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We show that the quantum-mechanical probability distribution involving complex probability amplitudes can be derived from three natural conditions imposed on a relativistically invariant probability function describing the motion of a particle that can take multiple paths simultaneously. The conditions are: (i) pairwise Kolmogorov additivity, (ii) time symmetry, and (iii) Bayes' rule. The resulting solution, parameterized by a single constant, is the squared modulus of a sum of complex exponentials of the relativistic action, thereby recovering the Feynman path-integral formulation of quantum mechanics.
The Feynman path integral formalism for non-relativistic quantum mechanics is revisited. A comparison is made with cases of light propagation (Huygens' principle) and Brownian motion. The difficulties for a physical model applying Feynman's formalism are pointed out. A reformulation is proposed, where the transition probability of a particle from one space-time point to another one is the sum of probabilities of the possible paths. As an application, Born approximation for scattering is derived within the formalism, which suggests an interpretation involving the stochastic motion of a particle rather than the square of a wavelike amplitude.
Feynman path integrals formalism for non-relativistic quantum mechanics is revisited. A comparison is made with the cases of light progagation (Huygens principle) and Brownian motion. The difficulties for a physical model behind Feynman formalism are pointed out. It is proposed a reformulation where the transition probability from one space-time point to another one is the sum of probabilities of the possible paths. The Born approximation for scattering is derived within the formalism, which suggests an interpretation in terms of particles, without the need of Born assumption that the modulus squared of the wavefunction is a probability density.
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In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
In a recent paper, (1) introduces a path related probability scheme in order to derive quantum mechanics probability. This probability is a real positive number between 0,1 and so one is classical, but the fact that it is linked to paths and not simply a position, time x,t suggests that this approach differs from strictly classical probability. In particular, the assumptions used by (1) are path related. (1)’s path based probability is P(nm) ( f(x,t)), where nm is a path and f(x,t), a Lorentz invariant. Bayesian composition is allowed and P(nm) = P(n)(path1) P(m)(path2). As a result, one may break a probability into a product of probabilities based on paths. For example, path1 might be from a starting point to a point P and path2 from P to a detector. For a linear path along x, this suggests the form exp(-iEt+ipx)=W(x,t). To obtain a real probability, one needs (1)’s assumption of time reversal invariant, i.e. P(-Et+px)=P(Et-px), so P = W*(x,t)W(x,t). We argue that (1)’s quantum form follows from the last two (of three assumptions). Formally, one suggests that quantum mechanics arises from (A) “Kolmogorov additivity restricted to one or two path contributions”, (B) time reversal symmetry and (C ) Bayesian composition. We go on to argue that the path related approach of (1) is equivalent to a momentum-energy conserving probability approach of (2).
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