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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

A statistical theory of measurements in unitary quantum mechanics

Leo Georgy Sapogin

Published Jul 1, 1982• 22 citations
Publisher
DOI
10.1007/bf02814013

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Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory.A total parameter space Φ, equipped with a group G of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter φ can be estimated. Choose an experiment/question a, and get from this a parameter space Λa, perhaps after some model reduction compatible with the group structure. As in statistics, it is important always to distinguish between observations and parameters, in particular between (minimal) observations ta and state variables (parameters) λa. Let Ka be the L2 ‐space of functions of ta, and let Ha be the image of Ka under the expectation operator of the model. The measure determining the L2‐spaces is the invariant measure under the maximal subgroup which induces a transformation on Λa.There is a unitary connection between Ka and Ha, and then under natural conditions between Ha and Hb for a and b are different. Thus there exists a common Hilbert space H such that Ha equals UaH for a unitary Ua. In agreement with the common formulation of quantum mechanics, the vectors of H are taken to represent the states of the system. The state interpretation is then: An ideal experiment Ea has been performed, and the result of this experiment, disregarding measurement errors, is that the parameter λa is equal to some fixed number λka. This essentially statistical construction leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born’s formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions.The theory is illustrated by a simple macroscopic example, and by the example of a spin 12 particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry. The result of this last example seems to have potential for being generalized considerably.

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The Measurement Problem as a Category Error: A Mathematical Proof of Wavefunction Unobservability

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We prove rigorously that the quantum mechanical wavefunction ψ is mathematically unobservable using information theory, set theory, and Hilbert space analysis. The proof establishes quantitative bounds: reconstruction error scales as ∥ψ − ψ̃_N∥² ≥ C/N² due to the uncertainty principle, making exact recovery impossible with finite measurements. This demonstrates that quantum mechanics is a statistical theory, firmly rooted in Unitary Mathematics — a framework previously instrumental in eliminating dark matter in galactic dynamics. The measurement problem dissolves when wavefunction realism (a category error) is replaced with statistical interpretation. We present three quantified falsifiable predictions, preemptively respond to objections (weak measurements, Bell inequalities, quantum computing), prove that wavefunction tomography cannot violate our bounds, and provide a constructive framework for doing quantum mechanics without wavefunctions. The parallel to dark matter elimination is exact: both are "Ghost Space" artifacts arising from mistaking mathematical tools for physical entities.

•Apr 17, 2026•0 citations
This chapter begins the final section of the book, which presents both review and new results of original research on decoherence and measurement theory. In this chapter, it is shown that normal quantum mechanics can lead to irreversible behavior in an open system, in contrast to the expectation of the Poincaré theorem that predicts repeating, cyclical behavior for all closed systems. The quantum Boltzmann equation, which implies the famous H-theorem that underlies all statistical mechanics, is derived.
Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory.A total parameter space Φ, equipped with a group G of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter φ can be estimated. Choose an experiment/question a, and get from this a parameter space Λa, perhaps after some model reduction compatible with the group structure. As in statistics, it is important always to distinguish between observations and parameters, in particular between (minimal) observations ta and state variables (parameters) λa. Let Ka be the L2 ‐space of functions of ta, and let Ha be the image of Ka under the expectation operator of the model. The measure determining the L2‐spaces is the invariant measure under the maximal subgroup which induces a transformation on Λa.There is a unitary connection between Ka and Ha, and then under natural conditions between Ha and Hb for a and b are different. Thus there exists a common Hilbert space H such that Ha equals UaH for a unitary Ua. In agreement with the common formulation of quantum mechanics, the vectors of H are taken to represent the states of the system. The state interpretation is then: An ideal experiment Ea has been performed, and the result of this experiment, disregarding measurement errors, is that the parameter λa is equal to some fixed number λka. This essentially statistical construction leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born’s formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions.The theory is illustrated by a simple macroscopic example, and by the example of a spin 12 particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry. The result of this last example seems to have potential for being generalized considerably.
We prove rigorously that the quantum mechanical wavefunction ψ is mathematically unobservable using information theory, set theory, and Hilbert space analysis. The proof establishes quantitative bounds: reconstruction error scales as ∥ψ − ψ̃_N∥² ≥ C/N² due to the uncertainty principle, making exact recovery impossible with finite measurements. This demonstrates that quantum mechanics is a statistical theory, firmly rooted in Unitary Mathematics — a framework previously instrumental in eliminating dark matter in galactic dynamics. The measurement problem dissolves when wavefunction realism (a category error) is replaced with statistical interpretation. We present three quantified falsifiable predictions, preemptively respond to objections (weak measurements, Bell inequalities, quantum computing), prove that wavefunction tomography cannot violate our bounds, and provide a constructive framework for doing quantum mechanics without wavefunctions. The parallel to dark matter elimination is exact: both are "Ghost Space" artifacts arising from mistaking mathematical tools for physical entities.