Quantum Groups, Coherent States, Squeezing and Lattice Quantum Mechanics

Abstract

By resorting to the Fock-Bargmann representation, we incorporate the quantum Weyl-Heisenberg algebra, q-WH, into the theory of entire analytic functions. The q-WH algebra operators are realized in terms of finite difference operators in the z plane. In order to exhibit the relevance of our study, several applications to different cases of physical interest are discussed: squeezed states and the relation between coherent states and theta functions on one side, lattice quantum mechanics and Bloch functions on the other, are shown to find a deeper mathematical understanding in terms of q-WH. The rôle played by the finite difference operators and the relevance of the lattice structure in the completeness of the coherent states system suggest that the quantization of the WH algebra is an essential tool in the physics of discretized (periodic) systems.

On the coherent behavior of pancreatic beta cell clusters

“…We recall indeed that squeezed coherent states with different squeezing parameters (different slopes in Figs. 3 and 6) belong to different dynamical regimes (phases) [24]. It is interesting to observe that the linear behavior in log-log PDS/frequency plots in specific intervals of frequencies found in the present system, also appears in brain studies where assemblies (clusters) of large number of neurons with phase locked amplitude modulated oscillations and self-similarity properties are observed as well [25,26,27,28,29,30,21].…”

“…Note that |qα is actually a squeezed coherent state [24,16] with ζ = ln q the squeezing parameter. Thus, q N acts in F as the squeezing operator.…”

“…Putting q = e ζ , with ζ a complex number, the q-deformed coherent state |qα is obtained by applying q N [24], called the fractal operator [21,20], to |α…”

Classical q-deformed dynamics

“…In fact it has been shown that it is possible to obtain a "coordinate" realization of the Fock space of the q-oscillators by using the deformed Jackson derivative (JD) or the so-called q-derivative operator [6,7,8].…”

Deformed quantum mechanics andq-Hermitian operators

“…From the seminal work of Biedenharn [2] and Macfarlane [3], it was clear that the q-calculus, originally introduced in the study of the basic hypergeometric series [4,5,6], plays a central role in the representation of the quantum groups with a deep physical meaning and not merely a mathematical exercise. Many physical applications have been investigated on the basis of the q-deformation of the Heisenberg algebra [7,8,9,10,11]. In Ref.…”