Adonai S. Sant'Anna

Quasi-Set-Theoretical Foundations of Statistical Mechanics: A Research Program

Quasi-set theory provides us a mathematical background for dealing with collections of indistinguishable elementary particles. In this paper, we show how to obtain the usual statistics (Maxwell–Boltzmann, Bose–Einstein, and Fermi–Dirac) into the scope of quasi-set theory. We also show that, in order to derive Maxwell–Boltzmann statistics, it is not necessary to assume that the particles are distinguishable or individuals. In other words, Maxwell–Boltzmann statistics is possible even in an ensamble of indistinguishable particles, at least from the theoretical point of view. The main goal of this paper is to provide the mathematical grounds of a quasi-set theoretical framework for statistical mechanics.

Time in Thermodynamics

We use Padoas principle of independence of primitive symbols in axiomatic systems in order to show that time is dispensable in continuum thermodynamics, according to the axiomatic formulation of Gurtin and Williams. We also show how to define time by means of the remaining primitive concepts of Gurtin and Williams system. Finally, we introduce thermodynamics without time as a primitive concept.

The mathematical role of time and space-time in classical physics

We use Padoas principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and spacetime in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that spacetime is also dispensable in Hamiltonian mechanics, Maxwells electromagnetic theory, the Dirac electron, classical gauge fields, and general relativity.

Axioma for Mach's Mechanics

We present an axiomatic framework for what we call Machs mechanics, inspired on the ideas by A. K. T. Assis and P. Graneau about relational mechanics. We show that, in contrast to a suggestion of these authors, Machs principle does not need to be committed with any Webers like gravitational force. Actually it is possible to settle an axiomatic framework for non-relativistic classical particle mechanics which incorporates Machs principle and is still consistent with any force that obeys the Newtonian action-reaction principle. We make further criticisms concerning relational mechanics.

Elementary particles, hidden variables, and hidden predicates

We recently showed that it is possible to deal withcollections of indistinguishable elementary particles (in thecontext of quantum mechanics) in a set-theoretical framework, byusing hidden variables. We propose in the presentpaper another axiomatics for collections of indiscernibleswithout hidden variables, where hidden predicates are implicitlyassumed. We also discuss the possibility of a quasi-settheoretical picture for quantum theory. Quasi-set theory, basedon Zermelo-Fraenkel set theory, was developed for dealing withcollections of indistinguishable, but, not identical objects.

INDISTINGUISHABLE PARTICLES AND HIDDEN VARIABLES: PART II

In a previous work it was shown that it is possible to deal with collections of indistinguishable elementary particles in a set-theoretical framework, by using hidden variables. We propose in the present paper a set-theoretical axiomatics for collections of indiscernibles with no explicit mention to hidden variables. We also show, in this context, the fundamental role of the (micro) state in the process of individuation of classical and quantum particles. Finally, we discuss the importance of the axiom of choice in Zermelo-Fraenkel set theory in the context of quantum distributions of bosons and fermions.

Spinor Representation of Electromagnetic Fields on Noncommutative Spaces

We apply Campolattaros spinor representation of the electromagnetic field to noncommutative spaces. The spinor representation of the (self-dual) electromagnetic field on noncommutative spaces is obtained.

Gravitation in Hertz Mechanics

In a previous paper, one of us presented an axiomatic framework for non-relativistic classical particle mechanics where the concept of force is not assumed as a primitive notion. Such a formulation was inspired by Hertzs mechanics, which considers only three primitive concepts: time, space, and mass. It is also emphasized that there is only one Fundamental Law: Every free system persists in its state of rest or of iniform motion in a straightest path. In the present paper we formulate a theory of gravitation which seems to be compatible with Hertzs mechanics, in the sense that it makes no explicit reference to the concepts of force or actions-at-a-distance. We show how to derive Keplers Laws in our axiomatic framework.