ar X iv : q - a lg / 9 70 30 15 v 1 3 M ar 1 99 7 Free coherent states and distributions on p - adic numbers

Free coherent states for a system with two degrees of freedom is defined. An isomorphism of the space of distributions on 2-adic disc and the space of free coherent states is constructed. 1 Construction of free (or Boltzmannian) coherent states Free (or Boltzmannian) Fock space has been considered in some recent works on quantum chromodynamics [1], [2], [3] and noncommutative probability [4], [5]. The subject of this work is free coherent states. We will introduce free coherent states and investigate the space of coherent states corresponding to a fixed eigenvalue of the operator of annihilation. In the paper [6] the subset of the set of free coherent states was constructed. The main result of the paper [6] is the construction of the homeomorphism from the ring of integer 2-adic numbers to the subset of the space of free coherent states with topology defined by the Hilbert metric. The result of the present paper is the generalization of the construction of the paper [6]. We introduce the space of coherent states. The main result of the present paper is the construction of the isomorphism of the space of coherent states to the space of distributions on the ring of integer 2-adic numbers. Coherent states introduced in the paper [6] correspond under this isomorphism to δ-functions. We will consider the system with two degrees of freedom. The system with one degree of freedom was investigated in [7]. The free commutation relations are particular case of q-deformed relations