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arXiv:hep-th/9305044

Statistics in the Propositional Formulation of Quantum Mechanics

Grigore, Dan Radu

Published May 7, 1993• 2 citations
arXivHTMLPDF

Abstract

We give a definition for the notion of statistics in the lattice-theoretical (or propositional) formulation of quantum mechanics of Birchoff, von Neumann and Piron. We show that this formalism is compatible only with two types of statistics: Bose-Einstein and Fermi-Dirac. Some comments are made about the connection between this result and the existence of exotic statistics (para-statistics, infinite statistics, braid statistics).

DOI
10.48550/arxiv.hep-th/9305044

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We give a definition for the notion of statistics in the lattice-theoretical (or propositional) formulation of quantum mechanics of Birchoff, von Neumann and Piron. We show that this formalism is compatible only with two types of statistics: Bose-Einstein and Fermi-Dirac. Some comments are made about the connection between this result and the existence of exotic statistics (para-statistics, infinite statistics, braid statistics).
Although usually only two kinds of statistics, namely Bose-Einstein and Fermi-Dirac statistics, are considered in Quantum Mechanics and in Quantum Field Theory, other kinds of statistics, called collectively parastatistics, are conceivable. We critically review theoretical studies of parastatistics to date, point out and clarify several confusions. We first study the "proofs" so far proposed for the symmetrization postulate which excludes parastatistics, emphasizing their ad hoc nature. Then, after exploring in detail the structure of the quantum mechanical theory of paraparticles, we clarify some confusions concerning the compatibility of parastatistics with the so-called cluster property, which has been an issue of controversy for several years. We show, following a suggestion of Greenberg, that the quantum mechanical theory of paraparticles can be formulated in terms of density matrix compatibly with the cluster property. We also discuss such topics as selection rules for systems with variable numbers of paraparticles, the connection between statistics and permutation characters, and the classification of paraparticles. For the quantum field theory of paraparticles, we study discrete representations of the para-commutation relations and illustrate in detail Greenberg and Messiah's theorem concerning Green's ansatzes. Also, fundamental topics such as the spin-statistic theorem, the TCP theorem and the observability of parafields are discussed on the basis of Green's ansatzes. Finally, we point out that the so-called particle permutation operators do not always define multi-dimensional representations of the permutation group both in first and second quantization theories. This questions the validity of the correspondence between the two theories which has recently been proposed.
Summary Interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially, in step with dramatic new developments in experimental techniques for studying small quantum systems. Furthermore, developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory. The present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context.
Main Claim I argue in this chapter that Einstein and von Neumann meet in algebraic relativistic quantum field theory in the following metaphorical sense: algebraic quantum field theory was created in the late 1950s/early 1960s and was based on the theory of “rings of operators,” which von Neumann established in 1935–1940 (partly in collaboration with J. Murray). In the years 1936–1949, Einstein criticized standard, nonrelativistic quantum mechanics, arguing that it does not satisfy certain criteria that he regarded as necessary for any theory to be compatible with a field theoretical paradigm. I claim that algebraic quantum field theory (AQFT) does satisfy those criteria and hence that AQFT can be viewed as a theory in which the mathematical machinery created by von Neumann made it possible to express in a mathematically explicit manner the physical intuition about field theory formulated by Einstein. The argument in favor of this claim has two components: 1. Historical : An interpretation of Einstein's (semi)formal wordings of his critique of nonrelativistic quantum mechanics. This interpretation results in mathematically explicit operational independence definitions, which, I claim, express independence properties of systems that are localized in causally disjoint spacetime regions. Einstein regarded these as necessary for a theory to comply with field theoretical principles. 2. Systematic : The presentation of several propositions formulated in terms of AQFT that state that the operational independence conditions in question do in fact typically hold in AQFT.
To my friend and mentor, Professor Attilio Rigamonti. Unperformed experiments have no results. Asher Peres Introduction More than a century after its birth, quantum mechanics (QM) remains mysterious. We still don't have general principles from which to derive its remarkable mathematical framework, as happened for the amazing Lorentz transformations, which were rederived by Einstein from the invariance of physical laws in inertial frames and from the constancy of the speed of light. Despite the utmost relevance of the problem of deriving QM from operational principles, research efforts in this direction have been sporadic. The deepest of the early attacks on the problem were the works of Birkhoff, von Neumann, Jordan, and Wigner, attempting to derive QM from a set of axioms with as much physical significance as possible. The general idea in Ref. is to regard QM as a new kind of prepositional calculus , a proposal that spawned the research line of quantum logic , which is based on von Neumann's observation that the two-valued observables – represented in his formulation of QM by orthogonal projection operators – constitute a kind of “logic” of experimental propositions. After a hiatus of two decades of neglect, interest in quantum logic was revived by Varadarajan, and most notably by Mackey, who axiomatized QM within an operational framework, with the single exception of an admittedly ad hoc postulate, which represents the propositional calculus mathematically in the form of an orthomodular lattice. The most significant extension of Mackey's work is the general representation theorem of Piron.