Hilbert space of the free coherent states and p – adic numbers

Rigged Hilbert space of the free coherent states is investigated. We prove that this rigged Hilbert space is isomorphous to the space of generalized functions over p–adic disk. We discuss the relation of the described isomorphism of rigged Hilbert spaces and noncommutative geometry and show, that the considered example realises the isomorphism of the noncommutative line and p–adic disk. In the present paper, continuing the investigations of [1], [2], we investigate the free coherent states (or shortly FCS), which are (unbounded) eigenvectors of the linear combination of annihilators in the free Fock space. In [1], [2] it was shown that the space of the free coherent states is highly degenerate for the fixed eigenvalue λ (and infinite dimensional), and this degeneracy is naturally described by the space D ′ (Z p) of generalized functions on p–adic disk (p is a number of independent creators in the free Fock space). In the present paper we reformulate the results of [1], [2] using the language of rigged Hilbert spaces and propose an interpretation of the relation between the free coherent states and p–adics using noncommutative geometry. We speculate that the isomorphism between the space of FCS and the space of generalized function on p–adic disk in the language of noncommutative geometry reduces to the isomorphism between the noncommutative (or quantum) line and p–adic disk. p–Adic mathematical physics studies the problems of mathematical physics with the help of p–adic analysis. p–Adic mathematical physics was studied in [3]–[15]. For instance in the book [3] the analysis of p–adic pseudodifferential operators was developed. In [4] a p–adic approach in the string theory was proposed. In [9] a theory of p–adic valued distributions was investigated. In [12], [13] it was shown that the Parisi matrix used in the replica method is equivalent, in the simplest case, to a p–adic pseudodifferential operator. In [14] it was shown that the wavelet basis in L 2 (R) after the p–adic change of variable (the continuous map of p–adic numbers onto real numbers conserving the measure) maps onto the basis of eigenvectors of the Vladimirov operator of p–adic fractional derivation. In [15] a procedure to generate the ultrametric space used in the replica approach was proposed.