Beginning with a review of the existing literature on the computation of spectral distances on noncommutative spaces like Moyal plane and fuzzy sphere, adaptable to Hilbert–Schmidt operatorial formulation, we carry out a correction, revision and extension of the algorithm provided in [1] i.e. [F. G. Scholtz and B. Chakraborty, J. Phys. A, Math. Theor. 46 (2013) 085204] to compute the finite Connes’ distance between normal states. The revised expression, which we provide here, involves the computation of the infimum of an expression which involves the “transverse” [Formula: see text] component of the algebra element in addition to the “longitudinal” component [Formula: see text] of [1], identified with the difference of density matrices representing the states, whereas the expression given in [1] involves only [Formula: see text] and corresponds to the lower bound of the distance. This renders the revised formula less user-friendly, as the determination of the exact transverse component for which the infimum is reached remains a nontrivial task, but under rather generic conditions it turns out that the Connes’ distance is proportional to the Hilbert-Schmidt norm of [Formula: see text], leading to considerable simplification. In addition, we can determine an upper bound of the distance by emulating and adapting the approach of [P. Martinetti and L. Tomassini, Commun. Math. Phys. 323 (2013) 107–141]. We then look for an optimal element for which the upper bound is reached. We are able to find one for the Moyal plane through the limit of a sequence obtained by finite-dimensional projections of the representative of an element belonging to a multiplier algebra, onto the subspaces of the total Hilbert space, occurring in the spectral triple and spanned by the eigen-spinors of the respective Dirac operator. This is in contrast with the fuzzy sphere, where the upper bound, which is given by the geodesic of a commutative sphere, is never reached for any finite [Formula: see text]-representation of [Formula: see text]. Indeed, for the case of maximal noncommutativity ([Formula: see text]), the finite distance is shown to coincide exactly with the above-mentioned lower bound, with the transverse component playing no role. This, however, starts changing from [Formula: see text] onwards and we try to improve the estimate of the finite distance and provide an almost exact result, using our revised algorithm. The contrasting features of these types of noncommutative spaces becomes quite transparent through the analysis, carried out in the eigen-spinor bases of the respective Dirac operators.
We present here a novel method of computing spectral distances in doubled Moyal plane in a noncommutative geometrical framework using Dirac eigen-spinors, while solving the Lipschitz ball condition explicitly through matrices. The standard results of longitudinal, transverse and hypotenuse distances between different pairs of pure states have been computed and Pythagorean equality between them have been re-produced. The issue of non-unital nature of Moyal plane algebra is taken care of through a sequence of projection operators constructed from Dirac eigen-spinors, which plays a crucial role throughout this paper. At the end, a toy model of "Higgs field" has been constructed by fluctuating the Dirac operator and the variation on the transverse distance has been demonstrated, through an explicit computation.
The Hilbert-Schmidt operator formulation of non-commutative quantum mechanics in 2D Moyal plane is shown to allow one to construct Schwinger's SU(2) generators. Using this the SU(2) symmetry aspect of both commutative and non-commutative harmonic oscillator are studied and compared. Particularly, in the non-commutative case we demonstrate the existence of a critical point in the parameter space of mass(µ) and angular frequency(ω) where there is a manifest SU(2) symmetry for a unphysical harmonic oscillator Hamiltonian built out of commuting (unphysical yet covariantly transforming under SU(2)) position like observable. The existence of this critical point is shown to be a novel aspect in non-commutative harmonic oscillator, which is exploited to obtain the spectrum and the observable mass (µ) and angular frequency (ω) parameters of the physical oscillator-which is generically different from the bare parameters occurring in the Hamiltonian. Finally, we show that a Zeeman term in the Hamiltonian of non-commutative physical harmonic oscillator, is solely responsible for both SU(2) and time reversal symmetry breaking.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.