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Mathematical foundations of measurement theory = 測定理論の数学的基礎

史郎石川

Published Jan 1, 2006• 0 citations

Abstract

ＴｈｅｐｈｉｌｏｓｏｐｈｙｏｆｍｅａｓｕｒｅｍｅｎｔｔｈｅｏｒｙＭｅａｓｕｒｅｍｅｎｔｓ（Ａｘｉｏｍ１）Ｔｈｅｒｅｌａｔｉｏｎａｍｏｎｇｓｙｓｔｅｍｓ（Ａｘｉｏｍ２）Ｂｏｌｔｚｍａｎｎ’ｓｅｑｕｉｌｉｂｒｉｕｍｓｔａｔｉｓｔｉｃａｌｍｅｃｈａｎｉｃｓＦｉｓｈｅｒ’ｓｓｔａｔｉｓｔｉｃｓＩ（ｕｎｄｅｒＡｘｉｏｍ１）Ｆｉｓｈｅｒ’ｓｓｔａｔｉｓｔｉｃｓＩＩ（ｒｅｌａｔｅｄｔｏＡｘｉｏｍｓ１ａｎｄ２）ＰｒａｃｔｉｃａｌｌｏｇｉｃＳｔａｔｉｓｔｉｃａｌｍｅａｓｕｒｅｍｅｎｔｓｉｎＣ＊‐ａｌｇｅｂｒａｉｃｆｏｒｍｕｌａｔｉｏｎＳｔａｔｉｓｔｉｃａｌｍｅａｓｕｒｅｍｅｎｔｓｉｎＷ＊‐ａｌｇｅｂｒａｉｃｆｏｒｍｕｌａｔｉｏｎＮｅｗｔｏｎｉａｎｍｅｃｈａｎｉｃｓｉｎｍｅａｓｕｒｅｍｅｎｔＴｈｅｏｒｙＭｅａｓｕｒｅｍｅｎｔｅｒｒｏｒＨｅｉｓｅｎｂｅｒｇ’ｓｕｎｃｅｒｔａｉｎｔｙｒｅｌａｔｉｏｎ

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ＴｈｅｐｈｉｌｏｓｏｐｈｙｏｆｍｅａｓｕｒｅｍｅｎｔｔｈｅｏｒｙＭｅａｓｕｒｅｍｅｎｔｓ（Ａｘｉｏｍ１）Ｔｈｅｒｅｌａｔｉｏｎａｍｏｎｇｓｙｓｔｅｍｓ（Ａｘｉｏｍ２）Ｂｏｌｔｚｍａｎｎ’ｓｅｑｕｉｌｉｂｒｉｕｍｓｔａｔｉｓｔｉｃａｌｍｅｃｈａｎｉｃｓＦｉｓｈｅｒ’ｓｓｔａｔｉｓｔｉｃｓＩ（ｕｎｄｅｒＡｘｉｏｍ１）Ｆｉｓｈｅｒ’ｓｓｔａｔｉｓｔｉｃｓＩＩ（ｒｅｌａｔｅｄｔｏＡｘｉｏｍｓ１ａｎｄ２）ＰｒａｃｔｉｃａｌｌｏｇｉｃＳｔａｔｉｓｔｉｃａｌｍｅａｓｕｒｅｍｅｎｔｓｉｎＣ＊‐ａｌｇｅｂｒａｉｃｆｏｒｍｕｌａｔｉｏｎＳｔａｔｉｓｔｉｃａｌｍｅａｓｕｒｅｍｅｎｔｓｉｎＷ＊‐ａｌｇｅｂｒａｉｃｆｏｒｍｕｌａｔｉｏｎＮｅｗｔｏｎｉａｎｍｅｃｈａｎｉｃｓｉｎｍｅａｓｕｒｅｍｅｎｔＴｈｅｏｒｙＭｅａｓｕｒｅｍｅｎｔｅｒｒｏｒＨｅｉｓｅｎｂｅｒｇ’ｓｕｎｃｅｒｔａｉｎｔｙｒｅｌａｔｉｏｎ

I have proposed OST (Observable System Theory) as a foundational theory of system theory.3 This theory adopts two principles, namely, the Born principle and the Causal principle, and furthermore explicitly uses the one-shot measurement principle as a rule of use. OST is one of the most fundamental scientific theories, alongside mathematics and physics, and it can provide a unified framework for discussing quantum mechanics (information), statistics (system theory), practical logic, epistemology (philosophy), statistical mechanics, and so on.In this paper, I clarify that Fisherian statistics, Bayesian statistics, and quantum mechanics share a common mathematical foundation, namely OST. In statistics, an OST-based understanding of the Kalman filter is presented. Moreover, the so-called “interpretation problem” of quantum mechanics after measurement is unnecessary, and the “Wigner’s friend paradox” is resolved within OST. In addition, the FR paradox (= Frauchiger–Renner paradox) is understood as a paradox arising from the violation of the one-shot measurement principle, and thus it suggests the validity of OST. Through these discussions, I claim the following two points:(i) OST-based statistics will, in the future, become the standard understanding of statistics.(ii) OST-based quantum mechanics is the final form of von Neumann-style quantum mechanics

This work develops a conceptual framework for the foundations of quantum physics, linking two main approaches: the algebraic formulation and quantum probability. Rather than proposing new axioms or theories, the text reorganizes and synthesizes existing models, highlighting their assumptions, conceptual structures, and operational significance. The analysis begins with von Neumann's measurement theory and its subsequent developments by Mackey, emphasizing the role of experimentally feasible procedures and the need for a statistical model grounded in laboratory practice. The work adopts an operational perspective, according to which physical quantities are defined solely through experimental measurement methods, and the corresponding probabilistic measures are derived from measurement outcomes. The introduction critically examines the limitations of purely mathematical formulations - such as the algebraic method - when separated from experimental interpretation. The text argues for a clear distinction between axioms, postulates, and presuppositions, and for a reconstruction of quantum theory that respects both empirical constraints and conceptual clarity. Overall, the goal is to provide a coherent path from operational principles to algebraic structures, offering a basis for an axiomatic reformulation of quantum mechanics that remains faithful to physical practice.

In this note, we shall propose a new measurement-axiom in nonrelativistic quantum theory, which can be considered a kind of generalized Copenhagen interpretation. This interpretation can assert rather radical statements, which may irritate our minds familiar with Copenhagen interpretation. We shall, under this interpretation, assert the following two statements: (i) the uncertainties (i.e. Δq and Δp) in Heisenberg's uncertainty relation "Δq・Δp≥■/2" can be characterized as (average) errors in the approximates simultaneous measurement, and so, Heisenberg's uncertainty relation can be clarified as the statement in physics for the first time, (ii) the (discrete) trajectory of a particle is enough significant (though this, of course, includes errors). Furthermore, concerning (ii) we shall show the numerical results of the trajectory of a particle in two slit experiment. (Though almost all parts of this note are composed of summaries of references [11], [12], [13] and [14], we add some discussions from other angles.)