A DIAGONAL REPRESENTATION OF QUANTUM DENSITY MATRIX USING q-BOSON OSCILLATOR COHERENT STATES

A q-analogue of Sudarshan’s diagonal representation of the Quantum Mechanical density matrix is obtained using q-boson coherent states. Earlier result of Mehta and Sudarshan on the self reproducing property of ρ(z, z) is also generalized and a self-consistent self-reproducing kernel K̃(z, z) is constructed. e-mail addresses: [email protected] ; [email protected] 1 A diagonal representation of the quantum mechanical density matrix using standard bosonic oscillator coherent states has been given by Sudarshan [1]. One of the remarkable features of this representation is that for any normal ordered operator, its ”average expectation value”, becomes the same as that of the classical function for a probability distribution over the complex plane. So, the classical complex representation and the quantum mechanical density matrices are in one-to-one correspondence. The emergence of quantum groups in the study of inverse scattering [2] resulted in q-Boson [3] and q-Fermion [4] oscillators and many usual quantum mechanical descriptions find their appropriate q-analogues [5]. It is possible that q-oscillators represent an effective way of dealing with non-ideal systems in which interactions are incorporated [6]. It is not obvious at present, whether nature makes use of q-Bosons (or q-Fermions) as some kind of nonlinear excitations of the electromagnetic (or electron) fields. Nevertheless, it is worthwhile to examine the q-anologue of important quantum mechanical descriptions. It is the purpose here to find a q-anologue of the diagonal representation of the Quantum Mechanical density matrix. Although Nelson and Fields [7] have outlined a q-anologue of the diagonal representation of the density matrix, our results, while in agreement with them, offer further insight into the problem, such as the relationship between q-coherent state matrix elements and Fock space matrix elements, self-reproducing property and self reproducing kernel. The q-Boson creation a and annihilation a operators satisfy aa − qaa = 1, ; [N, a] = −a, [N, a] = a, (1) where N( 6= aa) is the number operator. The q-Boson vacuum state | 0 > is an element of Fock Space F and a | 0 >= 0; < 0 | 0 >= 1. The n q-Boson normalized state | n > is given by | n >= 1