Wavelets in mathematical physics:q-oscillators

Abstract

We construct representations of a q-oscillator algebra by operators on Fock space on positive matrices. They emerge from a multiresolution scaling construction used in wavelet analysis. The representations of the Cuntz Algebra arising from this multiresolution analysis are contained as a special case in the Fock Space construction.In this paper we establish a connection between multiresolution wavelet analysis on one hand and representation theory for operator on Hilbert spaces depending on a real parameter on the other. These operators arise from a multiresolution wavelet analysis based on Bessel functions. We wish to develop a framework for the study of creation operators on Hilbert space, satisfying simple identities, and allowing a Hopf algebra structure. Examples will include oscillator algebras coming from physical models.In the first section of the paper, we review the background and the motivation for the study of the q-relations, both as it relates to problems in 1 mathematics and in physics. On the mathematical side, the problems concern wavelet analysis and transform theory, especially the Mellin transform, and on the physics side, they relate to the quon gas of statistical mechanics. For the construction of the representations, we then turn to the twisted Fock space and the q-oscillator algebra. Our approach is motivated by wavelet analysis, and it uses a certain loop group. Our main result is Theorem 6.

On Fractional Brownian Motion and Wavelets

“…The Cuntz algebra O N enters the picture as follows: formula (20) is not practical for computations, and the analysis of orthogonality relations is done better by reference to the Cuntz relations, see (15)- (16). Setting, for ξ j ∈ C, j ∈ Z…”

“…will yield a representation of the Cuntz relations; see (15)- (16). Conversely, if (22) is given to satisfy the Cuntz relations, then the matrix in (21) takes values in U N (C).…”

“…Later papers on this theme include [3][4][5][6] and references therein, to mention only a few. We will have occasion to use the following references regarding C * -algebras [7][8][9][10][11][12][13][14][15][16]; on multiresolutions and wavelets [4,10,[17][18][19][20][21][22][23][24][25][26][27][28][29]; and results from harmonic analysis, [30][31][32][33][34][35][36][37][38]; on Hilbert and reproducing kernels [3,[39][40][41][42]; on Markov measures [1,2,5,6,11,[43][44]…”

Radial Multiresolution, Cuntz Algebras Representations and an Application to Fractals

“…The operators monomials from (25) may be realized as multiplication operators on l 2 (Z). Thus an operator in the family (25) can be represented as multiplication by the indicator function of the set…”

“…Previously studied representations of the Cuntz algebras dependent on a parameter as [25] can be used towards an extension of our considerations to the construction of corresponding random walks on quantum groups.…”