Revisiting Connes’ finite spectral distance on noncommutative spaces: Moyal plane and fuzzy sphere

Abstract

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.

Connes distance of 2D harmonic oscillators in quantum phase space*

“…Actually, some researchers have already studied the Connes distance between the states of one-dimensional (1D) harmonic oscillators. [14,18,20] But we find that the calculations and results of the Connes distance of 2D harmonic oscillators are much more complicated than those of 1D oscillator system. So it is significant to study the explicit formulae of the Connes distance of 2D harmonic oscillator systems.…”

“…[17] Scholtz and his collaborators have done many works on the studies of Connes spectral distances in Moyal plane and fuzzy sphere. [18][19][20] They developed the Hilbert-Schmidt operatorial formulation, and obtained the distances of harmonic oscillator states and also coherent states. Kumar et al used Dirac eigen-spinor method to compute spectral distances in doubled Moyal plane.…”

Fingerprints of the quantum space-time in time-dependent quantum mechanics: An emergent geometric phase

“…In this context, note that |z) ∈ H q (35) saturates the space-time uncertainty: ∆ T ∆ X = θ 2 , implying that such a state represents a maximally localized "point", or rather an event in space-time. In fact this state being a pure density matrix can be regarded as a pure state of the algebra Âθ (29) and plays the role of a point, represented by Dirac's delta functional, in the corresponding commutative algebra C ∞ (R 2 ) describing (1+1) D commutative plane [34,35]. It can also be checked that the basis |z, z) ≡ |z) satisfies the over-completeness property:…”

“…Of course, we need to emphasize again that here t and x should not be identified as time and space coordinate; they are just the coherent state expectation values, as given in (34).…”

“…Interestingly, this X θ and T θ can be related to commutative x and t , defined in (34), by making use of similarity transformations as,…”