On the quantum dynamics of non-commutative systems

This is a review paper concerned with the global consistency of the quantum dynamics of non-commutative systems. Our point of departure is the theory of constrained systems, since it provides a unified description of the classical and quantum dynamics for the models under investigation. We then elaborate on recently reported results concerned with the sufficient conditions for the existence of the Born series and unitarity and turn, afterwards, into analyzing the functional quantization of non-commutative systems. The compatibility between the operator and the functional approaches is established in full generality. The intricacies arising in connection with the explicit computation of path integrals, for the systems under scrutiny, is illustrated by presenting the detailed calculation of the Feynman kernel for the non-commutative two dimensional harmonic oscillator.Comment: 19 pages, title changed, version to be published in Brazilian Journal of Physic

The three-dimensional noncommutative Gross-Neveu model

This work is dedicated to the study of the noncommutative Gross-Neveu model. As it is known, in the canonical Weyl-Moyal approach the model is inconsistent, basically due to the separation of the amplitudes into planar and nonplanar parts. We prove that if instead a coherent basis representation is used, the model becomes renormalizable and free of the aforementioned difficulty. We also show that, although the coherent states procedure breaks Lorentz symmetry in odd dimensions, in the Gross-Neveu model this breaking can be kept under control by assuming the noncommutativity parameters to be small enough. We also make some remarks on some ordering prescriptions used in the literature.Comment: 10 pages, IOP article style; v3: revised version, accepted for publication in J. Phys.

Canonical Quantization of the Maxwell-Chern-Simons Theory in the Coulomb Gauge

The Maxwell-Chern-Simons theory is canonically quantized in the Coulomb gauge by using the Dirac bracket quantization procedure. The determination of the Coulomb gauge polarization vector turns out to be intrincate. A set of quantum Poincar\'e densities obeying the Dirac-Schwinger algebra, and, therefore, free of anomalies, is constructed. The peculiar analytical structure of the polarization vector is shown to be at the root for the existence of spin of the massive gauge quanta.The Coulomb gauge Feynman rules are used to compute the M\"oller scattering amplitude in the lowest order of perturbation theory. The result coincides with that obtained by using covariant Feynman rules. This proof of equivalence is, afterwards, extended to all orders of perturbation theory. The so called infrared safe photon propagator emerges as an effective propagator which allows for replacing all the terms in the interaction Hamiltonian of the Coulomb gauge by the standard field-current minimal interaction Hamiltonian.Comment: 21 pages, typeset in REVTEX, figures not include

Whirling Waves and the Aharonov-Bohm Effect for Relativistic Spinning Particles

The formulation of Berry for the Aharonov-Bohm effect is generalized to the relativistic regime. Then, the problem of finding the self-adjoint extensions of the (2+1)-dimensional Dirac Hamiltonian, in an Aharonov-Bohm background potential, is solved in a novel way. The same treatment also solves the problem of finding the self-adjoint extensions of the Dirac Hamiltonian in a background Aharonov-Casher

Lorentz symmetry breaking in the noncommutative Wess-Zumino model: One loop corrections

In this paper we deal with the issue of Lorentz symmetry breaking in quantum field theories formulated in a non-commutative space-time. We show that, unlike in some recente analysis of quantum gravity effects, supersymmetry does not protect the theory from the large Lorentz violating effects arising from the loop corrections. We take advantage of the non-commutative Wess-Zumino model to illustrate this point.Comment: 9 pages, revtex4. Corrected references. Version published in PR

The coupling of fermions to the three-dimensional noncommutative CPN−1CP^{N-1} model: minimal and supersymmetric extensions

We consider the coupling of fermions to the three-dimensional noncommutative $CP^{N-1}$ model. In the case of minimal coupling, although the infrared behavior of the gauge sector is improved, there are dangerous (quadratic) infrared divergences in the corrections to the two point vertex function of the scalar field. However, using superfield techniques we prove that the supersymmetric version of this model with ``antisymmetrized'' coupling of the Lagrange multiplier field is renormalizable up to the first order in $\frac{1}{N}$. The auxiliary spinor gauge field acquires a nontrivial (nonlocal) dynamics with a generation of Maxwell and Chern-Simons noncommutative terms in the effective action. Up to the 1/N order all divergences are only logarithimic so that the model is free from nonintegrable infrared singularities.Comment: Minor corrections in the text and modifications in the list of reference

Superfield covariant analysis of the divergence structure of noncommutative supersymmetric QED4_4

Commutative supersymmetric Yang-Mills is known to be renormalizable for ${\cal N} = 1, 2$, while finite for ${\cal N} = 4$. However, in the noncommutative version of the model (NCSQED$_4$) the UV/IR mechanism gives rise to infrared divergences which may spoil the perturbative expansion. In this work we pursue the study of the consistency of NCSQED$_4$ by working systematically within the covariant superfield formulation. In the Landau gauge, it has already been shown for ${\cal N} = 1$ that the gauge field two-point function is free of harmful UV/IR infrared singularities, in the one-loop approximation. Here we show that this result holds without restrictions on the number of allowed supersymmetries and for any arbitrary covariant gauge. We also investigate the divergence structure of the gauge field three-point function in the one-loop approximation. It is first proved that the cancellation of the leading UV/IR infrared divergences is a gauge invariant statement. Surprisingly, we have also found that there exist subleading harmful UV/IR infrared singularities whose cancellation only takes place in a particular covariant gauge. Thus, we conclude that these last mentioned singularities are in the gauge sector and, therefore, do not jeopardize the perturbative expansion and/or the renormalization of the theory.Comment: 36 pages, 11 figures. Minor correction

Born series and unitarity in noncommutative quantum mechanics

This paper is dedicated to present model independent results for noncommutative quantum mechanics. We determine sufficient conditions for the convergence of the Born series and, in the sequel, unitarity is proved in full generality.Comment: 9 page

Noncommutative quantum mechanics: uniqueness of the functional description

The generalized Weyl transform of index $\alpha$ is used to implement the time-slice definition of the phase space path integral yielding the Feynman kernel in the case of noncommutative quantum mechanics. As expected, this representation for the Feynman kernel is not unique but labeled by the real parameter $\alpha$. We succeed in proving that the $\alpha$-dependent contributions disappear at the limit where the time slice goes to zero. This proof of consistency turns out to be intricate because the Hamiltonian involves products of noncommuting operators originating from the non-commutativity. The antisymmetry of the matrix parameterizing the non-commutativity plays a key role in the cancelation mechanism of the $\alpha$-dependent terms.Comment: 13 page