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Journal of Mathematical Physics

Quantum axiomatics, representation theorem, and communicability of observations in physics

Adolf Miklavc

Published Aug 1, 1977• 1 citation
Publisher

Abstract

We show how a fundamental assumption in the Dirac formulation of quantum mechanics, namely that the states of a physical system at a particular time are mathematically represented by unit vectors in Hilbert space, can be deduced from certain aspects of our experimental procedures and of the observed outcome of quantum mechanical experiments. Our assumptions have clear empirical meaning and the results hold true for any dimensionality of the system, without anomalies in low dimensions which exist in the two well-known axiomatic approaches to quantum mechanics. The propositional logic approach of Birkhoff and von Neumann does not work for quantum systems of dimension less than four and requires an assumption which does not have an empirical basis. Jordan algebra axioms, on the other hand, also lead to anomalies in low dimensions and, moreover, are formal and cannot be directly physically interpreted. In our work it was possible to avoid these shortcomings.

DOI
10.1063/1.523456

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We show how a fundamental assumption in the Dirac formulation of quantum mechanics, namely that the states of a physical system at a particular time are mathematically represented by unit vectors in Hilbert space, can be deduced from certain aspects of our experimental procedures and of the observed outcome of quantum mechanical experiments. Our assumptions have clear empirical meaning and the results hold true for any dimensionality of the system, without anomalies in low dimensions which exist in the two well-known axiomatic approaches to quantum mechanics. The propositional logic approach of Birkhoff and von Neumann does not work for quantum systems of dimension less than four and requires an assumption which does not have an empirical basis. Jordan algebra axioms, on the other hand, also lead to anomalies in low dimensions and, moreover, are formal and cannot be directly physically interpreted. In our work it was possible to avoid these shortcomings.
The well-known limitations on the field of the quantum mechanical vector space, which, within the framework of the propositional logic of Birkhoff and von Neumann, can be obtained only for systems of dimension greater than or equal to four, are obtained here for all dimensions. The propositional logic fails to provide any information about the quantum systems of dimension less than four and, moreover, one does not know the empirical meaning of one of its basic assumptions. The Jordan algebra approach, on the other hand, which provides the same limitations on the field, also suffers from anomalies in low dimensions and is altogether formal rather than physical. In the present work, which is based on a recent study of the topological properties of quantum states, there are no low-dimensional anomalies and the assumptions have clear empirical meaning.
The list of basic axioms of quantum mechanics as it was formulated by von Neumann includes only the mathematical formalism of the Hilbert space and its statistical interpretation. We point out that such an approach is too general to be considered as the foundation of quantum mechanics. In particular in this approach any finite-dimensional Hilbert space describes a quantum system. Though such a treatment might be a convenient approximation it can not be considered as a fundamental description of a quantum system and moreover it leads to some paradoxes like Bell's theorem. I present a list from seven basic postulates of axiomatic quantum mechanics. In particular the list includes the axiom describing spatial properties of quantum system. These axioms do not admit a nontrivial realization in the finite-dimensional Hilbert space. One suggests that the axiomatic quantum mechanics is consistent with local realism.
Abstract The object of this paper is to reformulate the principles of the quantum theory as a sequence of propositions which shall be either summary statements of standard experimental procedure or hypotheses concerning the results of experiment and having an immediate physical interpretation. It is shown that the standard process in micro-physics is a generalised spectral analysis, whose properties are simply expressible in symbolic form by means of projective or “idempotent” operators (Einzeloperatoren). It appears that only two hypotheses need be made and that these relate to the existence and properties of transition probabilities. From these fundamental principles, which have a direct physical significance, it is possible to deduce the subsidiary principles which form the accepted basis of the mathematical analysis of the quantum theory and which deal with the representation of quantum states and physical quantities by vectors and linear operators respectively. In this paper the emphasis is laid on the experimental process determining a state of a system and on the associated operators rather than on the state itself or the vector representing it in the system space. Projective operators, which represent actual processes of measurement, and unitary operators, which represent actual transformations of systems of measurement, are given priority over the (statistical) operators which represent physical variables. This method of representation makes the physical meaning of the theory fundamental, instead of leaving it to be extracted from a purely mathematical system of non-commutative algebra or differential equations.
The mathematical formulation of Quantum Mechanics in terms of complex Hilbert space is derived for finite dimensions, starting from a general definition of "physical experiment" and from five simple Postulates concerning "experimental accessibility and simplicity". For the infinite dimensional case, on the other hand, a C*-algebra representation of physical transformations is derived, starting from just four of the five Postulates via a Gelfand-Naimark-Segal (GNS) construction. The present paper simplifies and sharpens the previous derivation in version 1. The main ingredient of the axiomatization is the postulated existence of "faithful states" that allows one to calibrate the experimental apparatus. Such notion is at the basis of the operational definitions of the scalar product and of the "transposed" of a physical transformation. What is new in the present paper with respect to quant-ph/0603011 is the operational deduction of an involution corresponding to the "complex-conjugation" for effects, whose extension to transformations allows to define the "adjoint" of a transformation when the extension is composition-preserving.
The paper proposes a new approach in the foundations of Quantum Mechanics.It does not make any assumption about the physical world, but looks at the consequences of the formalism used in models.Whenever a system is represented by variables which meet precise, but common, mathematical properties, one can prove theorems which are very close to the axioms of Quantum Mechanics (Hilbert spaces, observables, eigen values,...).It is then possible to explore the conditions of the validity of these axioms and to give a firm ground to the usual computations.Moreover this approach sheds a new ligth on the issues of determism, and interacting systems.In the third edition of this paper developments have been added about the statistical procedures used to detect anomalies in Physical Laws.
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.