Perturbative Renormalization in Quantum Mechanics

Some quantum mechanical potentials, singular at short distances, lead to ultraviolet divergences when used in perturbation theory. Exactly as in quantum field theories, but much simpler, regularization and renormalization lead to finite physical results, which compare correctly to the exact ones. The Dirac delta potential, because of its relevance to triviality, and the Aharonov-Bohm potential, because ot its relevance to anyons, are used as examples here.Comment: LATEX, 13 pages, UB-ECM-PF 19-9

Second Virial Coefficient of Anyons without Hard Core

We calculate the second virial coefficient of anyons whose wave function does not vanish at coincidence points. This kind of anyons appear naturally when one generalizes the hard-core boundary condition according to self-adjoint extension method in quantum mechanics, and also when anyons are treated field theoretically by applying renormalization procedure to nonrelativistic Chern-Simons field theory. For the anyons which do not satisfy hard-core boundary condition, it is argued that the other scale-invariant limit is more relevant in high-temperature limit where virial expansion is useful. Furthermore, the cusp existing at the bosonic point for hard-core anyons disappears in all the other cases; instead it is shown that a new cusp is generated at the fermionic point. A physical explanation is given.Comment: 12 pages, RevTeX, One figure. Minor changes. A few references adde

Finite-size anyons and perturbation theory

We address the problem of finite-size anyons, i.e., composites of charges and finite radius magnetic flux tubes. Making perturbative calculations in this problem meets certain difficulties reminiscent of those in the problem of pointlike anyons. We show how to circumvent these difficulties for anyons of arbitrary spin. The case of spin 1/2 is special because it allows for a direct application of perturbation theory, while for any other spin, a redefinition of the wave function is necessary. We apply the perturbative algorithm to the N-body problem, derive the first-order equation of state and discuss some examples.Comment: 18 pages (RevTex) + 4 PS figures (all included); a new section on equation of state adde

Hidden nonlinear su(2|2) superunitary symmetry of N=2 superextended 1D Dirac delta potential problem

We show that the N=2 superextended 1D quantum Dirac delta potential problem is characterized by the hidden nonlinear $su(2|2)$ superunitary symmetry. The unexpected feature of this simple supersymmetric system is that it admits three different $\mathbb Z_2$-gradings, which produce a separation of 16 integrals of motion into three different sets of 8 bosonic and 8 fermionic operators. These three different graded sets of integrals generate two different nonlinear, deformed forms of $su(2|2)$, in which the Hamiltonian plays a role of a multiplicative central charge. On the ground state, the nonlinear superalgebra is reduced to the two distinct 2D Euclidean analogs of a superextended Poincar\'e algebra used earlier in the literature for investigation of spontaneous supersymmetry breaking. We indicate that the observed exotic supersymmetric structure with three different $\mathbb Z_2$-gradings can be useful for the search of hidden symmetries in some other quantum systems, in particular, related to the Lam\'e equation.Comment: 11 pages; comments and refs. added, version published in PL

Non-Abelian Chern-Simons Particles in an External Magnetic Field

The quantum mechanics and thermodynamics of SU(2) non-Abelian Chern-Simons particles (non-Abelian anyons) in an external magnetic field are addressed. We derive the N-body Hamiltonian in the (anti-)holomorphic gauge when the Hilbert space is projected onto the lowest Landau level of the magnetic field. In the presence of an additional harmonic potential, the N-body spectrum depends linearly on the coupling (statistics) parameter. We calculate the second virial coefficient and find that in the strong magnetic field limit it develops a step-wise behavior as a function of the statistics parameter, in contrast to the linear dependence in the case of Abelian anyons. For small enough values of the statistics parameter we relate the N-body partition functions in the lowest Landau level to those of SU(2) bosons and find that the cluster (and virial) coefficients dependence on the statistics parameter cancels.Comment: 35 pages, revtex, 3 eps figures include

Fucoid brown algae inject fucoidan carbon into the ocean

Peer reviewe

On the scattering amplitude in the Aharonov-Bohm gauge field

A general expression for the scattering amplitude of nonrelativistic spinless particles in the Aharonov-Bohm gauge potential is obtained within the time independent formalism. The result is valid also in the backward and forward directions as well as for any choice of the boundary conditions on the wave function at the flux tube position.Comment: 18 pages, plain TE

Green functions for generalized point interactions in 1D: A scattering approach

Recently, general point interactions in one dimension has been used to model a large number of different phenomena in quantum mechanics. Such potentials, however, requires some sort of regularization to lead to meaningful results. The usual ways to do so rely on technicalities which may hide important physical aspects of the problem. In this work we present a new method to calculate the exact Green functions for general point interactions in 1D. Our approach differs from previous ones because it is based only on physical quantities, namely, the scattering coefficients, $R$ and $T$, to construct $G$. Renormalization or particular mathematical prescriptions are not invoked. The simple formulation of the method makes it easy to extend to more general contexts, such as for lattices of $N$ general point interactions; on a line; on a half-line; under periodic boundary conditions; and confined in a box.Comment: Revtex, 9 pages, 3 EPS figures. To be published in PR

Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons

The quantum nonrelativistic spin-1/2 planar systems in the presence of a perpendicular magnetic field are known to possess the N=2 supersymmetry. We consider such a system in the field of a magnetic vortex, and find that there are just two self-adjoint extensions of the Hamiltonian that are compatible with the standard N=2 supersymmetry. We show that only in these two cases one of the subsystems coincides with the original spinless Aharonov-Bohm model and comes accompanied by the super-partner Hamiltonian which allows a singular behavior of the wave functions. We find a family of additional, nonlocal integrals of motion and treat them together with local supercharges in the unifying framework of the tri-supersymmetry. The inclusion of the dynamical conformal symmetries leads to an infinitely generated superalgebra, that contains several representations of the superconformal osp(2|2) symmetry. We present the application of the results in the framework of the two-body model of identical anyons. The nontrivial contact interaction and the emerging N=2 linear and nonlinear supersymmetries of the anyons are discussed.Comment: 18 pages, 1 figure, published versio