Proceedings Article | 28 March 2005
Proc. SPIE. 5813, Multisensor, Multisource Information Fusion: Architectures, Algorithms, and Applications 2005
KEYWORDS: Fourier transforms, Quantization, Quantum mechanics, Space operations, Matrices, Quantum information, Mechanics, Particles, Dynamical systems
We study classical and quantum harmonic analysis of phase space
functions (classical observes) on finite Heisenberg group HW<sub>2N+1</sub>(<b>Z</b><i><sup>N</sup><sub>m</sub>n</i>, <b>Z</b><i><sup>N</sup><sub>m</sub>n</i>, <b>Z</b><i><sub>m</sub>n</i>) over the ring <b>Z</b><i><sub>m</sub>n</i>. This group is the discrete version of the real Heisenberg group HW<sub>2N+1</sub>(<b>R</b><sup>N</sup>, <b>R</b><sup>N</sup>, <b>R</b>), where <b>R</b> is the real field. These functions have one dimensional and <i>m<sup>1</sup>, m<sup>2</sup>,...,m<sup>n</sup></i>-dimensions matrix valued spectral components (for irreducible representations of HW<sub>2N+1</sub>. The family of all 1D representations gives classical world world (<b>CW</b>). Various <i>m<sup>i</sup></i>-dimension representations (<i>i</i>=1,2,...,<i>n</i>) map classical world (<b>CW</b>) into quantum worlds <b>QW</b>}<i>m<sup>i</sup></i> of <i>i</i>th resolution <i>i</i>=1,2,...,<i>n</i>). Worlds <b>QW</b>(<i>m<sup>1</sup></i>) and <b>QW</b>(<i>m<sup>n</sup></i>) 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.
