Self-adjoint extensions and SUSY breaking in Supersymmetric Quantum Mechanics

We consider the self-adjoint extensions (SAE) of the symmetric supercharges and Hamiltonian for a model of SUSY Quantum Mechanics in $\mathbb{R}^+$ with a singular superpotential. We show that only for two particular SAE, whose domains are scale invariant, the algebra of N=2 SUSY is realized, one with manifest SUSY and the other with spontaneously broken SUSY. Otherwise, only the N=1 SUSY algebra is obtained, with spontaneously broken SUSY and non degenerate energy spectrum.Comment: LaTeX. 23 pages and 1 figure (minor changes). Version to appear in the Journal of Physics A: Mat. and Ge

Spectral functions of non essentially selfadjoint operators

One of the many problems to which J.S. Dowker devoted his attention is the effect of a conical singularity in the base manifold on the behavior of the quantum fields. In particular, he studied the small-$t$ asymptotic expansion of the heat-kernel trace on a cone and its effects on physical quantities, as the Casimir energy. In this article we review some peculiar results found in the last decade, regarding the appearance of non-standard powers of $t$, and even negative integer powers of $\log{t}$, in this asymptotic expansion for the selfadjoint extensions of some symmetric operators with singular coefficients. Similarly, we show that the $\zeta$-function associated to these selfadjoint extensions presents an unusual analytic structure.Comment: 57 pages, 1 figure. References added. Version to appear in the special volume of Journal of Physics A in honor of Stuart Dowker's 75th birthday. PACS numbers: 02.30.Tb, 02.30.Sa, 03.65.D

Boundaries in the Moyal plane

We study the oscillations of a scalar field on a noncommutative disc implementing the boundary as the limit case of an interaction with an appropriately chosen confining background. The space of quantum fluctuations of the field is finite dimensional and displays the rotational and parity symmetry of the disc. We perform a numerical evaluation of the (finite) Casimir energy and obtain similar results as for the fuzzy sphere and torus.Comment: 19 pages, 6 figures. Replaced by published versio

On the algebraic structure of rotationally invariant two-dimensional Hamiltonians on the noncommutative phase space

We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras $sl(2,\mathbb{R})$ or $su(2)$ according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential. PACS: 03.65.-w; 03.65.Fd MSC: 81R05; 20C35; 22E70Comment: 49 pages. No figures. Version to appear in JP

Confined two-dimensional fermions at finite density

We introduce the chemical potential in a system of two-dimensional massless fermions, confined to a finite region, by imposing twisted boundary conditions in the Euclidean time direction. We explore in this simple model the application of functional techniques which could be used in more complicated situations.Comment: 15 pages, LaTe

Inflation without inflaton: a model for dark energy

The interaction between two initially causally disconnected regions of the Universe is studied using analogies of noncommutative quantum mechanics and the deformation of Poisson manifolds. These causally disconnect regions are governed by two independent Friedmann-Lemaître-Robertson-Walker (FLRW) metrics with scale factors a and b and cosmological constants Λa and Λb, respectively. The causality is turned on by positing a nontrivial Poisson bracket [Pα,Pβ]=ϵαβκG, where G is Newton's gravitational constant and κ is a dimensionless parameter. The posited deformed Poisson bracket has an interpretation in terms of 3-cocycles, anomalies, and Poissonian manifolds. The modified FLRW equations acquire an energy-momentum tensor from which we explicitly obtain the equation of state parameter. The modified FLRW equations are solved numerically and the solutions are inflationary or oscillating depending on the values of κ. In this model, the accelerating and decelerating regime may be periodic. The analysis of the equation of state clearly shows the presence of dark energy. By completeness, the perturbative solution for κ << 1 is also studied.Facultad de Ciencias Exacta

Massless fermions in a bag at finite density and temperature

We introduce the chemical potential in a system of massless fermions in a bag by impossing boundary conditions in the Euclidean time direction. We express the fermionic mean number in terms of a functional trace involving the Green's function of the boundary value problem, which we study analytically. Numerical evaluations are made, and an application to a simple hadron model is discussed.Comment: 14 pages, 3 figures, RevTe

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

We study the pole structure of the $\zeta$-function associated to the Hamiltonian $H$ of a quantum mechanical particle living in the half-line $\mathbf{R}^+$, subject to the singular potential $g x^{-2}+x^2$. We show that $H$ admits nontrivial self-adjoint extensions (SAE) in a given range of values of the parameter $g$. The $\zeta$-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.Comment: 12 pages, 1 figure, RevTeX. References added. Version to appear in Jour. Phys. A: Math. Ge

Remarks on Screening in a Gauge-Invariant Formalism

In this paper we display a direct and physically attractive derivation of the screening contribution to the interaction potential in the Chiral Schwinger model and generalized Maxwell-Chern-Simons gauge theory. It is shown that these results emerge naturally when a correct separation between gauge-invariant and gauge degrees of freedom is made. Explicit expressions for gauge-invariant fields are found.Comment: 13 pages, 1 figure, to appear in PR