The relativistic reason for quantum probability amplitudes

arXiv:2512.10497 · quant-ph, gr-qc · Submitted 2025-12-11 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "The relativistic reason for quantum probability amplitudes".

Mira: Here's the gist of the paper. The gist 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, recovering the Feynman path-integral formulation of quantum mechanics<ref:2512.10497#pg2>.

Kai: Derivation Framework The derivation begins by imposing three physical requirements on a relativistically invariant family of probability functions Pn: (⋆) Kolmogorov additivity restricted to one- and two-path contributions (no irreducible higher-order interference), (⋆ ⋆) time symmetry, and (⋆ ⋆ ⋆) Bayesian composition<ref:2512.10497#pg2>.

Mira: The standard quantum-mechanical rule of summing complex amplitudes over paths and then taking the squared modulus—i.e., the Feynman path-integral prescription—is shown to emerge as the simplest nonclassical solution consistent with relativistic invariance and Bayesian composition rule<ref:2512.10497#pg2>.

Kai: This approach relies solely on relativistic invariance and minimal probabilistic assumptions applied directly within a minimally nonclassical extension of Kolmogorov probability theory<ref:2512.10497#pg2>.

Paper summary: Kai: To recap where we are, we're discussing "The relativistic reason for quantum probability amplitudes" and how the authors show that a relativistically invariant probability function satisfying pairwise Kolmogorov additivity, time symmetry, and Bayes’ rule leads directly to the Feynman path-integral formulation.

Mira: Right. The paper sets up these three structural conditions on a family of functions P n, which describe the motion of a particle taking multiple paths simultaneously >Page three# <ref:2512.10497#pg1,the motion of a particle>. They are checking if these conditions force the solution to be something specific.

Lev: And they claim that this specific solution is the squared modulus of a sum of complex exponentials, which is exactly what we use in the path-integral formulation >Page one# <ref:2512.10497#pg1,is the squared modulus of a sum of complex exponentials>. It's a direct derivation from those three rules applied consistently.

Kai: So, the main thesis here is that the quantum way of summing amplitudes isn't just an arbitrary rule we invented for convenience; it emerges as a necessary consequence of relativistic invariance combined with these minimal probabilistic constraints >Page one# <ref:2512.10497#pg1>.

Mira: Precisely. They show that if you start with those three conditions—pairwise additivity, time symmetry, and Bayesian composition—the only way to satisfy them is by having the probability function take this exponential form parameterized by one constant >Page one# <ref:2512.10497#pg1>.

Lev: From an error correction perspective, this means that the structure of the quantum state we are trying to describe is fundamentally tied to how probability must compose across spacetime segments in a relativistically consistent manner >Page three# <ref:2512.10497#pg1>. That's a deep structural claim.

Kai: It’s interesting because it shifts the focus from 'how do we calculate the path integral?' to 'what constraints must a physical probability function obey?'

Mira: Exactly. They show that these three conditions are the minimal set of assumptions needed to derive quantum mechanics, given relativistic invariance >Page one# <ref:2512.10497#pg1>. It’s about showing necessity rather than just sufficiency.

Lev: The paper doesn't delve into complex calculations or detailed experimental setups here; it's purely about the mathematical structure emerging from those abstract rules >Page one# <ref:2512.10497#pg1>. That’s a clean way to look at it.

Kai: It gives us a solid foundation for understanding why we see interference terms in quantum mechanics, even between paths that seem classically exclusive >Page two# <ref:2512.10497#pg1>.

Mira: Yes, because the paper explains how superluminal observers can interpret those classically exclusive trajectories as jointly contributing to the physical description due to relativistic invariance extending beyond standard notions of causality >Page two# <ref:2512.10497#pg1>.

Lev: If we were building a simulator for this, we’d need to ensure that our numerical implementation respects these three rules at every step, which would be tricky given the complexity >Page three# <ref:2512.10497#pg1>.

Kai: So the connection is clear: relativistic invariance plus those three probabilistic axioms forces the path integral structure >Page one# <ref:2512.10497#pg1>. That’s the big picture.

Conclusion: Kai: So, wrapping up our discussion on "The relativistic reason for quantum probability amplitudes," it seems the authors have shown that the quantum mechanical probability distribution involving complex probability amplitudes is derived from imposing pairwise Kolmogorov additivity, time symmetry, and Bayes’ rule on a relativistically invariant function.

Mira: That's the main finding. The result they get is the squared modulus of a sum of complex exponentials of the relativistic action >Page one# <ref:2512.10497#pg1,is the squared modulus of a sum of complex exponentials of the>. This directly recovers the Feynman path-integral formulation we already know in quantum mechanics.

Lev: What this implies for us is that adding complex amplitudes before taking the squared modulus isn't an arbitrary choice in quantum theory; it’s a mathematical consequence arising from combining relativistic invariance with basic probabilistic consistency >Page one# <ref:2512.10497#pg1>.

Kai: It means that if you start with those three conditions, the structure of the probability space itself dictates that we end up with this specific formulation >Page one# <ref:2512.10497#pg1>. It's not an extra layer of theory on top.

