Path integral formalism in a Lorentz invariant noncommutative space

TL;DR

This paper develops a new path integral formalism for noncommutative quantum mechanics within the DFRA framework, extending it to quantum field theory, and demonstrating how noncommutativity arises naturally without introducing it artificially.

Contribution

It introduces a novel path integral approach in the DFRA noncommutative space, applicable to quantum mechanics and field theory, emphasizing natural emergence of noncommutativity and dimensional reduction.

Findings

Formulated a path integral for NC quantum mechanics in DFRA space

Extended the formalism to NC quantum field theory with scalar fields

Demonstrated natural generation of NC parameters and dimensional reduction

Abstract

We introduced a new formulation for the path integral formalism for a noncommutative (NC) quantum mechanics defined in the recently developed Doplicher-Fredenhagen-Roberts-Amorim (DFRA) NC framework that can be considered an alternative framework for the NC spacetime of the early Universe. The operators formalism was revisited and we apply its properties to obtain a NC transition amplitude representation. Two DFRA's systems were discussed, the NC free particle and NC harmonic oscillator. Some temperature concepts in this NC space are also considered. The extension to NC DFRA quantum field theory is straightforward and we apply it to a massive scalar field. We construct the generating functional and the effective action to give rise one-particle-irreducible diagrams. As an example, we set the basis for a $n\;(n\geq 3)$ self-interaction $\phi^{n}$ to obtain the correction of the perturbation theory to the propagator and vertex of this model. The main concept that we would like to emphasize from the outset is that the formalism demonstrated here will not be constructed introducing a NC parameter in the system, as usual. It will be generated naturally from an already NC space. In this extra dimensional NC space, we presented also the idea of dimensional reduction to recover commutativity.