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arxiv: 2502.10523 · v21 · pith:WDUFRXLUnew · submitted 2025-02-14 · 🪐 quant-ph

Quantum Mechanics as a Reversible Diffusion Theory

Charalampos Antonakos This is my paper

Pith reviewed 2026-05-23 03:03 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum mechanicsstochastic diffusionBorn rulewave function interpretationreversible processescomplex probabilitieshidden variablestime symmetry
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The pith

The wave function and its conjugate act as complex probability distributions in forward and backward non-real stochastic diffusions whose set-theoretic intersection produces Born rule probabilities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper advances an interpretation in which quantum mechanics emerges from time-symmetric stochastic dynamics combined with non-classical probability. The wave function is treated as a complex probability distribution governing one diffusion process, while its conjugate governs the time-reversed counterpart. Only the reversible trajectories survive at the intersection of these two processes. When that intersection is expressed in set-theoretic terms, the resulting probabilities are exactly those given by the Born rule. This framework is offered as a way to derive the superposition principle from probability considerations alone and to account for the appearance of classical behavior at large scales.

Core claim

Quantum mechanics is the reversible diffusion that results when the forward and backward non-real stochastic processes described by the wave function and its conjugate intersect; translating that intersection via set theory directly yields the probabilities prescribed by the Born rule.

What carries the argument

The set-theoretic intersection of forward and backward non-real stochastic diffusion processes.

If this is right

  • The role of complex numbers in quantum mechanics follows from the need for separate forward and backward complex diffusion equations.
  • The superposition principle can be derived from probability theory rather than from the linearity of the Schrödinger equation.
  • Physical superposition is not required; only the probabilistic intersection of trajectories is needed.
  • Classical behavior emerges in large objects because stochastic processes make reversible trajectories statistically negligible at macroscopic scales.
  • A probabilistic, non-ontic view of the wave function combined with stochastic hidden variables supplies a consistent picture of quantum reality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the intersection mechanism holds, numerical simulation of quantum systems could be recast as sampling from paired forward and backward diffusion trajectories.
  • The approach suggests that any stochastic hidden-variable model must incorporate both time directions to recover the Born rule without additional postulates.
  • Extensions to relativistic or field-theoretic settings would require defining analogous forward and backward diffusions on spacetime.

Load-bearing premise

The intersection of the forward and backward non-real stochastic processes, when expressed in set-theoretic language, directly produces Born rule probabilities with no further assumptions required.

What would settle it

An explicit computation of the set-theoretic intersection for the ground state of the harmonic oscillator that fails to recover the Born-rule probability density.

read the original abstract

This paper proposes an interpretation of quantum mechanics, relying on the time-symmetric stochastic dynamics of quantum particles and on non-classical probability theory. Our main purpose is to demonstrate that the wave function and its complex conjugate can be interpreted as complex probability distributions in two complex diffusion equations related to non-real forward and backward in time stochastic motions respectively. We say non-real because Schroedinger forward and backward diffusions describe both reversible (real trajectories) and irreversible trajectories (non-real trajectories). The reversible trajectories are the only real trajectories and are given by the intersection of those forward and backward processes. It turns out that if we translate this intersection using set-theoretic language, we are led to a reversible diffusion described by Born rule probabilities. This proposal is useful also for explaining more about the role of complex numbers in quantum mechanics that produces this so-called "wave-like" nature of quantum reality. Our perspective also challenges the notion of physical superposition and aims at a derivation of superposition principle not based on the linearity of Schroedinger's equation but relying on pure probability theory. Moreover, it is suggested that, embracing the idea of stochastic processes in quantum theory, explains the reasons for the appearance of classical behavior in large objects, in contrast to the quantum behavior of small ones. In other words, we claim that a combination of a probabilistic and no-ontic view (neither epistemic though) of the wave function with a stochastic hidden-variables approach, may provide some insight into the quantum physical reality and potentially establish the groundwork for a novel interpretation of quantum mechanics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 0 minor

Summary. The manuscript proposes an interpretation of quantum mechanics as a reversible diffusion process arising from the intersection of two complex diffusion equations, one associated with the wave function ψ (forward in time) and one with its conjugate ψ* (backward in time). The reversible (real) trajectories are identified with this intersection, which, when translated via set-theoretic language, is claimed to yield Born-rule probabilities |ψ(x)|^2 without relying on the linearity of the Schrödinger equation. The work also aims to explain the role of complex numbers, derive superposition from probability theory, and account for the quantum-to-classical transition via stochastic hidden variables.

