A note on misunderstandings of Piron's axioms for quantum mechanics

Piron's axioms for a realistically interpreted quantum mechanics are analyzed in detail within the context of a formal mathematical structure expressed in the conventional set-theoretic idiom of mathematics. As a result, some of the serious misconceptions that have encouraged recent criticisms of Piron's axioms are exposed.     

Group-theoretic treatment of the axioms of quantum mechanics

This axiomatization is based on the observation that ifG is the group of automorphisms of the states (induced, e.g., by suitable evolutions), then we can define a spherical function by mapping each element ofG to the matrix of its transition probabilities. Starting from five physically conservative axioms, we utilize the correspondence between spherical functions and representations to apply the structure theory for compact Lie groups and their orbits in representation spaces to arrive at the standard complex Hilbert space structure of quantum mechanics.Supported by National Science Foundation under Grant MPS75-09371.     

A statistical approach to quantum mechanics

A Monte Carlo method is used to evaluate the Euclidean version of Feynman's sum over particle histories. Following Feynman's treatment, individual paths are defined on a discrete (imaginary) time lattice with periodic boundary conditions. On each lattice site, a continuous position variable x_i specifies the spacial location of the particle. Using a modified Metropolis algorithm, the low-lying energy eigenvalues, |ψ₀(x)|², the propagator, and the effective potential for the anharmonic oscillator are computed, in good agreement with theory. For a deep double-well potential, instantons were found in our computer simulations appearing as multi-kink configurations on the lattice.     

A differential structure for quantum mechanics

A method of obtaining a non-commutative analogue of a differential structure from the action of a Lie group on a C^*-algebra is proposed. The addition of this structure to the usual structure of quantum mechanics turns out to be equivalent to the replacement of the Hilbert space by a Gelfand triple (rigged Hilbert space).     

Axiomatic foundations of quantum physics: Critiques and misunderstandings. Piron's question-proposition system

Some of the most frequent misconceptions about axiomatic quantum physics are discussed with the aim of clarifying their true significance, taking Piron's approach as conceptual framework. In particular, we deal with the following topics: the wrong identification of Piron's questions and Mackey's questions, and some curious alleged empirical consequences; the role of propositions as suitable equivalence classes of questions, their partial order structure, and the paradoxical consequences of the erroneous assignment to questions of some lattice properties involving propositions; the logical and the empirical purport of some negative theorems; the standard Hilbert space model of the theory and the consequent metaphysical disasters related to some identifications, which are peculiar of this model. A controversy between Foulis-Piron-Randall and Hadjisavvas-Thieffine-Mugur-Schächter is analyzed on the basis of the proposed Hilbert space model (in which Piron's questions are realized by Hilbertian effects, i.e., linear bounded operatorsF such that which clarify the different point of views. As an example, we treat the unsharp localization operators inL ₂().     

Reggeon quantum mechanics: A critical discussion

The quantum-mechanical problem of reggeon field theory in zero transverse dimensions is re-examined in order to set up a precise mathematical framework for the case μ = α(0) ? 1 > 0. We establish a Hamiltonian formulation in a Hilbert space for (ifμ > 0) and we prove the equivalence of the related eigenvalue problem with a “radial” Schrödinger-type equation in an L²(0, ∞) space. We prove that the S-matrix and the pomeron Green functions, at fixed rapidity Y and triple-pomeron coupling λ ≠ 0, have a spectral decomposition and are analytic in μ for ?∞ < μ < + ∞. For μ > 0, we confirm most of the qualitative results found by previous authors, and in particular the tunnelling shift [～ exp(?μ²/2λ²)] setting the scale for the asymptotic behaviour in Y.In the classical limit of λ/μ small we find that the action, for μ > 0, develops a singularity in Y at some value Y_c. We give arguments to show that for Y ? Y_c the perturbative result is reached, while for $\text{Y ? Y}{}_{\mathrm{<mtext>c</mtext>}}$ perturbation theory breaks down. Most of these results are shown to be stable against the addition of a small quartic coupling of the simplest type [$\text{λ′(}\text{Ψ}\text{Ψ)}{}^2$] up to the “magic” vvalue λ′ = λ²/μ. The existence of a level crossing at this value is confirmed by an analytic continuation in λ′.     

