The relativistic reason for quantum probability amplitudes

TL;DR

The paper addresses the foundational origin of quantum probability amplitudes by deriving them from relativistic invariance and minimal probabilistic principles. It posits three conditions—pairwise Kolmogorov additivity (no higher-order interference), time symmetry, and Bayesian composition—and shows these yield a unique probability law for $n$ alternative relativistic paths given by $\mathcal{P}^n(\{\Phi_i\}) = \left| \sum_{i=1}^n e^{ i \kappa \Phi_i } \right|^2$, with $\kappa$ real. By setting $\kappa = 1/\hbar$ and $\Phi_i = -S_i$, this recovers the Feynman path-integral rule of adding complex amplitudes before squaring, framing it as a consequence of relativistic invariance and Bayesian consistency rather than an a priori axiom. The authors also discuss a Fourier-space solution that generalizes the result when the pairwise additivity condition is relaxed, highlighting the foundational link between relativistic probability and quantum amplitudes.

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.

Figures (2)

• Figure 1: Illustration of pairwise Kolmogorov additivity, using Venn diagrams for (a) two events $A,B$ and (b) three events $A,B,C$, with the overlap of three (or more) events taken to be empty.
• Figure 2: Illustration of the three conditions imposed on the relativistically invariant probability functions $\mathcal{P}^n$: ($\star$) Pairwise Kolmogorov additivity: The probability for $n$ alternatives is completely determined by the one-path and two-path contributions, with no irreducible higher-order overlaps. ($\star\,\star$) Time symmetry:$\mathcal{P}^n$ remains invariant under time reversal, so the direction of time between two spacetime points does not affect the probabilities. ($\star\star\star$) Bayesian composition: The probability for a transition from $S$ to $D$ via an intermediate point $P$ factorizes into the probabilities for $S \to P$ and $P \to D$, in analogy with the Bayesian chain rule.