Mira: They’re demonstrating necessity. The paper shows that these three rules are the minimal structural requirements to derive quantum mechanics when you include relativistic invariance >Page one# <ref:2512.10497#pg1>. It’s about showing what has to be there for the math to work consistently.

Lev: For hardware, this suggests that if we want a description of particle motion that is fundamentally consistent with relativity and basic probability, we should look at structures derived from these constraints >Page three# <ref:2512.10497#pg1>. That gives us a target structure to aim for in our simulations.

Kai: It really brings the focus back to the fundamental physical requirements. We’re not just dealing with numbers; we’re dealing with what must be true about how probability behaves across space and time, even in quantum settings >Page one# <ref:2512.10497#pg1>.

Mira: Yeah, so the title itself points to this: it's about a relativistic reason for those amplitudes, grounding them in physics rather than just abstract mathematics >Page one# <ref:2512.10497#pg1>. That’s a powerful way to frame the research.

Lev: The paper states its limitation is that it relies on minimal probabilistic assumptions within an extension of Kolmogorov probability theory >Page one# <ref:2512.10497#pg1>. So, while the derivation is tight, the applicability rests on how well those initial assumptions map onto real physical systems >Page one# <ref:2512.10497#pg1>.

Kai: That’s a good point. It’s a mathematical derivation based on some foundational choices about how we define probability in this context >Page one# <ref:2512.10497#pg1>. It doesn't solve all of quantum mechanics, but it explains its structure better.

Mira: Exactly. The paper shows the deep link between these three conditions and the Feynman prescription, which is what matters for understanding the nature of quantum probability amplitudes >Page one# <ref:2512.10497#pg1>.

Lev: It’s a strong result about consistency. If you can satisfy those three conditions, you get this specific output, which is a very concrete thing we can work with on hardware simulations >Page one# <ref:2512.10497#pg1>.

Kai: So, in short, the paper argues that the quantum path integral formulation is not just a convenient way to calculate probabilities; it’s what happens when you demand relativistic invariance and minimal probabilistic consistency >Page one# <ref:2512.10497#pg1>.

Karol Sajnok, *Kacper D˛ebski, +Andrzej Dragan

Institute of Theoretical Physics, University of Warsaw · Center for Theoretical Physics, Polish Academy of Sciences · Department of Optics, Palacký University · Centre for Quantum Technologies, National University of Singapore

quant-ph, gr-qc

Submitted: 2025-12-11

Updated: 2026-10-17

Comments: 6 pages, 2 figures, 1 appendix

Journal ref: New Journal of Physics, 28(3), 034511 (2026)

DOI: 10.1088/1367-2630/ae4d79

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 67/100

The gist: The gist The quantum-mechanical probability distribution involving complex probability amplitudes can be derived from three natural conditions imposed on a relativistically invariant probability

Key concepts

Pairwise Kolmogorov additivity: This condition restricts how probabilities combine when considering multiple paths. It states that the total probability for observing an outcome is completely determined by the probabilities of single paths and pairs of paths, with no irreducible interference terms involving three or more paths allowed.
Time symmetry: This requirement means the probability function must remain unchanged if time is reversed. Physically, it implies that the direction of time between two points in spacetime does not influence the calculated probabilities for a particle's motion.
Bayes' rule: This is the probabilistic analogue of a chain rule, dictating how probabilities factorize when moving through intermediate states. It enforces a specific composition law for transitions between different spacetime points, analogous to the Chapman–Kolmogorov equation in discrete systems.

Terminology

Additivity — 1× in this paper · explained in 1 episode

Summary

The gist 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, recovering the Feynman path-integral formulation of quantum mechanics<ref:2512.10497#pg2>.

Derivation Framework

The derivation begins by imposing three physical requirements on a relativistically invariant family of probability functions Pn: (⋆) Kolmogorov additivity restricted to one- and two-path contributions (no irreducible higher-order interference), (⋆ ⋆) time symmetry, and (⋆ ⋆ ⋆) Bayesian composition<ref:2512.10497#pg2>. The standard quantum-mechanical rule of summing complex amplitudes over paths and then taking the squared modulus—i.e., the Feynman path-integral prescription—is shown to emerge as the simplest nonclassical solution consistent with relativistic invariance and Bayesian composition rule<ref:2512.10497#pg2>. This approach relies solely on relativistic invariance and minimal probabilistic assumptions applied directly within a minimally nonclassical extension of Kolmogorov probability theory<ref:2512.10497#pg2>.

Key Conditions Imposed

The three structural conditions are explicitly defined as follows:

(⋆) Pairwise Kolmogorov additivity requires that the detection intensity for n alternatives is completely determined by the one-path intensities and the twopath contributions, with no additional, irreducible i-path terms with i ≥ 3 present<ref:2512.10497#pg2>. This is precisely the statement that all multi-slit patterns are fixed once all single-slit and doubleslit configurations are known<ref:2512.10497#pg2>.

