Symmetry, Structure, and Emergent Subsystems

TL;DR

This paper explores how symmetries in quantum models determine the structure of the Hilbert space, enabling the identification of emergent subsystems that aid in understanding system dynamics and interpretation.

Contribution

It introduces a framework for using irreducible representations of symmetry groups to partition Hilbert spaces into meaningful subsystems without ontological assumptions.

Findings

Subsystems change as the Hamiltonian transitions from integrable to chaotic.

Irreducible representations can serve as natural units for interpreting quantum models.

Partitioning Hilbert space reveals insights into quantum kinematics and dynamics.

Abstract

Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of interpretation. For models with symmetry, the properties of irreducible representations constrain the possibilities of Hilbert space arithmetic, i.e.\ how a Hilbert space can be decomposed into sums of subspaces and factored into products of subspaces. Partitioning the Hilbert space is equivalent to parsing the system into subsystems, and these emergent subsystems provide insight into the kinematics, dynamics, and informatics of a quantum model. This article provides examples of how complex models can be built up from irreducible representations that correspond to `natural' ontological units like spins and particles. It also gives examples of the reverse process in which complex models are partitioned into subsystems that are selected by the representations of the symmetries and require no underlying ontological commitments. These techniques are applied to a few-body model in one-dimension with a Hamiltonian depending on an interaction strength parameter. As this parameter is tuned, the Hamiltonian runs dynamical spectrum from integrable to chaotic, and the subsystems relevant for analyzing and interpreting the dynamics shift accordingly.