We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW 2N+1 (Z N m n,Z N m n, Z m n) over the ring Z m n. This group is the discrete version of the real Heisenberg group HW 2N+1 (R N , R N , R), where R is the real field. These functions have one dimensional and m 1 , m 2 ,..., m n -dimensions matrix valued spectral components (for irreducible representations of HW 2N+1 . The family of all ID representations gives classical world world (CW). Various m i -dimension representations (i = 1,2,...,n) map classical world (CW) into quantum worlds QW(m i ) of ith resolution (i = 1,2,....n). Worlds QW(m 1 ) and QW(m n ) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.
We study classical and quantum harmonic analysis of phase space functions (classical observes) on finite Heisenberg group HW2N+1(ZNmn, ZNmn, Zmn) over the ring Zmn. This group is the discrete version of the real Heisenberg group HW2N+1(RN, RN, R), where R is the real field. These functions have one dimensional and m1, m2,...,mn-dimensions matrix valued spectral components (for irreducible representations of HW2N+1. The family of all 1D representations gives classical world world (CW). Various mi-dimension representations (i=1,2,...,n) map classical world (CW) into quantum worlds QW}mi of ith resolution i=1,2,...,n). Worlds QW(m1) and QW(mn) contain rough information and fine details about quantum word, respectively. In this case the Fourier transform on the Heisenberg group can be considered as Weyl quantization multiresolution procedure. We call this transform the natural quantum Fourier transform.
Relative to an irreducible representation of the canonical commutation relations, convolutions between quantum mechanical operators and between functions and operators are defined, for which the usual Weyl transform acts as a Fourier transform. Basic properties of these operations are developed in close analogy to harmonic analysis on R2n. Using the quantum version of Wiener’s approximation theorem, a natural one-to-one correspondence between the closed, phase-space translation invariant subspaces of classical and quantum observables is established.
The Heisenberg-Weyl group
HW(d) related to a
d-dimensional Hilbert space
H(d), is enlarged into the Heisenberg-Weyl-parity group
HWP(d) that incorporates parity transformations. It consists of
2d3 elements, of which
d3 elements belong to the
HW(d) subgroup, and extra
d3 elements which are related through a Fourier transform with the former ones. It is shown that
HWP(d) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed.
HWP(d) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.
2d2 coherent states related to the
HWP(d) group are introduced, which consist of
d2 coherent states related to the
HW(d) subgroup, and extra
d2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the
2d2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the
d2 coherent states related to
HW(d). One of the consequences of the
HWP(d) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.
: n-dimensional coherent states systems generated by translations, modulations, rotations and dilations are described. Starting from unitary irreducible representations of the n-dimen- sional affine Weyl-Heisenberg group, which are not square-integrable, one is led to consider systems of coherent states labeled by the elements of quotients of the original group. Such systems can yield a resolution of the identity, and then be used as alternatives to usual wavelet or windowed Fourier analysis. When the quotient space is the phase space of the representation, different embeddings of it into the group provide different descriptions of the phase space. R'esum'e: On d'ecrit des syst`emes d"etats coh'erents engendr'es par des translations, dilatations et rotations en n-dimensions. Partant de repr'esentations unitaires irr'eductibles du groupe de Weyl-Heisenberg affine en dimension n, qui ne sont pas de carr'e integrable, on est naturellement conduit `a consid'erer des syst`emes d"etats coh...