The foundation of quantum theory and noncommutative spectral theory: Part II

The present paper comprises Sects. 5–8 of a work which proposes an axiomatic approach to quantum mechanics in which the concept of a filter is the central primitive concept. Having layed down the foundations in the first part of this work (which appeared in the last issue of this journal and comprises Sects. 0–4), we arrived at a dual pair Y, M consisting of abase norm space Y and anorder unit space M, being in order and norm duality with respect to each other. This is precisely the setting of noncommutative spectral theory, a theory which has been developed during the late nineteen seventies by Alfsen and Shultz. ^(2,3) In this part we add to the four axioms (Axioms S, DP, R, SP) of Sect. 3 three further axioms (Axioms E, O, L). These axioms are suggested by the work of Alfsen and Shultz and enable us to derive the JB-algebra structure of quantum mechanics (cf. Theorem 8.9).     

N=2 topological gauge theory,the Euler characteristic of moduli spaces,and the Casson invariant

We discuss gauge theory with a topologicalN=2 symmetry. This theory captures the de Rham complex and Riemannian geometry of some underlying moduli space and the partition function equals the Euler number () of . We explicitly deal with moduli spaces of instantons and of flat connections in two and three dimensions. To motivate our constructions we explain the relation between the Mathai-Quillen formalism and supersymmetric quantum mechanics and introduce a new kind of supersymmetric quantum mechanics based on the Gauss-Codazzi equations. We interpret the gauge theory actions from the Atiyah-Jeffrey point of view and relate them to supersymmetric quantum mechanics on spaces of connections. As a consequence of these considerations we propose the Euler number () of the moduli space of flat connections as a generalization to arbitrary three-manifolds of the Casson invariant. We also comment on the possibility of constructing a topological version of the Penner matrix model.From Oct. 1992: ictp, P.O. Box 586, I-34100 Trieste, Italy     

Maximal violation of Bell's inequalities is generic in quantum field theory

Under weak technical assumptions on a net of local von Neumann algebras {A(O)} in a Hilbert space , which are fulfilled by any net associated to a quantum field satisfying the standard axioms, it is shown that for every vector state in there exist observables localized in complementary wedge-shaped regions in Minkowski space-time that maximally violate Bell's inequalities in the state . If, in addition, the algebras corresponding to wedge-shaped regions are injective (which is known to be true in many examples), then the maximal violation occurs in any state on () given by a density matrix.     

Axiomatic unsharp quantum theory (From Mackey to Ludwig and Piron)

On the basis of Mackey's axiomatic approach to quantum physics or, equivalently, of a state-event-probability (SEVP) structure, using a quite standard fuzzification procedure, a set of unsharp events (or effects) is constructed and the corresponding state-effect-probability (SEFP) structure is introduced. The introduction of some suitable axioms gives rise to a partially ordered structure of quantum Brouwer-Zadeh (BZ) poset; i.e., a poset endowed with two nonusual orthocomplementation mappings, a fuzzy-like orthocomplementation, and an intuitionistic-like orthocomplementation, whose set of sharp elements is an orthomodular complete lattice. As customary, by these orthocomplementations the two modal-like necessity and possibility operators are introduced, and it is shown that Ludwig's and Jauch-Piron's approaches to quantum physics are interpreted in complete SEFP. As a marginal result, a standard procedure to construct a lot of unsharp realizations starting from any sharp realization of a fixed observable is given, and the relationship among sharp and corresponding unsharp realizations is studied.     

The modal interpretation of quantum mechanics and some of its relativistic aspects

The modal interpretation of quantum mechanics is an attempt to relate the mathematical formalism of quantum mechanics to physical properties (beables,existents) in such a way that the property attribution reflects the mathematical structure as much as possible—no additional structure is superimposed on the quantum mechanical formalism. In this article the main features of the modal interpretation are explained and the question is discussed of how this interpretation deals with some well-known problems of quantum measurement theory (relativistic covariance and the question of whether or not there is superluminal causation).     

