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Journal of Mathematical Physics

Axiomatic Foundations of Quantum Theories

M. Guenin

Published Feb 1, 1966• 24 citations
Publisher

Abstract

A formalism is developed on an axiomatic basis and is shown to contain as special cases classical mechanics, the usual quantum mechanics, as well as the quantum theory of systems with continuous superselection rules. The structure of the symmetries of a general physical theory is studied and a classification of observables is exhibited.
DOI
10.1063/1.1704929

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Motivated by Hilbert's sixth problem on axiomatization of physics, the author is proposing a rather provocative abstract axiomatic framework called S - formalization, where S is an arbitrary physical system .The proposal is an attempt to provide a general axiomatic framework , from which mathematical frameworks of new physical theories can be formulated on firm axiomatic basis and the mathematical frameworks of subjects such as standard (nonrelativistic) quantum mechanics are naturally derived as special cases of this general axiomatic framework. Introduction
We propose a geometric setting of the axiomatic mathematical formalism of quantum theory. Guided by the idea that understanding the mathematical structures of these axioms is of similar importance as was historically the process of understanding the axioms of geometry, we complete the spaces of observables and of states in a similar way as in classical geometry linear or affine spaces are completed by projective spaces. In this sense, our theory can be considered as a "completion of usual linear quantum theory" , such that the usual theory appears as the special case where a reference frame is fixed once and for all. In the present first part, this general setting is explained. Dynamics (time evolution) will be discussed in subsequent work.
An axiomatic mathematical structure is presented in terms of which arbitrary mathematical models of a physical system can be rigorously formulated and studied. Physical systems, observables, and states all arise in a natural way from the primitive notion of a physical operation. In particular, the structure is independent of the special mathematical completion defined by the quantum mechanical model, since no lattice theoretical or Hilbert space assumptions are used. The precise relationship between the present structure and the usual quantum mechanical model is investigated in a succeeding paper.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
An axiomatic model for quantum mechanics is formulated using physically significant axioms. The model contains a slight strengthening of Mackey's first six axioms, together with two axioms which ensure the existence of coordinate and momentum observables. The symmetries or rigid motions are an essential part of the structure, and a link is constructed between these and the quantum proposition system. Coordinate and momentum observables are defined in terms of abstract coordinate systems and one-parameter groups of motions. It is then shown that as far as the statistical properties of these observables in certain canonical states are concerned, the abstract model may be represented by the usual Hilbert space formulation. Spectral properties of σ-homomorphisms are also investigated. Also included are two appendices of a technical nature: the first considers one-parameter groups of derivables, and the second absolutely continuous σ-homomorphisms.