Inequivalent Quantization in the Field of a Ferromagnetic Wire

Abstract

We argue that it is possible to bind neutral atom (NA) to the ferromagnetic wire (FW) by inequivalent quantization of the Hamiltonian. We follow the well known von Neumann's method of self-adjoint extensions (SAE) to get this inequivalent quantization, which is characterized by a parameter Σ ∈ R(mod2π). There exists a single bound state for the coupling constant η 2 ∈ [0, 1). Although this bound state should not occur due to the existence of classical scale symmetry in the problem. But since quantization procedure breaks this classical symmetry, bound state comes out as a scale in the problem leading to scaling anomaly. We also discuss the strong coupling region η 2 < 0, which supports bound state making the problem re-normalizable.

Non-commutativity as a measure of inequivalent quantization

“…The ground state, therefore, has no lower bound, implying that it does not have any bound state. It is, however, known from some physical problems, for example binding of an electron in a polar molecule [5], the near horizon states of a black hole [6] and other [7][8][9][10] that inverse-square potential can bind particles. The theoretical interpretation of this binding can be obtained in terms of nontrivial quantization, which can be obtained by the von Neumann method of self-adjoint extensions.…”

“…. The solution of (10) has to be matched with (16) to get the relation of the non-commutativity parameter with the self-adjoint extension parameter . We see that there is exactly one bound state with the non-commutativity, 2 , and eigenfunction, ψ NC , being of the form…”

Localization at the energy threshold in non-commutative space

“…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 understood from the huge research works carried out so far, which in some stage can be described by an inverse square potential.…”

“…The importance of the inverse square potential in theoretical physics can be understood from the huge research works carried out so far, which in some stage can be described by an inverse square potential. Its presence is investigated in detail in molecular physics [52], atomic physics [50,52], black hole physics and mathematical physics. It shows that inverse square potential possesses bound state solution due to the scaling anomaly caused by quantization.…”