Propagator for the Free Relativistic Particle on Archimedean and Nonarchimedean Spaces

TL;DR

This paper explores the behavior of a free relativistic particle in both archimedean and nonarchimedean spaces, deriving propagators and analyzing their physical implications at very short distances.

Contribution

It introduces a method to treat the relativistic particle's dynamics in nonarchimedean spaces, extending the quadratic constraint approach to -adic spaces and calculating the corresponding Green function.

Findings

Derived the propagator for the relativistic particle in nonarchimedean space.

Compared properties of the propagator in archimedean and nonarchimedean settings.

Discussed potential physical implications of nonarchimedean space models.

Abstract

We consider dynamics of a free relativistic particle at very short distances, treating space-time as archimedean as well as nonarchimedean one. Usual action for the relativistic particle is nonlinear. Meanwhile, in the real case, that system may be treated like a system with quadratic (Hamiltonian) constraint. We perform similar procedure in -adic case, as the simplest example of a nonarchimedean space. The existence of the simplest vacuum state is considered and corresponding Green function is calculated. Similarities and differences between obtained results on both spaces are examined and possible physical implications are discussed.