Generalized Wigner theorem for non-invertible symmetries
Authors
Gerardo Ortiz
Chinmay Giridhar
Philipp Vojta
Andriy H. Nevidomskyy
Zohar Nussinov
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.
