H. Falomir

Testing spatial noncommutativity via the Aharonov-Bohm effect

The possibility of detecting noncommutative space relics is analyzed using the Aharonov-Bohm effect. We show that, if space is noncommutative, the holonomy receives nontrivial kinematical corrections that will produce a diffraction pattern even when the magnetic flux is quantized. The scattering problem is also formulated, and the differential cross section is calculated. Our results can be extrapolated to high energy physics and the bound

On the resolvent and spectral functions of a second order differential operator with a regular singularity

\ensuremath{\theta}\ensuremath{\sim}[10 \mathrm{TeV}{]}^{\ensuremath{-}2}

Pole structure of the Hamiltonian zeta-function for a singular potential

is found. If this bound holds, then noncommutative effects could be explored in scattering experiments measuring differential cross sections for small angles. The bound state Aharonov-Bohm effect is also discussed.

Unusual poles of the ζ-functions for some regular singular differential operators

We consider the resolvent of a second order differential operator with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents unusual powers of λ which depend on the singularity. The consequences for the pole structure of the ζ function, and for the small-t asymptotic expansion of the heat kernel, are also discussed.

Hamiltonian self-adjoint extensions for (2+1)-dimensional Dirac particles

We study the pole structure of the ζ-function associated with the Hamiltonian H of a quantum mechanical particle living in the half-line +, subject to the singular potential gx−2 + x2. We show that H admits nontrivial self-adjoint extensions (SAE) in a given range of values of the parameter g. The ζ-functions of these operators present poles which depend on g and, in general, do not coincide with half an integer (they can even be irrational). The corresponding residues depend on the SAE considered.

Magnetic-Dipole Spin Effects in Noncommutative Quantum Mechanics

We consider the resolvent of a system of first-order differential operators with a regular singularity, admitting a family of self-adjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents powers of λ which depend on the singularity, and can take even irrational values. The consequences for the pole structure of the corresponding ζ- and η-functions are also discussed.

Vortices, infrared effects and Lorentz invariance violation

We study the stationary problem of a charged Dirac particle in (2+1) dimensions in the presence of a uniform magnetic field B and a singular magnetic tube of flux Φ = 2πκ/e. The rotational invariance of this configuration implies that the subspaces of definite angular momentum l + 1/2 are invariant under the action of the Hamiltonian H. We show that for κ-l≥1 or κ-l≤0 the restriction of H to these subspaces, Hl, is essentially self-adjoint, while for 0<κ-l<1 Hl admits a one-parameter family of self-adjoint extensions (SAEs). In the latter case, the functions in the domain of Hl are singular (but square integrable) at the origin, their behaviour being dictated by the value of the parameter γ that identifies the SAE. We also determine the spectrum of the Hamiltonian as a function of κ and γ, as well as its closure.

Dirac operator on a disk with global boundary conditions

Abstract A general three-dimensional noncommutative quantum mechanical system mixing spatial and spin degrees of freedom is proposed. The analogous of the harmonic oscillator in this description contains a magnetic dipole interaction and the ground state is explicitly computed and we show that it is infinitely degenerated and implying a spontaneous symmetry breaking. The model can be straightforwardly extended to many particles and the main above properties are retained. Possible applications to the Bose–Einstein condensation with dipole–dipole interactions are briefly discussed.

Hidden superconformal symmetry of the spinless Aharonov-Bohm system

Abstract The Yang–Mills theory with non-commutative fields is constructed following Hamiltonian and Lagrangian methods. This modification of the standard Yang–Mills theory produces spatially localized solutions very similar to those of the standard non-Abelian gauge theories. This modification of the Yang–Mills theory contain in addition to the standard contribution, the term θ μ e μ ν ρ λ ( A ν F ρ λ + 2 3 A ν A ρ A λ ) where θ μ is a given space-like constant vector with canonical dimension of energy. The A μ field rescaling and the choice θ μ = ( 0 , 0 , 0 , θ ) , suggest the equivalence between the Yang–Mills–Chern–Simons theory in 2 + 1 dimensions and QCD in 3 + 1 dimensions in the heavy fermionic excitations limit. Thus, the Yang–Mills–Chern–Simons theory in 2 + 1 dimensions could be a codified way to QCD with only heavy quarks. The classical solutions of the modified Yang–Mills theory for the SU ( 2 ) gauge group are explicitly studied.

Divergencies in the Casimir energy for a medium with realistic ultraviolet behaviour

We compute the functional determinant for a Dirac operator in the presence of an Abelian gauge field on a bidimensional disk, under global boundary conditions of the type introduced by Atiyah–Patodi–Singer. We also discuss the connection between our result and the index theorem.