Localization at the energy threshold in non-commutative space

Abstract

The ground state energy of a scale symmetric system usually does not possess any lower bound, thus making the system quantum mechanically unstable. Self-adjointness and renormalization techniques usually provide the system a scale and thus making the ground state bounded from below. We on the other hand use noncommutative quantum mechanics and exploit the noncommutative parameter Θ as a scale for a scale symmetric system. The resulting Hamiltonian for the system then allows an unusual bound state at the threshold of the energy, E = 0. Apart from the Hamiltonian b H we also compute the other two generators of the so(2, 1) algebra, the dilation b D and the conformal generator b K in the noncommutative space. The so(2, 1) algebra is not closed in the noncommutative space, but the limit Θ → 0 smoothly goes to the so(2, 1) algebra restoring the conformal symmetry. We also discuss the system for large noncommutative parameter. The study of physics in noncommutative spacetime [1,2] or only in noncommutative space [3,4,5] has become an independent field of research work for a long time. It started with the work of Snyder [6,7], where Electromagnetic theory is considered in noncommutative spacetime. It is a well known fact that the coordinates of a plane become noncommutative when the quantum mechanical system in a magnetic field (perpendicular to plane) background is confined in lowest Landau level. However non-commutativity was present in theoretical physics before the concept of noncommutativity in spacetime or in space coordinates was introduced. For example, the canonically conjugate operators like coordinate x i and its conjugate momenta p i are noncommutative x i , p i = i , which leads to the uncertainty principle ∆x i ∆p i ≥ /2 in quantum mechanics. On the other hand although the different momentum components do commute, it is known that the components of a generalized momenta in the background magnetic field,k . Noncommutativity and its effect is studied in diverse fields starting from Quantum Field Theory [8,9,10,11,12,13,14], String theory [15,16,17,18] to quantum mechanics [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. In quantum mechanical context several models are studied in noncommutative space. The list includes harmonic oscillator [43], Hydrogen atom problem [44,45], Zeeman effect and Stark [44] effect. Even the effect of noncommutative space is studied for a general central potential and solutions are obtained in large noncommutative limit [3]. It is known that usually the inclusion of spacetime noncommutativity destroys the uniterity of a system but that can be restored by a different formulation of noncommutativity of Doplicher et at [46,47] The present letter is concerned with a scale invariant * Electronic address: pulakranjan.giri@saha.ac.in system in non-commutative space. In particular we consider a particle on a plane (2D) with an inverse square potential [48,49,50,51,52,53,54,55]. The importance of the inverse square potential in theoretical physics can be understo...

Non-commutativity as a measure of inequivalent quantization

“…In the present paper, we extend our discussion of [4] further and obtain a generic boundary condition for the zero-energy localized state. The paper is organized in the following fashion: first, we consider the inverse-square interaction on a plane and discuss briefly how it changes when the coordinates of the plane become non-commutative.…”

“…However, the length scale, , introduced in the problem due to the non-commutativity can be exploited to heal the ultraviolet divergence of the problem under study. In a recent paper [4], we investigated the inversesquare problem, H = p 2 + αr −2 , in non-commutative space in order to show how the length scale can be successfully used to regularize the problem. Since the inverse-square problem does not possess any dimensional parameter to start with, it is a scale-invariant problem.…”

“…for E NC = 0 and found a bound state with angular momentum m for ξ = √ α + m 2 > 1 [4]. For large values of the non-commutative parameter, , it is also possible to get the expectation values of the Hamiltonian.…”

“…Due to this inverse behavior of the eigenstate, we impose a nontrivial boundary condition for our problem at r = ∞. The operator H NC is essentially self-adjoint for ξ 2 1 which has been discussed in detail in [4]. Since any system is defined by a Hamiltonian and its corresponding domain, in our case H NC for ξ 2 1 acts on the domain…”