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AIP conference proceedings

From Born rule to large deviations

Kazuya Okamura

Published Jan 1, 2012• 1 citation
Publisher

Abstract

We formulate sector theory and quantum measurement theory in algebraic quantum theory, and derive Born rule on the basis of them. Then, a quantum version of Sanov's theorem is presented in order to show the consistency between the derivation of Born rule in the paper and large deviation type estimate.
DOI
10.1063/1.4773161

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Reports in Advances of Physical Sciences•Jan 1, 2026•0 citations
Abstract A concise review of the derivation of the Born rule and Schroedinger equation from first principles is provided. The starting point is a formalization of fundamental notions of measurement and composition, leading to a general framework for physical theories known as the positive formalism. Consecutively adding notions of spacetime, locality, absolute time and causality recovers the well established convex operational framework for classical and quantum theory. Requiring the partially ordered vector space of states to be an anti-lattice one obtains quantum theory in its standard formulation. This includes Born rule and Schroedinger equation.
Traditionally, some version of Born’s rule is always considered to be an indispensable part of any interpretation of quantum mechanics, either as a postulate, or as a result derived-not always on the basis of convincing reasoning-from other postulates. In this section, we have a close look at the possible forms of Born’s rule and discuss the limits of its validity. All traditional foundations of quantum mechanics heavily depend on the concept of (hypothetical, idealized) experiments-far too heavily. This is one of the reasons why these foundations are still unsettled, over 90 years after the discovery of the basic equations for modern quantum mechanics. No other theory has such controversial foundations. The main reason is that the starting point of the usual interpretations is an idealization of the measurement process that is taken too seriously, namely as the indisputable truth about everything measured. But in reality, this idealization is only a didactical trick for the newcomer to make the formal definitions of quantum mechanics a bit easier to swallow. Except in a few very simple cases, it is too far removed from experimental practice to tell much about real measurements, and hence about how quantum physics is used in real applications. In experimental physics, measurement is a very complex thing, far more complex than Born’s rule (the usual starting point) suggests. To measure the distance between two galaxies, the mass of the top quark, or the Lamb shift, just to mention three basic examples, cannot be captured by the idealistic measurement concept used there, nor by any of the refinements of it discussed in the literature. In each of the three cases mentioned, one assembles a lot of auxiliary information and ultimately calculates the measurement result from a best fit of a model to the data. Clearly, the theory must already be in place in order to do that. We do not even know what a top quark should be, whose mass we are measuring, unless we have a theory that tells us this! The two most accurately determined observables in the history of quantum physics, namely the anomalous magnetic moment of the electron and Lamb shift, are not even q-observables! To present the stage for the criticism of Born’s rule in Section 14.3, we first need to clarify the meaning of the term “Born’s rule”. To distinguish different useful meanings, we look in Sections 14.1-14.2 at the early history of Born’s rule.
This work presents a deductive solution to the Born rule problem within the framework of the Temporal Theory of the Universe (TTU) and Lie-Algebraic Observable-Dependent Renormalization (LA-ODR). Probability is derived as the energetic response of the temporal field τ, with quadratic scaling fixed by the global attractor u* ≈ 0.945531. The proof (Theorem D.0) establishes that P(Oᵢ) = Tr(ρ P̂ᵢ) emerges from pointer-sector decomposition, Lindblad decoherence, and a Gleason-type uniqueness argument supplemented by an operational proof of affine linearity. The derivation combines (i) dynamical sector selection via the temporal drift generator X̂_time, (ii) exponential suppression of orthogonal components via infrared Lindblad attractors, and (iii) CPTP-consistent measure uniqueness. The Born rule is thus a theorem of the TTU/LA-ODR measurement sector rather than an independent postulate. A single remaining model-specific condition — primitivity of the infrared LA-ODR measurement channel — is explicitly isolated and formulated as an open problem (Appendix W). Keywords: Born rule theorem; Temporal Theory of the Universe (TTU); LA-ODR renormalization; temporal susceptibility χτ; global attractor u*; pointer sectors; Lindblad semigroup; CPTP consistency; Gleason uniqueness; disformal geometry; nucleon mass scaling; FPGA/TPLL; SQUID detection.
In this paper, we derive the Born rule in Chronon Field Theory (CFT) from first principles under realistic measurement assumptions. A recent work in CFT shows that Lorentzian signature and causal structure emerge dynamically from a unit-norm constraint on the chronon field, providing the geometric foundation for the time-asymmetric alignment dynamics studied here. First, we obtain a diffusion limit for alignment order parameters on the outcome simplex and prove that the overlaps with apparatus eigen-domains form martingales up to the absorption time; optional stopping then yields single-shot Born probabilities. Second, we derive the stochastic limit from noisy chronon gradient flow with boundary coupling by a hydrodynamic limit (tightness, identification of the generator, and boundary layer analysis). Third, we establish a large-deviation principle (LDP) for empirical frequencies in repeated measurements via Sanov’s theorem, with rate function minimized at the Born vector. We quantify robustness to finite temperature, imperfect interfaces and basis degeneracies and outline falsifiable predictions for alignment timescales and drift bounds. Conceptually, the analysis shows that measurement is not a collapse of the system into one of its own eigenstates, but stochastic absorption of the system field into pre-stabilized apparatus eigen-domains. This does not directly contradict the standard textbook view — since apparatus domains are engineered to correspond to system eigenbases — but offers a deeper dynamical interpretation of why outcomes are definite and Born-weighted.