In the author's opinion, two main features are characteristic for modern applications of mathematics to physics: the use of symmetries and the use of unobservable quantities (antioperationalistic approach). He gives a brief history of symmetry in physics: first, symmetry as a tool for describing useful features of given theories, then symmetry as a tool for restricting possible equations (e.g. the Dirac equation was thus obtained) and, finally, symmetry as a creative tool allowing one to obtain equations almost uniquely (Yang-Mills, etc.). Whenever equations are invariant with respect to some group, in the case that our observation data are invariant with respect to some subgroup $G$ , one cannot tell the state $x$ from the state $gx$ , where $g\\in G$ , so the parameters telling $x$ from $gx$ become unobservable---of this kind are potentials in electrodynamics, coordinate effects in general relativity, etc. This unobservability can be ultimate (e.g. phase of wave function in quantum theory) or only approximate---in all other theories where symmetry is only approximate. \n \n \nIn the author's opinion, two main features are characteristic for modern applications of mathematics to physics: the use of symmetries and the use of unobservable quantities (antioperationalistic approach). He gives a brief history of symmetry in physics: first, symmetry as a tool for describing useful features of given theories, then symmetry as a tool for restricting possible equations (e.g. the Dirac equation was thus obtained) and, finally, symmetry as a creative tool allowing one to obtain equations almost uniquely (Yang-Mills, etc.). Whenever equations are invariant with respect to some group, in the case that our observation data are invariant with respect to some subgroup $G$, one cannot tell the state $x$ from the state $gx$, where $g\\in G$, so the parameters telling $x$ from $gx$ become unobservable---of this kind are potentials in electrodynamics, coordinate effects in general relativity, etc. This unobservability can be ultimate (e.g. phase of wave function in quantum theory) or only approximate---in all other theories where symmetry is only approximate. \n \n \n

Feb 1, 19870 citations

Observables, Symmetries and Constraints

Michael J. W. Hall, Marcel Reginatto

Fundamental theories of physicsJan 1, 20160 citations

Symmetry reduction induced by anyon condensation: A tensor network approach

José Garre-Rubio, S. Iblisdir, David Pérez-Garcı́a

Topological ordered phases are related to changes in the properties of their quasiparticle excitations (anyons). We study these relations in the framework of projected entanglement pair states and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states. The case of ${\mathbb{Z}}_{2}$ on-site symmetry is studied in detail.Topological ordered phases are related to changes in the properties of their quasiparticle excitations (anyons). We study these relations in the framework of projected entanglement pair states and show how condensing and confining anyons reduces a local gauge symmetry to a global on-site symmetry. We also study the action of this global symmetry over the quasiparticle excitations. As a byproduct, we observe that this symmetry reduction effect can be applied to one-dimensional systems as well and brings about appealing physical interpretations on the classification of phases with symmetries using matrix product states. The case of ${\mathbb{Z}}_{2}$ on-site symmetry is studied in detail.

arXiv:1702.08759Physical review. B./Physical review. BOct 16, 201715 citations

Quantum Physics: States, Observables and Their Time Evolution

Arno Böhm, Piotr Kielanowski, G. B. Mainland

Nov 6, 20190 citations

Symmetry structure dependent diagnostic of the Quantum Mpemba Effect

Tamizhselvan S, Manju C, Bijay Kumar Agarwalla +1 more

Understanding symmetry restoration in isolated quantum many-body systems is an important problem in nonequilibrium many-body quantum physics. Recent studies have shown that the quantum Mpemba effect can be characterized through entanglement asymmetry, where states with stronger initial symmetry breaking restore symmetry faster. However, it remains unclear whether conventional energy-based measures, such as the trace distance, capture the same phenomenon. We investigate this question in closed spin- $1/2$ quantum systems with different symmetries by analyzing the dynamics of symmetry-breaking initial states. Combining numerical simulations with an analytical decomposition of the trace distance into symmetry-coherence and residual contributions, we identify the conditions under which trace distance tracks entanglement asymmetry and reproduces the Mpemba--like behavior observed in it. For charge symmetry, the residual contribution is negligible, making the trace distance effectively governed by symmetry-sector coherences. In contrast, for permutation symmetry, a significant residual contribution leads to qualitatively different relaxation dynamics. Our results establish when conventional energy-based diagnostics reliably capture symmetry-restoration dynamics and clarify the distinct physical information encoded by entanglement asymmetry and trace distance.Understanding symmetry restoration in isolated quantum many-body systems is an important problem in nonequilibrium many-body quantum physics. Recent studies have shown that the quantum Mpemba effect can be characterized through entanglement asymmetry, where states with stronger initial symmetry breaking restore symmetry faster. However, it remains unclear whether conventional energy-based measures, such as the trace distance, capture the same phenomenon. We investigate this question in closed spin-$1/2$ quantum systems with different symmetries by analyzing the dynamics of symmetry-breaking initial states. Combining numerical simulations with an analytical decomposition of the trace distance into symmetry-coherence and residual contributions, we identify the conditions under which trace distance tracks entanglement asymmetry and reproduces the Mpemba--like behavior observed in it. For charge symmetry, the residual contribution is negligible, making the trace distance effectively governed by symmetry-sector coherences. In contrast, for permutation symmetry, a significant residual contribution leads to qualitatively different relaxation dynamics. Our results establish when conventional energy-based diagnostics reliably capture symmetry-restoration dynamics and clarify the distinct physical information encoded by entanglement asymmetry and trace distance.

quant-pharXiv:2609.10470Sep 10, 20260 citations

G

, one cannot tell the state

x

from the state

gx

, where

g\\in G

, so the parameters telling

x

from

gx

become unobservable---of this kind are potentials in electrodynamics, coordinate effects in general relativity, etc. This unobservability can be ultimate (e.g. phase of wave function in quantum theory) or only approximate---in all other theories where symmetry is only approximate. \n \n \n

{\mathbb{Z}}_{2}

on-site symmetry is studied in detail.

1/2

quantum systems with different symmetries by analyzing the dynamics of symmetry-breaking initial states. Combining numerical simulations with an analytical decomposition of the trace distance into symmetry-coherence and residual contributions, we identify the conditions under which trace distance tracks entanglement asymmetry and reproduces the Mpemba--like behavior observed in it. For charge symmetry, the residual contribution is negligible, making the trace distance effectively governed by symmetry-sector coherences. In contrast, for permutation symmetry, a significant residual contribution leads to qualitatively different relaxation dynamics. Our results establish when conventional energy-based diagnostics reliably capture symmetry-restoration dynamics and clarify the distinct physical information encoded by entanglement asymmetry and trace distance.