Orientational Order Parameters for Arbitrary Quantum Systems

The concept of quantum-mechanical nematic order, which is important in systems such as superconductors, is based on an analogy to classical liquid crystals, where order parameters are obtained through orientational expansions. This method is generalized to quantum mechanics based on an expansion of Wigner functions. This provides a unified framework applicable to arbitrary quantum systems. The formalism recovers the standard definitions for spin systems. For Fermi liquids, the formalism reveals the nonequivalence of various definitions of the order parameter used in the literature. Moreover, new order parameters for quantum molecular systems with low symmetry are derived, which cannot be properly described with the usual nematic tensors.     

Foundations of quantum probability

This paper presents an overview of the foundations of quantum probability. The main concepts in this theory are measurements and generalized actions. These concepts correspond to the usual quantum observables and states. Probabilities are computed by means of a universal influence function. We first derive the form of the universal influence function and then construct the amplitude and probability of a measurement with respect to a given generalized action. It is shown that traditional quantum mechanics can be derived as a special case of this theory and moreover the theory gives a complete realistic interpretation of quantum mechanics. It is demonstrated that spins of any order can be described within this framework and a realistic solution to the EPR problem can be achieved.     

Relativistic quantum theory without quantized fields

Peculiarities of symmetrical quantum systems are considered with the aid of the Mackey's induced representations theory. The four-dimensional coordinate representation of the relativistic quantum mechanics suggested by Stueckelberg in 1941 is rederived, reinterpreted and generalized for an arbitrary spin. Then it is applied to introduce the causal propagator as a particleantiparticle transition amplitude without consideration of a field equation. Finally the theory of relativistic quantum particles interaction is reformulated without an appeal to the concept of quantized fields.     

AN ALTERNATIVE ONE-PARTICLE QUANTUM THEORY WITH A NEW EXPECTATION VALUE POSTULATE

Our recent treatment of some single-particle systems implies that if the expectation value postulate in usual quantum theory is changed in some proposed way, we can also have a self-consistent and reasonable one-particle quantum theory. In this paper, the framework of the theory barmonized with a newly proposed self-energy quantum electrodynamics is given. Our theory is compatible with Einstein's completeness. It in statistical ensemble reproduces the results given by the usual theory and in classical limit reduces to the classical mechanics for the single-particle systems.     

A new differential calculus on a complex banach space with application to variational problems of quantum theory

It is proved that a semilinear function on a complex banach space is not differentiable according to the usual definition of differentiability in the calculus on banach spaces. It is shown that this result makes the calculus largely inapplicable to the solution of variational problems of quantum mechanics. A new concept of differentiability called semi-differentiability is defined. This generalizes the standard concept of differentiability in a banach space and the resulting calculus is particularly suitable for optimizing real-valued functions on a complex banach space and is directly applicable to the solution of quantum mechanical variational problems. As an example of such application a rigorous proof of a generalized version of a result due to Sharma (1969) is given. In the course of this work a new concept of prelinearity is defined and some standard results in the calculus on banach spaces are extended and generalized into more powerful ones applicable directly to prelinear functions and hence yielding the standard results for linear functions as particular cases.     

On the reality of spin and helicity

The possibilities of a realistic interpretation of quantum mechanics are investigated by means of a statistical analysis of experiments performed on the simplest type of quantum systems carrying spin or helicity. To this end, fundamental experiments, some new, for measuring polarization are reviewed and (re)analyzed. Theunsharp reality of spin is essential in the interpretation of some of these experiments and represents a natural motivation for recent generalizations of quantum mechanics to a theory incorporating effect-valued measures as unsharp observables and generalized systems of imprimitivity.     

New formulation of quantum mechanics

A reformulation of the mathematical foundations of quantum mechanics is presented. This new framework is based on the concepts of measurement, generalized action, and a unique universal influence function. The main axiom is that the probability of a measurement outcome is the sum (or integral) of the influences between pairs of alternatives that result in the outcome when the measurement is executed. The framework provides answers to various puzzling questions of traditional quantum mechanics. Moreover, it gives a realistic model that extends the usual quantum mechanical formalism.     