(⋆ ⋆) Time symmetry requires invariance of the probability functional under time reversal of all paths, meaning Pn remains invariant under time reversal, so the direction of time between two spacetime points does not affect the probabilities<ref:2512.10497#pg2>.

(⋆ ⋆ ⋆) Bayesian composition expresses the probabilistic analogue of the Bayesian chain rule, stating that a transition from S to D via an intermediate point P factorizes into the probabilities for S → P and P → D, in analogy with the Chapman–Kolmogorov equation for a single discrete intermediate state<ref:2512.10497#pg2>.

The Resulting Solution

The main claim is that the only family of bounded and smooth probability functions Pn satisfying these three conditions is given by the expression:

P n(Φi) = Xn i=1 e(iκΦi)/2, where κ is an arbitrary real constant with dimensions inverse to those of the chosen relativistic invariant Φ (proper time, action, etc.)<ref:2512.10497#pg2>. This form Pn should be interpreted as an unnormalized probability density<ref:2512.10497#pg2>. Choosing κ = 1/ħ and Φi = -Si reproduces the Feynman path-integral rule, showing that the prescription of adding complex amplitudes before squaring is not an arbitrary feature of quantum theory, but a mathematical consequence of Bayes’ rule and relativistic invariance combined with the absence of interference beyond second order<ref:2512.10497#pg2>.

Mathematical Proof Outline

The proof proceeds by analyzing the constraints derived from these conditions. The one-argument Bayes rule reduces to Cauchy’s multiplicative functional equation, leading to exponential solutions for P1(Φ) under composition of segments<ref:2512.10497#pg2>. For the many-argument case, a specific rearrangement leads to Pn factorizing into an n-argument function of the deviations of its arguments from their arithmetic mean, and an exponential function of the arithmetic mean itself<ref:2512.10497#pg2>. Time symmetry further constrains P1(Φ) to be 1, which implies ∂P1(0) = 0<ref:2512.10497#pg2>. Pairwise Kolmogorov additivity then yields Pn(Φi) = Xn i=1 D(Φi - Φj) - n(n − 2), where D is an even function of a single real variable<ref:2512.10497#pg2>. The final determination of the function D, using the Bayes composition law for n=m=2, leads to a functional equation that reduces to the d’Alembert equation in Fourier space<ref:2512.10497#pg2>. The choice of trigonometric solutions arises from the normalizability of the probabilistic expression in the limit of infinitely many paths, yielding Eq. (6) as the most general bounded solution derived from conditions (⋆), (⋆ ⋆), and (⋆ ⋆ ⋆)<ref:2512.10497#pg2>.

Conclusion

The three natural requirements—pairwise Kolmogorov additivity, time-reversal symmetry, and Bayesian composition—uniquely lead to a nontrivial solution: the squared modulus of a sum of complex exponentials of these invariants<ref:2512.10497#pg2>. The complex-amplitude framework emerges directly from minimal probabilistic assumptions together with relativistic invariance<ref:2512.10497#pg2>. Our result shows that the familiar Feynman prescription of adding complex amplitudes and then squaring is not an ad hoc rule, but the natural consequence of combining relativistic invariance with basic probabilistic consistency<ref:2512.10497#pg2>.

Acknowledgments

We acknowledge a very useful observation made by ChatGPT-5 that helped us complete the proof presented in this work<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg2>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg2>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg2>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg3>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg3>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg3>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg4>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg4>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg4>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg5>. Section III provides a general derivation of all probability functions that satisfy these conditions<ref:2512.10497#pg5>. Concluding remarks are presented in Sec. IV<ref:2512.10497#pg5>. The paper is organized as follows. In Sec. II, we formulate the problem by introducing three physical requirements: pairwise Kolmogorov additivity, time symmetry, and Bayesian composition<ref:2512.10497#pg6>.

Improvements for AI systems

textbfGeneral Quantum Probability Reconstruction from Relativistic Invariance and Minimal Axioms: Enables derivation of quantum probability amplitudes directly from physical principles rather than assuming a complex Hilbert space or lattice-theoretic proxy, as shown by deriving Eq. (6) from relativistic invariance and Bayesian composition rule.
textbfPath Integral Simulation via Complex Amplitude Summation: Allows AI systems to compute transition probabilities for a particle traversing multiple paths simultaneously by using the derived solution, the squared modulus of a sum of complex exponentials of the relativistic action, thus recovering the Feynman path-integral formulation.
textbfMulti-Slit Pattern Prediction and Verification: Enables the prediction and verification of quantum interference patterns in experiments by showing that the detection intensity for n alternatives is completely determined by the one-path intensities and the twopath contributions, a direct consequence of pairwise Kolmogorov additivity.
textbfModel Selection via Axiomatic Consistency: Allows AI to identify physical models consistent with relativistic invariance and minimal probabilistic assumptions, specifically favoring solutions where the structure emerges from minimal probabilistic assumptions together with relativistic invariance, rather than being constrained by prior formalism like the standard quantum-mechanical rule of summing complex amplitudes over paths.