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.

The issue of thermal correlation functions and the associated effective statistical potential in twodimensional Moyal space, arising in the twisted approach to implement rotational symmetry, has been revisited in an operatorial formulation where no explicit star product is used initially. The corresponding results using Moyal and Voros star products are then easily obtained by taking the corresponding overlap with Moyal and Voros bases. in contrast to the Moyal case where the concept of distance and, in particular, the relative separation between a pair of particles remain ambiguous when the Moyal star product is used, the Voros basis is more physical and the inter-particle distance can be introduced unambiguously. The forms of the correlation function and the effective potential are found to be same as the Moyal case except that the thermal wavelength undergoes a non-commutative deformation, ensuring that it has a lower bound of the order of √ θ. It is shown that in a suitable basis (called here quasi-commutative basis) in the multiparticle sector the thermal correlation function coincides with the commutative result both in the Moyal and Voros cases along with the restoration of the Pauli principle, except that in the Voros case the thermal wavelength, again, gets a non-commutative correction. Finally, we extend our result to three-dimensional non-commutative space and compute the correlation function and effective potential using both twisted and quasi-commutative bases in the Moyal and Voros cases. We find that there is SO(3) → SO(2) symmetry breaking in the effective potential, which also violates the Pauli principle, even for a pair of free particles, despite the fact that a deformed co-product is used to construct twisted symmetric/anti-symmetric basis. However, this SO(3) symmetry, along with Pauli principle, is restored once we use the quasi-commutative bases.

An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the su(2) algebra. This has been computed for both the discrete and the Perelemov’s SU(2) coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by n ∈ ℤ/2.

An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the su(2) algebra. This has been computed for both the discrete, as well as for the Perelemov's SU (2) coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by n ∈ Z/2.