A Discussion on Particle Number and Quantum Indistinguishability

The concept of individuality in quantum mechanics shows radical differences from the concept of individuality in classical physics, as E. Schr?dinger pointed out in the early steps of the theory. Regarding this fact, some authors suggested that quantum mechanics does not possess its own language, and therefore, quantum indistinguishability is not incorporated in the theory from the beginning. Nevertheless, it is possible to represent the idea of quantum indistinguishability with a first-order language using quasiset theory (Q). In this work, we show that Q cannot capture one of the most important features of quantum non-individuality, which is the fact that there are quantum systems for which particle number is not well defined. An axiomatic variant of Q, in which quasicardinal is not a primitive concept (for a kind of quasisets called finite quasisets), is also given. This result encourages the searching of theories in which the quasicardinal, being a secondary concept, stands undefined for some quasisets, besides showing explicitly that in a set theory about collections of truly indistinguishable entities, the quasicardinal needs not necessarily be a primitive concept. Graciela Domenech — Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).     

Vertex calculation for a composite particle

The classical treatment and the quantization of composite relativistic systems is given a manifestly covariant formulation in presence of constraints. A particular formulation of Feynman's quantum mechanics is used to treat the scattering of composite relativistic systems. A covariant harmonic oscillator model is employed to calculate vertices of interactions: the results are similar to the corresponding ones in the usual field theories, but the presence of some convergence factors gives hope that a theory with composite particles may be finite.     

Intrinsic Properties of Quantum Systems

A new realist interpretation of quantum mechanics is introduced. Quantum systems are shown to have two kinds of properties: the usual ones described by values of quantum observables, which are called extrinsic, and those that can be attributed to individual quantum systems without violating standard quantum mechanics, which are called intrinsic. The intrinsic properties are classified into structural and conditional. A systematic and self-consistent account is given. Much more statements become meaningful than any version of Copenhagen interpretation would allow. A new approach to classical properties and measurement problem is suggested. A quantum definition of classical states is proposed.     

Axioms for quantum theory

The first three of these axioms describe quantum theory and classical mechanics as statistical theories from the very beginning. With these, it can be shown in which sense a more general than the conventional measure theoretic probability theory is used in quantum theory. One gets this generalization defining transition probabilities on pairs of events (not sets of pairs) as a fundamental, not derived, concept. A comparison with standard theories of stochastic processes gives a very general formulation of the non existence of quantum theories with hidden variables. The Cartesian product of probability spaces can be given a natural algebraic structure, the structure of an orthocomplemented, orthomodular, quasi-modular, not modular, not distributive lattice, which can be compared with the quantum logic (lattice of all closed subspaces of an infinite dimensional Hubert space). It is shown how our given system of axioms suggests generalized quantum theories, especially Schrödinger equations, for phase space amplitudes.     

Stochastic interpretations of nonrelativistic quantum theory

A critique of the causla and classical stochastic interpretations of nonrelativistic quantum mechanics is presented. The only way that the classical stochastic formulation can be made compatible with the theory of quantum measurement is to extend the probability measure density for fluctuating paths to the complex domain. In doing so, we obtain the generalized stochiastic formulation in which the methods of classical probability theory can be used to describe the quantum mechanical phenomenon of interfering alternatives. Illustrative examples from quantum theory are used to show the complete compatibility between the traditional and generalized stochastic interpretations of quantum mechanics. Work supported in part by a contribution from the CNR.     

Many-times formalism and Coulomb interaction

We extend here the many-times formalism, formerly used mainly for particles moving in given classical fields, to interacting particles. In order to minimize the difficulties associated with an equal-time interaction, we limit ourselves to nonrelativistic quantum mechanics and a two-particle interaction, such as that corresponding to the Coulomb force between charged particles. We obtain a set of differential equations which are really not consistent, but they serve as a guide to a formulation in terms of integral equations that has the same perturbation expansion as the usual theory for the scattering of particles. The integral equation for two-particle amplitudes can be modified to give the correct theory for bound states, but this is not the case for more than two particles. We expect that this theory can be generalized to a formulation of relativistic quantum mechanics of interacting particles.     

A new interpretation of Bell's inequalities

Bell's inequalities are always derived assuming that local hidden-variable theories give a set of positive-definite probabilities for detecting a particle with a given spin orientation. The usual claim is that quantum mechanics, by its very nature, cannot produce a set of such probabilities. We show that this is not the case if one allows for generalized (nonpositive-definite) master probability distributions. The master distributions give the usual quantum mechanical violation of Bell's inequalities. Consequences for the interpretation of quantum mechanics are discussed.