In this paper we show that the existence of a primarily discrete spacetime may be a fruitful assumption from which we may develop a new approach of statistical thermodynamics in pre-relativistic conditions. The discreteness of spacetime structure is determined by a condition that mimics the Heisenberg uncertainty relations, and the motion in this spacetime model is chosen as simple as possible. From these two assumptions we define a path entropy that measures the number of closed paths associated with a given energy of the system preparation. This entropy has a dynamical character and depends on the time interval on which we count the paths. We show that there exists a like-equilibrium condition for which the path entropy corresponds exactly to the usual thermodynamic entropy and, more generally, the usual statistical thermodynamics is reobtained. This result derived without using the Gibbs-ensemble method shows that the standard thermodynamics is consistent with a motion that is time irreversible at a microscopic level. From this change of paradigm it becomes easy to derive an H-theorem. A comparison with the traditional Boltzmann approach is presented. We also show how our approach can be implemented in order to describe reversible processes. By considering a process defined simultaneously by initial and final conditions, a well-defined stochastic process is introduced and we are able to derive a Schrodinger equation, an example of time-reversible equation.

Entropy, time-irreversibility and the Schrödinger equation in a primarily discrete spacetime

In this paper we show that the existence of a primarily discrete space-time may be a fruitful assumption from which we may develop a new approach of statistical thermodynamics in pre-relativistic conditions. The discreetness of space-time structure is determined by a condition that mimics the Heisenberg uncertainty relations and the motion in this space-time model is chosen as simple as possible. From these two assumptions we define a path-entropy that measures the number of closed paths associated with a given energy of the system preparation. This entropy has a dynamical character and depends on the time interval on which we count the paths. We show that it exists an like-equilibrium condition for which the path-entropy corresponds exactly to the usual thermodynamic entropy and, more generally, the usual statistical thermodynamics is reobtained. This result derived without using the Gibbs ensemble method shows that the standard thermodynamics is consistent with a motion that is time-irreversible at a microscopic level. From this change of paradigm it becomes easy to derive a $H-theorem$. A comparison with the traditional Boltzmann approach is presented. We also show how our approach can be implemented in order to describe reversible processes. By considering a process defined simultaneously by initial and final conditions a well defined stochastic process is introduced and we are able to derive a Schroedinger equation, an example of time reversible equation.

Entropy, time irreversibility and Schroedinger equation in a primarily discrete space-time

We study some quantum systems described by noncanonical commutation relations formally expressed as , where is the associated (harmonic-oscillator-like) Hamiltonian of the system and is a Hermitian (constant) operator, i.e. . In passing, we also consider a simple ( canonical) model, in the framework of a relativistic Klein?Gordon-like wave equation.

https://iopscience.iop.org/article/10.1088/1751-8113/40/49/019/pdf

Supersymmetry, quark confinement and the harmonic oscillator

We present a nonrelativistic wave equation for the electron in (3+1)-dimensions which includes negative-energy eigenstates. We solve this equation for three well-known instances, reobtaining the corresponding Pauli equation (but including negative-energy eigenstates) in each case.

The Schrödinger Equation and Negative Energies

The occurrence of a fundamental length, a lower bound to any output of a position measurement, seems to be an attractive possibility in classical and quantum physics. In this regard, we investigate a simple model in the context of the classical Maxwell-Lorentz theory where a basic length can be identified. We then propose a possible experiment to indirectly detect the existence of this length.

Fundamental length in classical electrodynamics

The Dirac equation for the Coulomb-like problem is modified by incorporating minimal interactions into the Dirac Hamiltonian, that keep the 1/r potential dependence We determine the general energy eigenvalues and the corresponding eigenfunctions

Coulomb-Like Interactions in the Dirac Equation

Motivated by previous work on high-energy quantum mechanics, a simple model is devised to study the internal geometry of hadron resonances. In this model we assume new basic canonical commutation relations between the (internal) coordinate and momentum operators of the hadronic quantum system. By systematically imposing Lie algebra commutation relations between these and other observables, we discuss the free and bound particle problems, identifying in each case the corresponding internal symmetries. For the bound particle problem, which models quark confinement, this symmetry turns out to be characterized by Dirac's two-oscillator representation of theO(3, 2) de Sitter group.

Internal Geometry of Hadron Resonances

Electron rest-frame internal canonical coordinates are reobtained by the free-particle Foldy-Wouthuysen transformation: Schrodinger “microscopic momentum”, Barut-Bracken “microscopic coordinate”, and the rest Hamiltonian, which describe Zitterbewegung in this frame.SO(4, 1) Snyder space-time invariant quantization is considered in order to construct a dynamical group for Zitterbewegung. The electron's internal structure appears associated with its secondorder self-energy process and governed by the 15-parameter dynamical groupSO(5, 1). This is a generalization of Barut-Bracken symmetry which describes Zitterbewegung as generated by an algebra of the rotation groupSO(5). This noncompact symmetrySO(5, 1) permits a natural interpretation for the operators of its algebra and introduces a generalization to higher-dimensional fermionic representations.

SO(5, 1) Dynamical Symmetry for Electron Zitterbewegung

We consider the possibility that simultaneously time and intrinsic length can be regarded as discrete real parameters. We study the dynamics of the free particle. For both scattering and bound states there are configurations where the energy is bounded from above and from below even for positive wave-function solutions. For the case of continuous evolution we show that the wave equation with a linear scalar coupling describes an oscillator that has built-in hidden supersymmetry.

Discrete Time and Intrinsic Length in Quantum Mechanics

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Papers

Coulomb-Like Interactions in the Dirac Equation

Internal Geometry of Hadron Resonances

SO(5, 1) Dynamical Symmetry for Electron Zitterbewegung

Discrete Time and Intrinsic Length in Quantum Mechanics