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 SIGMA, 2010, Volume 6, 083, 37 pages (Mi sigma541)

The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space

Everton M. C. Abreuab, Albert C. R. Mendesc, Wilson Oliveirac, Adriano O. Zangirolamic

a Grupo de Física Teórica e Matemática Física, Departamento de Física, Universidade Federal Rural do Rio de Janeiro, BR 465-07, 23890-971, Seropédica, RJ, Brazil
b Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Xavier Sigaud 150, Urca, 22290-180, RJ, Brazil
c Departamento de Física, ICE, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil

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. Namely, for instance, in NC quantum mechanics we will show that $\theta^{ij}$ $(i,j=1,2,3)$ is an operator in Hilbert space and we will explore the consequences of this so-called “operationalization”. The DFRA formalism is constructed in an extended space-time with independent degrees of freedom associated with the object of noncommutativity $\theta^{\mu\nu}$. We will study the symmetry properties of an extended $x+\theta$ space-time, given by the group $\mathcal P'$, which has the Poincaré group $\mathcal P$ as a subgroup. The Noether formalism adapted to such extended $x+\theta$ $(D=4+6)$ space-time is depicted. 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. In this framework it is also possible to give dynamics to the NC operator sector, resulting in new features. 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. Next, we explain the dynamical spacetime symmetries in NC relativistic theories by using the DFRA algebra. 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 $\mathcal P'$. 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.

Keywords: noncommutativity; quantum mechanics; gauge theories

DOI: https://doi.org/10.3842/SIGMA.2010.083

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ArXiv: 1003.5322
MSC: 70S05; 70S10; 81Q65; 81T75
Received: March 28, 2010; in final form October 2, 2010; Published online October 10, 2010
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Citation: Everton M. C. Abreu, Albert C. R. Mendes, Wilson Oliveira, Adriano O. Zangirolami, “The Noncommutative Doplicher–Fredenhagen–Roberts–Amorim Space”, SIGMA, 6 (2010), 083, 37 pp.

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Abreu E.M.C., Amorim R., Guzman Ramirez W., “Noncommutative particles in curved spaces”, Journal of High Energy Physics, 2011, no. 3, 135
2. Abreu E.M.C., Neves M.J., “Green Functions in Lorentz Invariant Noncommutative Space-Time”, Int. J. Mod. Phys. A, 27:20 (2012), 1250109
3. Abreu E.M.C., Marcial M.V., Mendes A.C.R., Oliveira W., Oliveira-Neto G., “Noncommutative Cosmological Models Coupled to a Perfect Fluid and a Cosmological Constant”, J. High Energy Phys., 2012, no. 5, 144
4. Abreu E.M.C., Neves M.J., “Some Aspects of Quantum Mechanics and Field Theory in a Lorentz Invariant Noncommutative Space”, Int. J. Mod. Phys. A, 28:7 (2013), 1350017
5. Abreu E.M.C., Marcial M.V., Mendes A.C.R., Oliveira W., “Analytical and Numerical Analysis of a Rotational Invariant D=2 Harmonic Oscillator in the Light of Different Noncommutative Phase-Space Configurations”, J. High Energy Phys., 2013, no. 11, 138
6. Rasouli S.M.M., Ziaie A.H., Marto J., Moniz P.V., “Gravitational Collapse of a Homogeneous Scalar Field in Deformed Phase Space”, Phys. Rev. D, 89:4 (2014), 044028
7. Abreu E.M.C., Mendes A.C.R., Oliveira W., “N=2 Supersymmetric Radiation Damping Problem on a Noncommutative Plane”, Ann. Phys.-Berlin, 526:5-6 (2014), 227–234
8. Abreu E.M.C., Neves M.J., “Self-Quartic Interaction For a Scalar Field in An Extended Dfr Noncommutative Space-Time”, Nucl. Phys. B, 884 (2014), 741–765
9. Neves M.J., Abreu E.M.C., “Spontaneous symmetry breaking and masses numerical results in Doplicher-Fredenhagen-Roberts noncommutative space-time”, EPL, 114:2 (2016), 21001
10. Shi, T.; Kong, J.; Wang, X.; Liu, Z.; Xiong, J., “Improved Roberts operator for detecting surface defects of heavy rails with superior precision and efficiency”, High Technology Letters, 22:2 (2016), 207-214
11. Neves M.J., Abreu E.M.C., “The Standard Electroweak Model in the Noncommutative Dfr Space-Time”, Int. J. Mod. Phys. A, 32:33 (2017), 1750190
12. Abreu E.M.C., Neves M.J., “Strong Interaction Model in Dfr Noncommutative Space-Time”, Int. J. Mod. Phys. A, 32:17 (2017), 1750099
13. Abreu E.M.C., Neves M.J., Nikoofard V., “Classical Mechanics and Quantum Fields in Noncommutative Phase Spaces”, Acta Phys. Pol. B, 48:4 (2017), 773–792
14. Rasouli S.M.M., Saba N., Farhoudi M., Marto J., Moniz V P., “Inflationary Universe in Deformed Phase Space Scenario”, Ann. Phys., 393 (2018), 288–307
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