The Relativistic Particle: Dirac observables and Feynman propagator

TL;DR

This paper explores the algebra of Dirac observables for the relativistic particle, revealing non-commutative geometry similar to DSR, and introduces a five-dimensional framework to regularize divergences and formulate quantum amplitudes.

Contribution

It introduces a five-dimensional formalism for the relativistic particle, connecting Dirac observables, non-commutative geometry, and DSR, and formulates Feynman propagators within this context.

Findings

Position observables become non-commutative

DSR emerges as a natural regularization

Feynman amplitudes are formulated in terms of Dirac observables

Abstract

We analyze the algebra of Dirac observables of the relativistic particle in four space-time dimensions. We show that the position observables become non-commutative and the commutation relations lead to a structure very similar to the non-commutative geometry of Deformed Special Relativity (DSR). In this framework, it appears natural to consider the 4d relativistic particle as a five dimensional massless particle. We study its quantization in terms of wave functions on the 5d light cone. We introduce the corresponding five-dimensional action principle and analyze how it reproduces the physics of the 4d relativistic particle. The formalism is naturally subject to divergences and we show that DSR arises as a natural regularization: the 5d light cone is regularized as the de Sitter space. We interpret the fifth coordinate as the particle's proper time while the fifth moment can be understood as the mass. Finally, we show how to formulate the Feynman propagator and the Feynman amplitudes of quantum field theory in this context in terms of Dirac observables. This provides new insights for the construction of observables and scattering amplitudes in DSR.