Fuzzy spheres from inequivalent coherent states quantizations

Abstract

We present a new procedure which allows a coherent state (CS) quantization of any set with a measure. It is manifest through the replacement of classical observables by CS quantum observables, which acts on a Hilbert space of prescribed dimension N . The algebra of CS quantum observables has the finite dimension N 2 . The application to the 2-sphere provides a family of inequivalent CS quantizations, based on the spin spherical harmonics (the CS quantization from usual spherical harmonics appears to give a trivial issue for the cartesian coordinates). We compare these CS quantizations to the usual (Madore) construction of the fuzzy sphere. The difference allows us to consider our procedures as the constructions of new type of fuzzy spheres. The very general character of our method suggests applications to construct fuzzy versions of a variety of sets. Some ideas on quantizationA classical description of a set of data, say X, is usually carried out by considering sets of real or complex functions on X. Depending on the context (data handling, signal analysis, mechanics. . . ) the set X will be equipped with a definite structure (topological space, measure space, symplectic manifold. . . ) and the set of functions on X which will be considered as classical observables must be restricted with regard to the structure on X; for instance, signals should be square integrable with respect to the measure assigned to set X.How to provide instead a "quantum description" of the same set X? As a first characteristic, the latter replaces -this is a definition -the classical observables by quantum observables, which do not commute in general. As usual, these quantum observables will be realized as operators acting on some Hilbert space H, whose projective version will be considered as the set of quantum states. This Hilbert space will be constructed as a subset in the set of functions on X.The advantage of the coherent states (CS) quantization procedure, in a standard sense [1,2,3] as in recent generalizations [4] and applications [5] is that it requires a minimal significant structure on X, namely the only existence of a measure µ(dx), together with a σ-algebra of measurable subsets. As a measure space, X will be given the name of an observation set in the present context, and the existence of a measure provides us with a statistical reading of the set of measurable real or complex valued functions on X: computing for instance average values on subsets with bounded measure. The quantum states will correspond to measurable and square integrable functions on the set X, but not all square integrable functions are eligible as quantum states. The construction of H is equivalent to the choice of a class of eligible quantum states, together with a technical condition of continuity. This provides a correspondence between classical and quantum observables by defining a generalization of the so-called coherent states.Although the procedure appears mathematically as a quantization, it may also be considered as a change of point ...

The physical meaning of the de Sitter invariants

“…is written with the help of the five-dimensional vector-operator W (ρ) whose components read [7][8][9][10] …”

“…The best example is the de Sitter spacetime [5,6] which has SO (1,4) isometries and, moreover, allows one to analytically solve the equations of the usual matter fields variously coupled to the de Sitter gravity. These opportunities encouraged many authors to develop either a de Sitter quantum mechanics [7][8][9][10] or a quantum field theory [11][12][13][14][15] in an axiomatic manner, exploiting the high symmetry of this manifold but avoiding the standard steps of the canonical theory.…”

“…[7][8][9][10] where the tetrad-gauge covariance is neglected focusing only on the usual linear representations of the isometry group. In this way one obtains a quantum mechanics on the five-dimensional Minkowski manifold in which the de Sitter spacetime is embedded.…”

“…Unfortunately, these representations do not play the main role in our gauge-covariant field theory since this is a four-dimensional theory in the non-holonomic frames of the de Sitter spacetime where the CRs of the isometry group are induced by the finite-dimensional representations of the gauge group. For this reason we believe that the study of the Casimir operators of our CRs could complete the analyze of the de Sitter invariants offered by the five-dimensional theory [7][8][9][10].…”

Infinite quantum well: A coherent state approach

“…The approach via coherent state quantization Coherent state quantization [11,12,13,14,15,16,17] is an alternative way of representing classical observables into a quantum system. The states used in it include Glauber and Perelomov CS's but lie in a wider definition that admits a large range of state families resolving the identity.…”