Standard and generalized formalisms of quantum mechanics

François David

Vladan Panković

In this work a general relativistic generalization of Bell inequality is suggested. Namely,it is proved that practically in any general relativistic metric there is a generalization of Bell inequality.It can be satisfied within theories of local (subluminal) hidden variables, but it cannot be satisfied in the general case within standard quantum mechanical formalism or within theories of nonlocal (superluminal) hidden variables. It is shown too that within theories of nonlocal hidden variables but not in the standard quantum mechanical formalism a paradox appears in the situation when one of the correlated subsystems arrives at a Schwarzschild black hole. Namely, there is no way that black hole horizon obstructs superluminal influences between spin of the subsystem without horizon and spin of the subsystem within horizon,or simply speaking,there is none black hole horizon nor "no hair" theorem for subsystems with correlated spins. It implies that standard quantum mechanical formalism yields unique consistent and complete description of the quantum mechanical phenomenons.

Pavel Bóna

We are dealing in this work with such formal and conceptual extensions of nonrelativistic quantum mechanics (QM) which contain QM with its standard formalism and interpretation as a subtheory. QM is here primarily equivalently reformulated in the form of a Poisson system on the infinite-dimensional phase space consisting of all density matrices. It is shown that inclusion of additional ("nonlinear") symmetry generators (i.e. "Hamiltonians") into this reformulation of (linear) QM leads to a considerable extension of the theory: two kinds of quantum "mixed states" should be distinguished, and operator - valued functions of density matrices should be used in the role of "nonlinear observables". A general framework for physical theories is obtained in this way, containing various nonlinear versions of QM, as well as some of "approximations" to standard QM as subtheories. Questions of interpretation are considered and formally consistently solved. Specifications and illustrations, as well as mathematical Appendices and two functioning Indexes are included, making the text almost selfcontained.

Nuno Costa Dias, João Nuno Prata

The Weyl-Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations is still missing. In this paper we present a covariant generalization of the Weyl order prescription and of the Weyl-Wigner map and use them to derive Vey quantum mechanics directly from the standard operator formulation. The procedure displays some interesting features: it yields all the key ingredients and provides a more straightforward interpretation of the Vey theory including a direct implementation of unitary operator transformations as phase space coordinate transformations in the Vey idiom. These features are illustrated through a simple example. 1

Sol Wieder