Generalization of Wigner's theorem on symmetries in the C*-algebraic approach

Gerardo Ortíz, Chinmay Giridhar, Philipp Vojta +2 more

Aritra K. Mukhopadhyay

L. Bracci, G. Morchio, Franco Strocchi

Chinmay Giridhar, Philipp Vojta, Zohar Nussinov +2 more

The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's original paradigm. Here, we develop an algebraic framework for systematically deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries, and focus on a (2+1)-dimensional quantum Ising plaquette model as a concrete illustration. We show that bond-algebraic automorphisms, when combined with the so-called half-gauging procedure, reveal the structure of the non-invertible duality symmetry operators, which can be explicitly represented as a sequential quantum circuit. The resulting duality defects are constrained by the model's rigid higher symmetries (lower-dimensional subsystem symmetries), leading to restricted mobility. We establish the fusion algebra of these defects. Finally, in constructing the non-invertible duality transformation, we explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.

R. Simon, Narasimhaiengar Mukunda, S. Chaturvedi +1 more

Huzihiro Araki

Abstract In this chapter we explain how we can describe quantum theory within the general framework given in Chapter 1. We do not discuss various theories concerning the question of how quantum theory can (or cannot) be deduced from experimental facts or from a verifiable axiomatic system. Instead we start by introducing the assumptions of quantum theory. The main content is the algebraic viewpoint of quantum theory based on the theory of operator algebras, the GNS construction which supplies the fundamental relation between the description of the states by Hilbert space vectors and the description of the states in this general framework, the Wigner theorem which is basic for the mathematical description of quantum mechanical symmetries, and its generalizations. The basic definitions and theorems about Hilbert spaces and operator algebras are collected in the appendix.