The Noncommutative Doplicher-Fredenhagen-Roberts-Amorim Space

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 ($θ^{μν}$) 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 $θ^{μν}$. 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 $θ^{ij}$ 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 $θ^{μν}$ 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, $θ^{μν}$ and its canonical momentum $π_{μν}$ 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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