Quantification of Symmetry

TL;DR

This paper introduces a continuous measure called degree of symmetry (DoS) using group theory, enabling nuanced quantification of symmetry and symmetry breaking in physical systems.

Contribution

It proposes a novel, computable continuous metric for symmetry, overcoming the traditional binary perspective, and links it to group theoretical properties.

Findings

DoS effectively detects accidental degeneracy.

DoS characterizes spontaneous symmetry breaking.

The measure is computable via irreducible representations.

Abstract

Symmetry is conventionally described in a contrariety manner that the system is either completely symmetric or completely asymmetric. Using group theoretical approach to overcome this dichotomous problem, we introduce the degree of symmetry (DoS) as a non-negative continuous number ranging from zero to unity. DoS is defined through an average of the fidelity deviations of Hamiltonian or quantum state over its transformation group G, and thus is computable by making use of the completeness relations of the irreducible representations of G. The monotonicity of DoS can effectively probe the extended group for accidental degeneracy while its multi-valued natures characterize some (spontaneous) symmetry breaking.