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Journal of Mathematical Physics

Operator Algebras and Axioms of Measurements

Etang Chen

Published Nov 1, 1971• 5 citations
Publisher

Abstract

The axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras. It is shown that a concrete C*-algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra; although a von Neumann algebra satisfies the axiom of decompossability of ensembles, however, the axiom of components of the mixtures of two ensembles is true only if a von Neumann algebra is a factor of type In (n < + ∞). It is also verified that the set of decision effects, which is proved to be a subset of projections of a von Neumann algebra, has similar lattice structure of quantum mechanics, and its connection with quantum logic in the sense of Varadarjan is also figured out.

DOI
10.1063/1.1665544

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The axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras. It is shown that a concrete C*-algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra; although a von Neumann algebra satisfies the axiom of decompossability of ensembles, however, the axiom of components of the mixtures of two ensembles is true only if a von Neumann algebra is a factor of type In (n < + ∞). It is also verified that the set of decision effects, which is proved to be a subset of projections of a von Neumann algebra, has similar lattice structure of quantum mechanics, and its connection with quantum logic in the sense of Varadarjan is also figured out.
We study the sequential product the operation p∗q=pqp on the set of effects, [0, 1]𝒜, of a von Neumann algebra 𝒜 that represents sequential measurement of first p and then q. In their work [J. Math. Phys. 49(5), 052106 (2008)], Gudder and Latrémolière give a list of axioms based on physical grounds that completely determines the sequential product on a von Neumann algebra of type I, that is, a von Neumann algebra ℬ(ℋ) of all bounded operators on some Hilbert space ℋ. In this paper we give a list of axioms that completely determines the sequential product on all von Neumann algebras simultaneously (Theorem 4).
Chapter 6 Von Neumann algebras 6.1 The algebras l (X) In this chapter we discuss quantum measure theory. This is formally rather similar to quantum topology; in both cases we treat the given structure (topological or measure theoretic) in terms of the algebra of complex-valued functions which are compatible with that structure. The algebras which arise in the latter case are actually a special case of the former: every von Neumann algebra is a C*-algebra (although this is not always the best way to think about them).
Following Birkhoff and von Neumann, quantum logic has traditionally been based\non the lattice of closed linear subspaces of some Hilbert space, or, more generally,\non the lattice of projections in a von Neumann algebra A. Unfortunately, the logical\ninterpretation of these lattices is impaired by their nondistributivity and by various\nother problems. We show that a possible resolution of these difficulties, suggested by\nthe ideas of Bohr, emerges if instead of single projections one considers elementary\npropositions to be families of projections indexed by a partially ordered set C(A) of\nappropriate commutative subalgebras of A. In fact, to achieve both maximal generality\nand ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra\nand that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such\nfamilies of projections form a Heyting algebra in a natural way, so that the associated\npropositional logic is intuitionistic: distributivity is recovered at the expense of the\nlaw of the excluded middle.\nSubsequently, generalizing an earlier computation for n×n matrices, we prove that\nthe Heyting algebra thus associated to A arises as a basis for the internal Gelfand\nspectrum (in the sense of Banaschewski–Mulvey) of the "Bohrification" A of A, which\nis a commutative Rickart C*-algebra in the topos of functors from C(A) to the category\nof sets. We explain the relationship of this construction to partial Boolean algebras\nand Bruns–Lakser completions. Finally, we establish a connection between probability\nmeasure on the lattice of projections on a Hilbert space H and probability valuations\non the internal Gelfand spectrum of A for A = B(H).
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.
We apply the algebraic theory of infinite classical lattices from Part I to write an axiomatic theory of measurements, based on Mackey's axioms for quantum mechanics. The axioms give a complete theory of measurements in the sense of Haag and Kastler, taking the traditional form of a logic of propositions provided with a classical spectral theorem. The results are expressed in terms of probability distributions of individual measurements. As applications, we give a separation theorem for states by the set of observables and discuss its relationship to the equivalence of ensembles in the thermodynamic-limit program. We also introduce a weak equivalence of states based on the theory.