Mathematical Modeling of Finite Quantum Systems

TL;DR

This paper proposes a finite, symmetry-based mathematical framework for quantum systems, avoiding infinities and emphasizing the role of permutation invariants in quantum behavior and phenomena.

Contribution

It introduces a finite, constructive approach to quantum modeling, linking quantum phenomena to permutation symmetries and invariant subspaces.

Findings

Quantum dynamics can be embedded into permutation dynamics.

Interference arises from invariant subspaces of symmetry groups.

Observable quantities are expressed via permutation invariants.

Abstract

We consider the problem of quantum behavior in the finite background. Introduction of continuum or other infinities into physics leads only to technical complications without any need for them in description of empirical observations. The finite approach makes the problem constructive and more tractable. We argue that quantum behavior is a natural consequence of symmetries of dynamical systems. It is a result of fundamental impossibility to trace identity of indistinguishable objects in their evolution - only information about invariant combinations of such objects is available. We demonstrate that any quantum dynamics can be embedded into a simple permutation dynamics. Quantum phenomena, such as interferences, arise in invariant subspaces of permutation representations of the symmetry group of a system. Observable quantities can be expressed in terms of the permutation invariants.