Born’s Rule Arises From Reversible Change Physics

Oskar Axelsson and colleagues demonstrate that Born’s rule, a key principle of quantum mechanics describing the probability of measurement outcomes, arises from fundamental principles without requiring explicit probabilistic assumptions. The rule emerges naturally from combining linear reversible evolution, describing how systems change before measurement, with the irreversible formation of persistent records when a measurement occurs. This combination establishes Born’s rule as the unique measure consistent with these two structural features of physical processes, linking reversible and irreversible dynamics without relying on a pre-defined quantum formalism.

Operational regimes of reversible dynamics and irreversible record formation

Researcher Oskar Axelsson, have long sought to derive the standard formulation of quantum mechanics from more primitive physical principles. Several approaches have been proposed, including decision-theoretic arguments within the Everett interpretation and derivations based on envariance and symmetry properties of entangled states. While illuminating, these arguments often rely on assumptions about probability, decision theory, or the Hilbert-space formalism.

This work pursues a different route, considering physical processes exhibiting two operational regimes: reversible evolution prior to persistent record formation and irreversible outcome selection upon record formation. Reversible evolution combines alternatives additively at the amplitude level, while irreversible record formation composes multiplicatively through sequential refinement of outcomes. This compatibility uniquely selects a quadratic assignment of outcome weights, establishing the Born rule as the only weighting compatible with reversible linear evolution and multiplicative irreversible refinement.

Measurements are commonly described as consisting of two distinct regimes: reversible unitary evolution governed by the Schrödinger equation and effectively irreversible record formation associated with measurement outcomes. Prior to persistent record formation, interactions are reversible, with configurations associated with different potential outcomes remaining operationally interconvertible. This derivation avoids probabilistic assumptions, decision theory, and the Hilbert-space formalism.

Once a record forms, configurations corresponding to distinct outcomes become operationally distinguishable and are no longer reversibly transformable. Physical processes assume only the coexistence of these two regimes, alternating between periods of reversible evolution and events involving persistent record formation. Irreversibility here refers to operational irreversibility, meaning recorded configurations cannot be returned to their pre-record state by any physically accessible reversible transformations, despite the underlying microscopic dynamics remaining time symmetric.

At the microscopic level, dynamical laws governing physical systems are typically reversible. A deterministic dynamical law allows reconstruction of a configuration from its later state using the same law. The Schrödinger equation, iħ∂ψ/∂t = Hψ, generates unitary evolution, enabling the retrieval of an earlier state via the inverse unitary operator U−1(t) = U(−t). This equation contains no intrinsic direction of time. This microscopic reversibility contrasts with macroscopic experience, where measurement or record formation appears irreversible due to the creation of persistent records practically impossible to erase through local physical interactions.

Such irreversibility reflects practical limitations on accessible transformations rather than a fundamental asymmetry of the underlying dynamical laws. Noether’s theorem establishes the connection between reversibility and conservation laws, stating that time-translation symmetry implies conserved energy. These considerations motivate the operational distinction employed in this work. Interactions are governed by reversible dynamics consistent with time-translation symmetry prior to record formation.

Once a record forms, resulting configurations are no longer mutually reachable through reversible evolution. The compatibility between these two regimes imposes strong constraints on how outcome weights must be assigned. In the reversible regime, the parameter α labels configurations prior to record formation and combines additively under reversible evolution. Reversible transformations change the configuration without altering any recorded outcome, meaning outcome weights must be invariant under transformations leaving the reversible configuration operationally indistinguishable.

These transformations form a continuous symmetry of the reversible regime. The simplest nontrivial continuous representation preserving additive composition is a phase rotation α → eiθα. Configurations related by such transformations cannot be distinguished prior to record formation and must therefore be assigned the same outcome weight. As a result, outcome weights depend only on the magnitude |α|, μ = f(|α|). Once a persistent record forms, configurations become operationally distinguishable.

Subsequent evolution proceeds within the configuration consistent with the recorded outcome until a further irreversible record may form. A non-negative weight μ is associated with each realised record, representing a physically persistent configuration distinguishing alternatives. Physical processes typically consist of alternating stages of reversible evolution and irreversible record formation. Consider two successive irreversible record formation events, R1 and R2, resulting in a refined record R12 encoding both distinctions.

The weight assigned to a record depends only on its physical configuration, ensuring that identical records receive identical weights regardless of the description used to obtain them. Consistency of refinement requires that the assignment of weights respects the compositional structure of refinements, implying a consistent multiplicative structure for the mapping from reversible configurations to record weights. For successive refinements corresponding to independent stages of distinction, the combined weight satisfies μ(R12) = μ(R1) μ(R2). This relation expresses the consistency of sequential record formation as a structural property of refinement, not an assumption about probabilistic independence.

