Combinatorial Approach to Modeling Quantum Systems

TL;DR

This paper introduces a combinatorial framework for quantum systems, explaining quantum phenomena through invariance and indistinguishability, and deriving classical mechanics principles from quantum trajectories.

Contribution

It provides a constructive, combinatorial model of quantum behavior that naturally explains complex numbers, unitarity, and the emergence of classical mechanics from quantum evolution.

Findings

Quantum behavior explained by invariance and indistinguishability.

Derivation of least action principle from combinatorial quantum models.

Construction of models linking quantum trajectories to classical physics.

Abstract

Using the fact that any linear representation of a group can be embedded into permutations, we propose a constructive description of quantum behavior that provides, in particular, a natural explanation of the appearance of complex numbers and unitarity in the formalism of quantum mechanics. In our approach, the quantum behavior can be explained by the fundamental impossibility to trace the identity of indistinguishable objects in their evolution. Any observation only provides information about the invariant relations between such objects. The trajectory of a quantum system is a sequence of unitary evolutions interspersed with observations -- non-unitary projections. We suggest a scheme to construct combinatorial models of quantum evolution. The principle of selection of the most likely trajectories in such models via the large numbers approximation leads in the continuum limit to the principle of least action with the appropriate Lagrangians and deterministic evolution equations.