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arXiv:2606.02504

Closed-loop Structure of Quantum Probabilities from Unitarity

M. J. Rave

quant-phSubmitted Jun 1, 2026• Announced Jun 2, 2026• 0 citations
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Abstract

In previous work (Rave, 2008) it was proposed that closed loops should be treated as fundamental quantum entities, and such loops were presented in a quasi-probability framework. We demonstrate that the closed-loop decomposition of quantum probabilities is a direct consequence of unitarity, and that Bargmann invariants arise naturally as the phase-invariant quantities associated with these loops, rather than being introduced independently. This identifies interference not as mysterious cross terms, but as contributions from distinct classes of closed loops weighted by their associated Bargmann phases. Additionally, the Born rule is seen to reflect the fundamental quadratic structure arising from the product of forward and reverse amplitudes, which together define such loops.

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In previous work (Rave, 2008) it was proposed that closed loops should be treated as fundamental quantum entities, and such loops were presented in a quasi-probability framework. We demonstrate that the closed-loop decomposition of quantum probabilities is a direct consequence of unitarity, and that Bargmann invariants arise naturally as the phase-invariant quantities associated with these loops, rather than being introduced independently. This identifies interference not as mysterious cross terms, but as contributions from distinct classes of closed loops weighted by their associated Bargmann phases. Additionally, the Born rule is seen to reflect the fundamental quadratic structure arising from the product of forward and reverse amplitudes, which together define such loops.
Decoherence theory has played a central role in explaining the suppression of interference and the emergence of classicality in quantum systems. However, it remains unclear whether decoherence alone can universally account for the origin of quantum probabilities themselves. In the present work, we examine the consequences of interpreting decoherence not merely as an interference-suppression mechanism, but as the fundamental generator of probabilistic measurement outcomes. If quantum probabilities are produced through system–apparatus–environment interactions, the resulting probabilities would generally be expected to depend on the detailed structure of those interactions. Such dependence would imply the possibility of interaction-dependent corrections to the Born-rule probabilities. However, the experimentally observed universality and robustness of the Born rule strongly constrain such a possibility. We therefore argue that decoherence successfully explains the effective suppression of interference terms, but does not by itself universally determine quantum probabilities without additional principles ensuring probability invariance. Based on this distinction, we propose that the origin of quantum probabilities should be conceptually separated from decoherence-induced classicality. Within the Intrinsic Observability Framework (IOF), this separation arises naturally through structural limitations on observability rather than through environment-generated probability formation. The present analysis suggests a conceptual reorganization of the role of decoherence in the quantum measurement problem.
I present a derivation of quantum probabilities (Born rule) from a minimal set of spacetime composition axioms. By treating boundary states on spacetime slices as fundamental and imposing linearity under sequential composition, tensor factorization for spacelike-separated regions, and unitary invariance, I show that the quadratic probability measure arises uniquely. My derivation interprets quantum amplitudes as intrinsic features of spacetime: probabilities naturally emerge from how amplitudes on boundaries glue together, without introducing additional operators or postulates. This submission establishes priority for a specific, elegant argument linking the Born rule to the symmetries of spacelike factorization and foliation invariance, providing a first-principles foundation for quantum probabilities from spacetime considerations.
This paper examines the geometric structure underlying amplitudes, interference, and probability in quantum mechanics. Building on the interpretation of quantum phase as holonomy, amplitudes are analyzed in terms of parallel transport in complex line bundles. Interference is interpreted as arising from the composition of alternative transport paths, and probability emerges from the invariant comparison of amplitudes under phase transformations. Under general assumptions of linearity and phase invariance, the squared-norm structure associated with probability is shown to arise naturally from consistency conditions on amplitude composition. The analysis focuses on kinematic structure and does not assume specific dynamical models.
We show that the quadratic measure associated with the Born rule need not be introduced as an independent probabilistic postulate, but follows, under the stated structural and regularity assumptions, from the compatibility between reversible linear evolution and the formation of stable records. Physical processes alternate between reversible dynamics and episodes of record formation. Quantities that remain well defined throughout such processes must therefore admit a representation that remains consistent across both regimes. Prior to record formation, alternatives combine additively at the level of complex amplitudes, whereas successive record-forming distinctions admit a multiplicative representation through refinement. Requiring compatibility between these compositional structures constrains the admissible weight assignment to the family of weights proportional to the pth power of the amplitude magnitude. The finite-dimensional specialization of Lamperti's isometry theorem then singles out p = 2 as the unique value compatible with continuous reversible linear mixing while preserving total weight. The Born rule therefore emerges as the unique weight assignment compatible with reversible linear composition, multiplicative record refinement, phase invariance, and continuous reversible invariance, without invoking probabilistic assumptions or a specific interpretation of quantum measurement.
A pedagogical introduction is given to the quantum mechanics of closed systems, most generally the universe as a whole. Quantum mechanics aims at predicting the probabilities of alternative coarse-grained time histories of a closed system. Not every set of alternative coarse-grained histories that can be described may be consistently assigned probabilities because of quantum mechanical interference between individual histories of the set. In “Copenhagen” quantum mechanics, probabilities can be assigned to histories of a subsystem that have been “measured”. In the quantum mechanics of closed systems, containing both observer and observed, probabilities are assigned to those sets of alternative histories for which there is negligible interference between individual histories as a consequence of the system's initial condition and dynamics. Such sets of histories are said to decohere. We define decoherence for closed systems in the simplified case when quantum gravity can be neglected and the initial state is pure. Typical mechanisms of decoherence that are widespread in our universe are illustrated. Copenhagen quantum mechanics is an approximation to the more general quantum framework of closed subsystems. It is appropriate when there is an approximately isolated subsystem that is a participant in a measurement situation in which (among other things) the decoherence of alternative registrations of the apparatus can be idealized as exact. Since the quantum mechanics of closed systems does not posit the existence of the quasiclassical domain of everyday experience, the domain of the approximate aplicability of classical physics must be explained.