# Relativistic particle: Dirac observables and Feynman propagator

## Abstract

We analyze the algebra of Dirac observables of the relativistic particle in four space-time dimensions. We show that the position observables become noncommutative and the commutation relations lead to a structure very similar to the noncommutative 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 (due to the 5D representation), 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.

- Authors:

- Perimeter Institute, 31 Caroline St North, Waterloo, ON, N2L 2Y5 (Canada)
- (Italy)
- (France)

- Publication Date:

- OSTI Identifier:
- 20935273

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 10; Other Information: DOI: 10.1103/PhysRevD.75.105016; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ALGEBRA; COMMUTATION RELATIONS; COORDINATES; DE SITTER SPACE; GEOMETRY; LIGHT CONE; MANY-DIMENSIONAL CALCULATIONS; MASS; MASSLESS PARTICLES; PROPAGATOR; QUANTIZATION; QUANTUM FIELD THEORY; RELATIVISTIC RANGE; RELATIVITY THEORY; SCATTERING AMPLITUDES; WAVE FUNCTIONS

### Citation Formats

```
Freidel, Laurent, Girelli, Florian, Livine, Etera R., SISSA, Via Beirut 2-4, 34014 Trieste, INFN, Sezione di Trieste, and Laboratoire de Physique, ENS Lyon, CNRS UMR 5672, 46 Allee d'Italie, 69364 Lyon Cedex 07.
```*Relativistic particle: Dirac observables and Feynman propagator*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.105016.

```
Freidel, Laurent, Girelli, Florian, Livine, Etera R., SISSA, Via Beirut 2-4, 34014 Trieste, INFN, Sezione di Trieste, & Laboratoire de Physique, ENS Lyon, CNRS UMR 5672, 46 Allee d'Italie, 69364 Lyon Cedex 07.
```*Relativistic particle: Dirac observables and Feynman propagator*. United States. doi:10.1103/PHYSREVD.75.105016.

```
Freidel, Laurent, Girelli, Florian, Livine, Etera R., SISSA, Via Beirut 2-4, 34014 Trieste, INFN, Sezione di Trieste, and Laboratoire de Physique, ENS Lyon, CNRS UMR 5672, 46 Allee d'Italie, 69364 Lyon Cedex 07. Tue .
"Relativistic particle: Dirac observables and Feynman propagator". United States.
doi:10.1103/PHYSREVD.75.105016.
```

```
@article{osti_20935273,
```

title = {Relativistic particle: Dirac observables and Feynman propagator},

author = {Freidel, Laurent and Girelli, Florian and Livine, Etera R. and SISSA, Via Beirut 2-4, 34014 Trieste and INFN, Sezione di Trieste and Laboratoire de Physique, ENS Lyon, CNRS UMR 5672, 46 Allee d'Italie, 69364 Lyon Cedex 07},

abstractNote = {We analyze the algebra of Dirac observables of the relativistic particle in four space-time dimensions. We show that the position observables become noncommutative and the commutation relations lead to a structure very similar to the noncommutative 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 (due to the 5D representation), 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.},

doi = {10.1103/PHYSREVD.75.105016},

journal = {Physical Review. D, Particles Fields},

number = 10,

volume = 75,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}