The possibility of detecting noncommutative space relics is analyzed using the Aharonov-Bohm effect. We show that, if space is noncommutative, the holonomy receives non-trivial 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 θ ∼ [10 TeV] −2 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.
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.
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 = 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, H l , is essentially self-adjoint, while for 0 < κ − l < 1 H l admits a one-parameter family of self-adjoint extensions (SAEs). In the latter case, the functions in the domain of H l 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.
We study the pole structure of the ζ-function associated to the Hamiltonian H of a quantum mechanical particle living in the half-line R + , subject to the singular potential gx −2 + x 2 . 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.
In this article we considered models of particles living in a
three-dimensional space-time with a nonstandard noncommutativity induced by
shifting canonical coordinates and momenta with generators of a unitary
irreducible representation of the Lorentz group. The Hilbert space gets the
structure of a direct product with the representation space, where we are able
to construct operators which realize the algebra of Lorentz transformations. We
study the modified Landau problem for both Schr\"odinger and Dirac particles,
whose Hamiltonians are obtained through a kind of non-Abelian Bopp's shift of
the dynamical variables from the ones of the usual problem in the normal space.
The spectrum of these models are considered in perturbation theory, both for
small and large noncommutativity parameters. We find no constraint between the
parameters referring to no-commutativity in coordinates and momenta but they
rather play similar roles. Since the representation space of the unitary
irreducible representations SL(2,R) can be realized in terms of spaces of
square-integrable functions, we conclude that these models are equivalent to
quantum mechanical models of particles living in a space with an additional
compact dimension.Comment: PACS: 03.65.-w; 11.30.Cp; 02.40.Gh, 19 pages, no figures. Version to
appear in Physical Review
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