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arXiv:0805.0960 · Physical Review A

von Neumann lattices in finite-dimensional Hilbert spaces

M. Revzen, F. C. Khanna

Published Jul 8, 2008• 1 citation
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Abstract

The prime number decomposition of a finite-dimensional Hilbert space reflects itself in the representations that the space accommodates. The representations appear in conjugate pairs for factorization to two relative prime factors which can be viewed as two distinct degrees freedom. These, Schwinger's quantum degrees of freedom, are uniquely related to a von Neumann lattices in the phase space that characterizes the Hilbert space and specifies the simultaneous definitions of both (modular) positions and (modular) momenta. The area in phase space for each quantum state in each of these quantum degrees of freedom, is shown to be exactly $h$, Planck's constant.

DOI
10.1103/physreva.78.012101

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The prime number decomposition of a finite-dimensional Hilbert space reflects itself in the representations that the space accommodates. The representations appear in conjugate pairs for factorization to two relative prime factors which can be viewed as two distinct degrees freedom. These, Schwinger's quantum degrees of freedom, are uniquely related to a von Neumann lattices in the phase space that characterizes the Hilbert space and specifies the simultaneous definitions of both (modular) positions and (modular) momenta. The area in phase space for each quantum state in each of these quantum degrees of freedom, is shown to be exactly hhh, Planck's constant.
A Hilbert space in M dimensions is shown explicitly to accommodate representations that reflect the decomposition of M into prime numbers. Representations that exhibit the factorization of M into two relatively prime numbers: the kq representation (Zak J 1970 Phys. Today 23 51), and related representations termed q 1 q 2 representations (together with their conjugates) are analysed, as well as a representation that exhibits the complete factorization of M . In this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up M .
We propose to address in a natural manner, the modular variable concept, which was introduced in 1969 by Aharonov, Pendleton and Petersen to explain certain non-local properties of Quantum Mechanics. We have argued elsewhere, with the use of some elementary number theoretic properties, that a nite dimensional tensor product of two spaces with relatively prime dimensions cannot be regarded as two truely separate degrees of freedom, but eectively, only as one single degree of freedom. We show in this present work that a physical application of this remarkable mathematical fact can be that of clarifying the notion of modular variables in Quantum Mechanics.
The Ndistinct prime numbers that make up a composite number M allow its bi-partitioning into pairs of two relatively prime factors. Each such pair defines a pair of conjugate representations. An example of such pairs of conjugate representations, each of which spans the M-dimensional space, are the kq representations, which are the most natural representations for periodic systems. Here, we emphasize their relevance to factorizations: the number of prime numbers that make up M relates directly to the number of conjugate pairs of kq representations. It is also shown how Schwinger's factorization procedure is used for reducing M-dimensional space into subspaces according to the prime numbers that make up M.
In 1960 Schwinger (J Schwinger 1960 Proc. Natl Acad. Sci. 46 570–9) proposed the algorithm for factorization of unitary operators in the finite M -dimensional Hilbert space according to a coprime decomposition of M . Using a special permutation operator A we generalize the Schwinger factorization to every decomposition of M . We obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as Schwinger's. We apply the new factorization to two problems. First, we show how to generate two kq -like mutually unbiased bases for any composite dimension. Then, using a Harper-like Hamiltonian model in the finite dimension M = M 1 M 2 , we show how to design a physical system with M 1 energy levels, each having degeneracy M 2 .
In 1960 Schwinger [J. Schwinger, Proc.Natl.Acad.Sci. 46 (1960) 570- 579] proposed the algorithm for factorization of unitary operators in the finite M dimensional Hilbert space according to a coprime decomposition of M. Using a special permutation operator A we generalize the Schwinger factorization to every decomposition of M. We obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as Schwinger's. We apply the new factorization to two problems. First, we show how to generate two kq-like mutually unbiased bases for any composite dimension. Then, using a Harper-like Hamiltonian model in the finite dimension M = M1M2, we show how to design a physical system with M1 energy levels, each having degeneracy M2.
Weyl's displacement operators for position and momentum commute if the product of the elementary displacements equals Planck's constant. Then, their common eigenstates constitute the Zak basis, each state specified by two phase parameters. Upon enforcing a periodic dependence on the phases, one gets a one-to-one mapping of the Hilbert space on the line onto the Hilbert space on the torus. The Fourier coefficients of the periodic Zak bases make up the discrete Zak bases. The two bases are mutually unbiased. We study these bases in detail, including a brief discussion of their relation to Aharonov's modular operators, and mention how they can be used to associate with the single degree of freedom of the line a pair of genuine qubits.