Logics for quantum mechanics

The two concepts of probability used in physics are analyzed from the formal and the material points of view. The standard theory corresponds toprob ₁ (probability of the coexistence of two properties). A general logicomathematical theory ofprob ₂ (probability of transition between states) is presented in axiomatic form. The underlying state algebra is neither Boolean nor Birkhoff-von Neumann but partial Boolean. In the Boolean subalgebras,prob ₁ theory holds. The theory presented contains the logicomathematical foundations of quantum mechanics and, as degenerate cases, the theories of stochastic games and of Markov chains.     

A discretization ofp-adic quantum mechanics

We show that some compact subgroups ( _n,m ) of thep-adic Heisenberg group act irreducibly on corresponding finite dimensional spaces of test-functions (S _m,n ). Under certain conditions, a compact group (A _m+n ) of linear canonical transformations, isomorphic toSL(2,Z _p ), can be represented unitarily onS _m,n as a group of automorphisms of _n,m . The restriction toS _m,n can be considered as a discretization because an invariant subgroup (I n,m) ofA _m+n is represented trivially. It is possible to take a limit whereI m,n becomes an arbitrarily small neighborhood of the identity, while the dimension ofS _m,n becomes arbitrarily large. This is a possible definition of the continuum limit that we relate to other projective limits appearing naturally in the present context.Address after August 1 1990; University of Iowa, Iowa City, Iowa 52242, USA     

Three approaches to quantum mechanics

A recently introduced variational principle for quantum mechanics is compared with some aspects of stochastic mechanics and of the orthodox approach to quantum mechanics.     

Wall-crossing invariants: from quantum mechanics to knots

Journal of Experimental and Theoretical Physics - We offer a pedestrian-level review of the wall-crossing invariants. The story begins from the scattering theory in quantum mechanics where the...     

On the quantum logic approach to quantum mechanics

A quantum logic structure for quantum mechanics which contains the concepts of a physical space, localizability, and symmetry groups is formulated. It is shown that there is an underlying Hilbert space which mirrors much of this axiomatic structure. Quantum fields are defined and shown to arise naturally from the quantum logic structure. The fields ofHaag andWightman are generalized to this theory and an attempt is made to find a local equivalence for these fields.     

A constructive approach to the foundations of quantum mechanics

An axiomatic theory is formulated which describes a class of yes-no experiments, involving a fixed basic source, a fixed basic detector, and various filters. It is assumed that all filters considered can be constructed from a setP of primitive filters by composition and stochastic selection. Two physically plausible axioms are formulated which allow us to define the concept of asystem in the present context (cf. Definition2.4). To each system we can attach anorder unit module ( ) (cf. Definition5.1) whereby ( ) is acomplete, separable order unit space. Two additional axioms are proposed which have the effect that the space ( ) becomes isomorphic to the order unit space underlying a JB-algebra, at least in the case where isfinite dimensional (cf. Corollary7.9).     

The transitions among classical mechanics,quantum mechanics,and stochastic quantum mechanics

Various formalisms for recasting quantum mechanics in the framework of classical mechanics on phase space are reviewed and compared. Recent results in stochastic quantum mechanics are shown to avoid the difficulties encountered by the earlier approach of Wigner, as well as to avoid the well-known incompatibilities of relativity and ordinary quantum theory. Specific mappings among the various formalisms are given.     

An improved formulation of some theorems and axioms in the axiomatic foundation of the Hilbert space structure of quantum mechanics

In Ref. [1] the axiomatic foundation of the Hilbert space structure of quantum mechanics was outlined. Apart from a set of physically plausible axioms, the (mathematical) assumption (V 1) of the minimal-decomposition property of the basenorm spaceB was incorporated into the axiomatic scheme of the theory.It is shown in the present paper that the assumption (V 1) is superfluous. In the first part of the paper we give a short summary of the axioms; in the second part the main theorems are proved without using assumption (V 1).     

Axioms for nonrelativistic quantum mechanics

On the basis of the axioms assumed it is proved that the logic of propositions concerning any quantum-mechanical system may be endowed with the structure of an orthomodular atomistic complete lattice satisfying the covering postulate, and hence, as a consequence of these axioms, the Piron-MacLaren representation theorem for the logic is obtained.     

Topological quantum mechanics for physicists

This text is an attempt to write an introduction and outline of main results of topological quantum mechanics for readers with a physical background. Instead of presenting rigorous mathematical formulations, we concentrate on explanation of the physical ideas that underline most of the constructions. We review here topological quantum mechanics, since it is the simplest in the diverse family of topological theories, which contains most of their common properties.