The formulation does not assume context-independent pre-existing outcome values, assigning weights only to physically realised records including their formation context. Reversible evolution combines configurations additively at the level of compatibility parameters, while irreversible record formation induces a multiplicative structure on weights associated with physical records. A consistent assignment of weights must therefore respect both structures simultaneously.

Lemma 1 establishes that if α denotes the compatibility parameter describing a configuration prior to record formation, and μ = f(α) denotes the weight assigned to the resulting physical record, then f(α1 + α2) = f(α1) f(α2). This demonstrates that compatibility between the additive structure of reversible evolution and the multiplicative structure induced by record refinement constrains the weight function. Determining the class of functions satisfying f(α1 + α2) = f(α1)f(α2) requires considering that configurations differing only by a global phase transformation are operationally indistinguishable before record formation. Therefore, weights assigned after record formation must depend only on the magnitude |α|, leading to μ = f(|α|). Successive scaling of amplitudes must respect the same multiplicative structure governing record refinement, implying f(|α1||α2|) = f(|α1|)f(|α2|). This is Cauchy’s multiplicative functional equation, with continuous non-negative solutions being power laws, μ = |α|p, for some real p > 0. Reversible evolution transforms compatibility parameters linearly, α′i = Σj Uijαj. Since reversible evolution does not alter the set of physically accessible configurations prior to record formation, the total weight assigned to records must remain invariant under such transformations.

If weights take the form μ = |α|p, then the total weight Σi |αi|p must be preserved by reversible evolution. Thus, reversible transformations must act as linear isometries of the p-norm. A result due to Lamperti shows that for p = 2, the linear isometries of Lp spaces consist only of coordinate permutations and multiplicative factors, and therefore do not form a continuous group. Continuous reversible dynamics therefore requires p = 2, meaning the only weight compatible with reversible linear evolution is μ = |α|2.

Deriving the Born rule from reversible evolution and irreversible measurement

Outcome weight assignments now demonstrate a unique solution at a power law of 2, departing from previous derivations requiring postulated quadratic measures. This threshold signifies a fundamental shift, enabling the derivation of the Born rule without reliance on probabilistic assumptions or the Hilbert-space formalism. The compatibility between reversible evolution, where configurations combine additively, and irreversible record formation, inducing multiplicative weighting, uniquely constrains the admissible measure.

Researchers have shown that a consistent framework emerges from combining reversible processes with irreversible record formation, explaining how outcome weights are assigned. Specifically, the additive combination of configurations during reversible evolution, before any measurement, contrasts with the multiplicative weighting that arises when a persistent record, a lasting outcome, is created; this distinction is key. This work clarifies that the weighting isn’t arbitrary, but uniquely constrained to a quadratic form, validating the mathematical underpinnings of the Born rule without needing to assume probabilities or complex quantum structures like Hilbert space.

Deriving probability in quantum mechanics from dynamics under separation of reversible and

Establishing the Born rule as a consequence of physical dynamics, rather than a starting assumption, offers a compelling resolution to long-standing debates about the foundations of quantum mechanics. However, this derivation presently concentrates on systems exhibiting specific structural characteristics, namely, the clear separation between reversible and irreversible processes, leaving open the question of broader applicability. The research acknowledges that alternative derivations may prove necessary when considering physical regimes where these features do not consistently hold, or where the distinction between reversible evolution and record formation becomes blurred.

This work establishes the Born rule not as a postulate, but as a logical consequence of fundamental physical principles. By demonstrating compatibility between reversible dynamics, predictable system changes, and irreversible record formation, the derivation bypasses reliance on probabilistic assumptions or complex mathematical structures like Hilbert space. This derivation clarifies that the weighting of measurement outcomes isn’t arbitrary, but uniquely determined by how systems evolve and register results; a consistent mathematical framework emerges from combining these processes.

The research demonstrates that the Born rule, which governs the weighting of outcomes in quantum mechanics, arises from the consistent combination of two physical processes: reversible evolution and irreversible record formation. This means the rule isn’t an assumption, but a natural consequence of how systems change predictably before a measurement and then register a lasting result. By linking these dynamics, scientists validated the mathematical basis of the Born rule without needing to invoke probabilities or complex structures. The authors note that further work may be needed to apply this derivation to systems where the distinction between reversible and irreversible processes is less clear.