The superposition of the states and the logic approach to quantum mechanics

An axiomatic approach to quantum mechanics is proposed in terms of a logic scheme satisfying a suitable set of axioms. In this context the notion of pure, maximal, and characteristic state as well as the superposition relation and the superposition principle for the states are studied. The role the superposition relation plays in the reversible and in the irreversible dynamics is investigated and its connection with the tensor product is studied. Throughout the paper, theW ^*-algebra model, which satisfies our axioms, is used to exemplify results and properties of the general scheme.     

Quantum theory as a universal physical theory

The problem of setting up quantum theory as a universal physical theory is investigated. It is shown that the existing formalism, in either the conventional or the Everett interpretation, must be supplemented by an additional structure, the interpretation basis. This is a preferred ordered orthonormal basis in the space of states. Quantum measurement theory is developed as a tool for determining the interpretation basis. The augmented quantum theory is discussed.     

Nondemolition principle of quantum measurement theory

We give an explicit axiomatic formulation of the quantum measurement theory which is free of the projection postulate. It is based on the generalized nondemolition principle applicable also to the unsharp, continuous-spectrum and continuous-in-time observations. The collapsed state-vector after the objectification is simply treated as a random vector of the a posterioristate given by the quantum filtering, i.e., the conditioning of the a prioriinduced state on the corresponding reduced algebra. The nonlinear phenomenological equation of continuous spontaneous localization has been derived from the Schrödinger equation as a case of the quantum filtering equation for the diffusive nondemolition measurement. The quantum theory of measurement and filtering suggests also another type of the stochastic equation for the dynamical theory of continuous reduction, corresponding to the counting nondemolition measurement, which is more relevant for the quantum experiments.     

The problem of time in parametrized theories

A common feature of reparametrization invariant theories is the difficulty involved in identifying an appropriate evolution parameter and in constructing a Hilbert space on states. Two well known examples of such theories are the relativistic point particle and the canonical formulation of quantum gravity. The strong analogy between them (specially for minisuperspace models) is considered in order to stress the correspondence between the localization problem and the problem of time, respectively. A possible solution for the first problem was given by the proper time formulation of relativistic quantum mechanics. Thus, we extrapolate the main outlines of such a formalism to the quantum gravity framework. As a consequence, a proposal to solve the problem of time arises.     

From where do quantum groups come?

The phase space realizations of quantum groups are discussed using *-products. We show that on phase space, quantum groups appear necessarily as two-parameter deformation structures, one parameter (v) being concerned with the quantization in phase space, the other () expressing the quantum groups as deformation of their Lie counterparts. Introducing a strong invariance condition, we show the uniqueness of the -deformation. This suggests that the strong invariance condition is a possible origin of the quantum groups.Dedicated to Asim Barut with all our friendship.     

Making Sense of a World of Clicks

In a recent article, O. Ulfbeck and A. Bohr [Found. Phys. 31, 757 (2001)] have stressed the genuine fortuitousness of detector clicks, which has also been pointed out, in different terms, by the present author [Am. J. Phys. 68, 728 (2000)]. In spite of this basic agreement, the present article raises objections to the presuppositions and conclusions of Ulfbeck and Bohr, in particular their rejection of the terminology of indefinite variables, their identification of reality with the world of experience, their identification of experience with what takes place on the spacetime scene, and the claim that their interpretation of quantum mechanics is entirely liberated from classical notions. An alternative way of making sense of a world of uncaused clicks is presented. This does not invoke experience but deals with a free-standing reality, is not fettered by classical conceptions of space and time but introduces adequate ways of thinking about the spatiotemporal aspects of the quantum world, and does not reject indefinite variables but clarifies the implications of their existence.     

