Generalized Wigner theorem for non-invertible symmetries

We establish the conditions under which a conservation law associated with a non-invertible operator may be realized as a symmetry in quantum physics. 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. Our generalized theorem applies irrespective of the origin of the (non)invertible symmetry, holds in arbitrary spatial dimensions, and is independent of the Hamiltonian or action. We explore its physical consequences and, using simple model systems, illustrate how the distinction between invertible and non-invertible symmetries can sometimes be tied to the choice of boundary conditions.

💡 Research Summary

The authors revisit one of the most fundamental results in quantum theory—Wigner’s theorem, which states that any transformation preserving transition probabilities between quantum states must be implemented by a unitary or anti‑unitary operator. The theorem implicitly assumes that the symmetry operator is invertible, an assumption that is violated by the growing class of “non‑invertible” or categorical symmetries that appear in modern condensed‑matter and high‑energy contexts (e.g., topological quantum field theories, duality defects, and higher‑form symmetries). The paper asks: can such non‑invertible symmetries coexist with the probability‑conservation principle that underlies quantum mechanics?

To answer this, the authors formulate a generalized Wigner theorem that does not require invertibility. They first enlarge the physical Hilbert space ( \mathcal H ) to a “gauged” Hilbert space ( \widetilde{\mathcal H}= \mathcal H \oplus \mathcal H^\perp ). On this extended space any symmetry transformation ( \tilde D ) that leaves all transition probabilities invariant must be of the form \