Significance. If the asserted derivation of Born-rule probabilities from the intersection of non-real forward/backward diffusions were explicitly demonstrated and shown to be independent of standard quantum postulates, the paper would offer a novel stochastic interpretation that addresses the origin of probabilities and the emergence of classicality. As written, however, the central claims remain unsupported by any derivations or calculations.

major comments (3)
  1. [Abstract] Abstract: The statement that 'translating this intersection using set-theoretic language' leads to a reversible diffusion described by Born rule probabilities is presented as a direct consequence, yet no equations, definitions of the complex probability distributions, or explicit combination rules are supplied showing how the forward and backward complex densities produce the real measure |ψ(x)|^2.
  2. [Abstract] Abstract: The claim that superposition can be derived from 'pure probability theory' rather than the linearity of the Schrödinger equation is asserted without any construction, axioms, or steps that would replace the standard linear superposition with a set-theoretic or stochastic operation on the forward/backward processes.
  3. [Abstract] Abstract: The forward and backward diffusions are said to be 'related to' the Schrödinger equation, but the text provides no demonstration that the resulting intersection probability is independent of the very linearity the paper seeks to circumvent; any such derivation must exhibit this independence explicitly.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and for identifying areas where the presentation of our derivations could be strengthened. We respond to each major comment below, clarifying the content of the full manuscript while agreeing that the abstract would benefit from additional explicit steps. Revisions will be made accordingly.

read point-by-point responses

  1. Referee: The statement that 'translating this intersection using set-theoretic language' leads to a reversible diffusion described by Born rule probabilities is presented as a direct consequence, yet no equations, definitions of the complex probability distributions, or explicit combination rules are supplied showing how the forward and backward complex densities produce the real measure |ψ(x)|^2.

    Authors: Sections 2 and 3 of the manuscript define the complex densities via the forward and backward stochastic differential equations and formalize their intersection as the set of trajectories sharing non-zero real measure. The combination rule is the normalized product of these real parts, directly producing |ψ(x)|^2. We will revise the abstract to include a concise outline of these definitions and the set operation. revision: yes

  2. Referee: The claim that superposition can be derived from 'pure probability theory' rather than the linearity of the Schrödinger equation is asserted without any construction, axioms, or steps that would replace the standard linear superposition with a set-theoretic or stochastic operation on the forward/backward processes.

    Authors: Section 4 derives superposition as the set-theoretic union of supports from independent forward and backward processes under extended probability axioms for complex measures. This construction does not invoke Schrödinger linearity. We will add an explicit axiomatic list and step-by-step construction in a new subsection to make the replacement operation fully transparent. revision: yes

  3. Referee: The forward and backward diffusions are said to be 'related to' the Schrödinger equation, but the text provides no demonstration that the resulting intersection probability is independent of the very linearity the paper seeks to circumvent; any such derivation must exhibit this independence explicitly.

    Authors: The diffusions originate from time-symmetric probability conservation and stochastic dynamics alone. The intersection measure is computed directly from the real parts of these processes. We will insert a dedicated subsection proving independence by deriving the probability using only set operations on the stochastic measures, without reference to the Schrödinger equation's linearity. revision: yes

Circularity Check

1 steps flagged

Set-theoretic intersection of forward/backward diffusions asserted to yield Born rule without explicit equations or independence from Schrödinger linearity

specific steps
  1. renaming known result [Abstract]
    "The reversible trajectories are the only real trajectories and are given by the intersection of those forward and backward processes. It turns out that if we translate this intersection using set-theoretic language, we are led to a reversible diffusion described by Born rule probabilities."

    Forward/backward diffusions are constructed from Schrödinger; the 'translation' is asserted to produce |ψ|^2 probabilities with no shown combination rule, normalization, or independence from the linearity the paper claims to replace. The result is therefore the input QM re-labeled as an output of set theory.

full rationale

The paper defines forward and backward complex diffusions from the Schrödinger equation, then claims their intersection (translated set-theoretically) produces Born-rule probabilities on real trajectories. No equations are exhibited showing how the complex densities combine into |ψ(x)|^2 independently of the input QM structure or linearity. This reduces the central claim to re-expressing the known Born rule in new language rather than deriving it from the stochastic processes alone.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The proposal rests on several domain assumptions and invented concepts introduced without independent evidence or derivation; no free parameters are explicitly fitted but the framework is constructed around matching known QM outcomes.

axioms (2)
  • domain assumption Quantum particles obey time-symmetric stochastic dynamics with non-real forward and backward motions
    Stated in the abstract as the foundational dynamics for the interpretation.
  • ad hoc to paper The intersection of forward and backward processes can be translated via set theory into Born rule probabilities
    Invoked as the step that connects the diffusion picture to standard QM predictions.
invented entities (2)
  • non-real trajectories no independent evidence
    purpose: To describe the irreversible components of the forward and backward diffusions while isolating real reversible trajectories
    Introduced in the abstract to distinguish observable paths from the complex diffusion processes.
  • complex probability distributions for the wave function no independent evidence
    purpose: To reinterpret the wave function as probabilities in the diffusion equations
    Proposed as the core interpretive move linking QM to stochastic theory.

pith-pipeline@v0.9.0 · 5795 in / 1730 out tokens · 44948 ms · 2026-05-23T03:03:59.304772+00:00 · methodology

discussion (0)

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Reference graph

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