Quantum mechanics and the direction of time

In recent papers the authors have discussed the dynamical properties of large Poincaré systems (LPS), that is, nonintegrable systems with a continuous spectrum (both classical and quantum). An interesting example of LPS is given by the Friedrichs model of field theory. As is well known, perturbation methods analytic in the coupling constant diverge because of resonant denominators. We show that this Poincaré catastrophe can be eliminated by a natural time ordering of the dynamical states. We obtain then a dynamical theory which incorporates a privileged direction of time (and therefore the second law of thermodynamics). However, it is only in very simple situations that this time ordering can be performed in an extended Hilbert space. In general, we need to go to the Liouville space (superspace) and introduce a time ordering of dynamical states according to the number of particles involved in correlations. This leads then to a generalization of quantum mechanics in which the usual Heisenberg's eigenvalue problem is replaced by a complex eigenvalue problem in the Liouville space.     

Contribution to the theory of single-mode laser modulation II. Conduction modulation

The problem of conduction modulation of laser radiation is considered. It is shown that acoustic wave interacting with photon field in active medium causes a periodic modulation of the number of quanta in the single-mode gas laser. The problem of modulation is solved quantum mechanically and it is shown that dielectric and conduction modulation give similar results.     

Hidden variables with directionalization

A hidden-variables model is presented which, by using a hypothesis of directionalization of the photons at the deflectors, is able to reproduce all the quantum mechanical predictions for the Orsay realization of the Einstein-Podolsky-Rosen-Bohm experiment, even for ideal polarizers, detectors, time-coincidence windows, and event-ready setups. The model also holds for the no-enhancement assumption. The requirements for an experiment aimed to discriminate between quantum mechanics and the new model are discussed. Under some plausible assumptions, such experiment is achievable with the current technology. It would be essentially a remake of the Orsay one with improved deflectors and a time-resolved measurement of the photon flows toward each detector.     

Faux-boolean algebras and classical models

A certain substructure of the lattice of projections on a Hilbert space is defined, and it is shown under what conditions such a structure can be said to have a classical model (in a sense made precise). The results have implications for the interpretation of quantum mechanics.     

Differential calculus on quantum spheres

Three-dimensional differential calculus on quantum spheres S _infc ^sup2 ,]–1, 1[{0}, c[0, ], is introduced and investigated. Spectra of generalized Laplacians are found. These operators are expressed by generalized directional derivatives. Classical limits of these objects are obtained and a simple approach to quantum mechanics on a quantum sphere is presented.     

Quantum probabilities,operators of state preparation,and the principle of superposition

An integrated view concerning the probabilistic organization of quantum mechanics is first obtained by systematic confrontation of the Kolmogorov formulation of the abstract theory of probabilities with the quantum mechanical representationand its factual counterparts. Because these factual counterparts possess a peculiar space-time structure stemming from the operations by which the observer produces the studied states (operations of state preparation) and the qualifications of these (operations of measurement), the approach brings forth probability-trees, complex constructs with treelike space-time support. Though it is strictly entailed by confrontation with the abstract theory of probabilities as it now stands, the construct of a quantum mechanical probability treetransgresses this theory. It indicates the possibility of an extended abstract theory of probabilities: Quantum mechanics appears to be neither a normal probabilistic theory nor an abnormal one, but a pioneering particular realization of afuture extended abstract theory of probabilities. The integrated perception of the probabilistic organization of quantum mechanics removes the current identifications of spectral decompositions of one state vector, with superpositions of several state vectors. This leads to the definition of operators of state preparation and of the calculus with these and to a clear understanding of the physical significance of the principle of superposition. Furthermore, a complement to the quantum theory of measurements is obtained.     

Geometrization of quantum mechanics

Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional manifold of instantaneous pure states. This geometrical structure can accommodate generalizations of quantum mechanics, including the nonlinear relativistic models recently proposed. It is shown that any such generalization satisfying a few physically reasonable conditions would reduce to ordinary quantum mechanics for states that are near the vacuum. In particular the origin of complex structure is described.     

Beam Optics and Signal Analysis in a Quantumlike Approach

The application of the tomographic map introduced recently in quantum mechanics is discussed in connection with classical problems that are described by equations formally coinciding with the equations of quantum theory. Examples of the classical problems such as chargedparticlebeam transport and analyticsignal analysis are reviewed. Some aspects of possible applications in quantum computing and quantum communication are discussed.