The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space

TL;DR

This paper reviews the DFRA noncommutative space, extending the DFR model by treating the noncommutativity parameter as a dynamic variable with conjugate momentum, and explores its classical and quantum formulations.

Contribution

It introduces a minimal extension of the DFR space where the noncommutativity parameter is dynamic, and develops a consistent algebra, classical mechanics, and quantum theories within this framework.

Findings

Constructed a consistent algebra with extended canonical operators.

Developed a classical mechanics formulation leading to a NC quantum theory.

Analyzed fermionic and scalar fields within the extended noncommutative space.

Abstract

This work is an effort in order to compose a pedestrian review of the recently elaborated Doplicher, Fredenhagen, Roberts and Amorim (DFRA) noncommutative (NC) space which is a minimal extension of the DFR space. In this DRFA space, the object of noncommutativity ($\theta^{\mu\nu}$) is a variable of the NC system and has a canonical conjugate momentum. The DFRA formalism is constructed in an extended space-time with independent degrees of freedom associated with the object of noncommutativity $\theta^{\mu\nu}$. A consistent algebra involving the enlarged set of canonical operators is described, which permits one to construct theories that are dynamically invariant under the action of the rotation group. A consistent classical mechanics formulation is analyzed in such a way that, under quantization, it furnishes a NC quantum theory with interesting results. The Dirac formalism for constrained Hamiltonian systems is considered and the object of noncommutativity $\theta^{ij}$ plays a fundamental role as an independent quantity. It is also explained about the generalized Dirac equation issue, that the fermionic field depends not only on the ordinary coordinates but on $\theta^{\mu\nu}$ as well. The dynamical symmetry content of such fermionic theory is discussed, and we show that its action is invariant under ${\cal P}'$. In the last part of this work we analyze the complex scalar fields using this new framework. As said above, in a first quantized formalism, $\theta^{\mu\nu}$ and its canonical momentum $\pi_{\mu\nu}$ are seen as operators living in some Hilbert space. In a second quantized formalism perspective, we show an explicit form for the extended Poincar\'e generators and the same algebra is generated via generalized Heisenberg relations. We also consider a source term and construct the general solution for the complex scalar fields using the Green function technique.