Quantum models based on finite groups

TL;DR

This paper proposes a constructive approach to quantum mechanics using finite groups and permutation representations, aiming to clarify physical concepts while preserving empirical accuracy.

Contribution

It introduces a method to reformulate quantum formalism with finite groups, replacing continuous groups and providing a clearer conceptual framework.

Findings

Quantum formalism can be reconstructed using finite group representations.

Permutation representations enable formulation of quantum problems in finite group terms.

The approach preserves the empirical content of standard quantum mechanics.

Abstract

We consider a constructive modification of quantum-mechanical formalism. Replacement of a general unitary group by unitary representations of finite groups makes it possible to reproduce quantum formalism without loss of its empirical content. Since any linear representation of a finite group can be implemented as a subrepresentation of a permutation representation, quantum-mechanical problems can be formulated in terms of permutation groups. Reproducing quantum behavior in the framework of permutation representations of finite groups makes it possible to clarify the meaning of a number of physical concepts.