von Neumann Lattices in Finite Dimensions Hilbert Spaces

Details von Neumann Lattices in Finite Dimensions Hilbert Spaces von Neumann Lattices in Finite Dimensions Hilbert Spaces

2008-05-07

arxiv.org/abs/0805.0960v1

Physics

Quantum Physics

16 pages

Scientific paper

10.1103/PhysRevA.78.012101

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.

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