Symmetries and Transformations

Weiguang Cao, Linhao Li, Masahito Yamazaki

Lattice non-invertible symmetries have rich fusion structures and play important roles in understanding various exotic topological phases. In this paper, we explore methods to generate new lattice non-invertible transformations/symmetries from a given non-invertible seed transformation/symmetry. The new lattice non-invertible symmetry is constructed by composing the seed transformations on different sites or sandwiching a unitary transformation between the transformations on the same sites. In addition to known non-invertible symmetries with fusion algebras of Tambara-Yamagami \mathbb Z_N×\mathbb Z_N ℤ N × ℤ N type, we obtain a new non-invertible symmetry in models with \mathbb Z_N ℤ N dipole symmetries. We name the latter the dipole Kramers-Wannier symmetry because it arises from gauging the dipole symmetry. We further study the dipole Kramers-Wannier symmetry in depth, including its topological defect, its anomaly and its associated generalized Kennedy-Tasaki transformation.

Edgard Elbaz

David Wallace

This chapter provides a fairly systematic analysis of when quantities that are variant under a dynamical symmetry transformation should be regarded as unobservable, or redundant, or unreal; of when models related by a dynamical symmetry transformation represent the same state of affairs; and of when mathematical structure that is variant under a dynamical symmetry transformation should be regarded as surplus. In most of these cases, the answer is ‘it depends’: that is, it depends on the details of the symmetry in question. A central feature of the analysis is that in order to draw any of these conclusions for a dynamical symmetry, it needs to be understood in terms of its possible extensions to other physical systems, in particular to measurement devices.

M. Sozzi

This chapter reviews the general theory of transformations in quantum theory. Topics covered include symmetry and invariance, transformations and symmetries in quantum theory, symmetries and conservation laws, discrete transformations and inversions, and symmetry violations. Suggested further reading and exercise problems are provided at the end of the chapter.

K. Tamvakis