von Neumann Lattices in Finite Dimensions Hilbert Spaces

TL;DR

This paper explores the structure of finite-dimensional Hilbert spaces through prime decomposition, revealing how von Neumann lattices in phase space encode quantum degrees of freedom with areas equal to Planck's constant.

Contribution

It demonstrates the connection between prime factorization of Hilbert spaces and von Neumann lattices, linking quantum degrees of freedom to phase space geometry.

Findings

Quantum states occupy areas of exactly h in phase space.

Hilbert space representations decompose into conjugate pairs related to prime factors.

Von Neumann lattices characterize modular position and momentum in finite dimensions.

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.