The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim 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 ${\mathcal{P}}^{\prime }$. 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 Poincaré 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.

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