M.Sc Thesis
M.Sc Student Simkhovich Boris Factorization Properties of Finite Spaces Department of Physics Professor Emeritus Michael Revzen Professor Ady Mann Professor Emeritus Joshua Zak

Abstract

A finite phase space of dimension M, where coordinate and momentum have M possible values, is a frequent component of various physical and mathematical problems. In this work we consider problems of Fast Fourier Transform, Schwinger factorization of unitary operators and generation of Zak bases which all relate to a finite phase space. In particular, we consider the possibility of performing operations in M=M1M2 - dimensional phase space as acting in factorized dimensions M1 and M2. For coprime numbers M1 and M2 all the three listed problems have solutions. The FFT problem was solved by Good and achieves fast operation by factorization of the M - dimensional Discrete Fourier Transform (DFT) into smaller M1 and M2 - dimensional DFTs. The problem of factorization of unitary operators was solved by Schwinger through the use of the Fermat-Euler theorem. The two mutually unbiased Zak bases in the coprime case can be generated using the two complete sets of commuting operators. In our work we show that all these solutions are interconnected by the use of the Chinese Remainder Theorem from number theory. For non-coprime numbers M1 and M2, only in the case of the Fast Fourier Transform there exists the solution proposed by Cooley and Tukey. The solution is based on the use of “conjugate” Division Algorithm Representations for coordinate and momentum variables. Following this solution, we propose a permutation operator, which transforms a single coordinate variable between “conjugate” Division Algorithm Representations of it. This special choice of the permutation operator allows us to provide the solution for the Schwinger factorization of unitary operators for all possible factorizations of M. We derive complementary pairs of unitary operators, each pair describing one of the factorized dimensions M1 or M2. Then we apply them to the Harper-like Hamiltonian model. We generalize the idea of previous work for energy spectrum design. For each possible factorization of the dimension M=M1M2 we propose Harper-like Hamiltonians which have M1 energy levels, each M2 degenerate. Such energy spectra control may be of interest for practical realization in solid state physics for electrons in magnetic field. We also apply the new factorization to the problem of Zak bases and generate a pair of mutually unbiased bases, one of which is the original Zak basis and the other is a Zak-like basis. The new Zak-like basis has particular quasi-periodicity properties in comparison to the original Zak bases.