Generalized Wigner theorem for non-invertible symmetries

TL;DR

This work resolves the tension between Wigner's theorem and non-invertible (generalized) symmetries by showing that any such symmetry preserving transition probabilities must be a partial isometry realized as ${D} = \mathcal{U} \tilde{P}$ on an enlarged gauged Hilbert space, with ${\mathcal{U}}$ unitary or antiunitary and $\tilde{P}$ positive semidefinite and non-invertible but acting as the identity on the original space. The authors develop a two-pronged proof via polar decomposition, treating linear and antilinear cases, and demonstrate the construction explicitly in a minimally gauged transverse-field Ising chain (TFIC) where a ${ m Z}_2$ gauge field yields a gauge-enlarged Hamiltonian $H_G$ and projection operators $\tilde{P}_\pm$ implementing the non-invertible symmetry. The results show that non-invertible symmetries are, in essence, partial isometries linked to dualities in an enlarged Hilbert space, with their invertibility controlled by boundary conditions. This reframes physical states as equivalence classes in a gauged bundle and connects generalized symmetries to gauge-theoretic and duality structures, offering a robust framework for incorporating non-invertible symmetries into quantum theory.

Abstract

We establish the conditions under which a conservation law associated with a non-invertible operator may be realized as a symmetry in quantum mechanics. As established by Wigner, all quantum symmetries must be represented by either unitary or antiunitary transformations. Relinquishing an implicit assumption of invertibility, we demonstrate that the fundamental invariance of quantum transition probabilities under the application of symmetries mandates that all non-invertible symmetries may only correspond to {\it projective} unitary or antiunitary transformations, i.e., {\it partial isometries}. This extends the notion of physical states beyond conventional rays in Hilbert space to equivalence classes in an {\it extended, gauged Hilbert space}, thereby broadening the traditional understanding of symmetry transformations in quantum theory. We discuss consequences of this result and explicitly illustrate how, in simple model systems, whether symmetries be invertible or non-invertible may be inextricably related to the particular boundary conditions that are being used.