Generalized Wigner theorem for noninvertible symmetries

Journal Article · Thu Mar 12 00:00:00 EDT 2026 · Physical Review. B

DOI:https://doi.org/10.1103/cg2d-mqq3· OSTI ID:3028664

Ortiz, Gerardo ^[1]; Giridhar, Chinmay ^[2]; Vojta, Philipp ^[3]; Nevidomskyy, Andriy H. ^[4]; Nussinov, Zohar ^[5]

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1. Indiana Univ., Bloomington, IN (United States); Inst. for Advanced Study, Princeton, NJ (United States)
2. Rice Univ., Houston, TX (United States)
3. Washington Univ., St. Louis, MO (United States)
4. Rice Univ., Houston, TX (United States); Brookhaven National Laboratory (BNL), Upton, NY (United States). Condensed Matter Physics
5. Washington Univ., St. Louis, MO (United States); Technische Universitat Chemnitz (Germany); Indian Inst. of Technology (IIT), Madras (India)

In this article, 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 projective unitary or antiunitary transformations, i.e., partial isometries. This extends the notion of physical states beyond conventional rays in Hilbert space to equivalence classes in an 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 Organization: Rice Univ., Houston, TX (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Basic Energy Sciences (BES). Materials Sciences & Engineering Division (MSE)
• Grant/Contract Number: SC0025047
• Other Award/Contract Number: PHY2210452
• OSTI ID: 3028664
• Journal Information: Physical Review. B, Journal Name: Physical Review. B; ISSN 2469-9969; ISSN 2469-9950
• Publisher: American Physical Society (APS)
• Country of Publication: United States
• Language: English

Related Subjects

75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
FOS: Physical sciences
High Energy Physics - Theory (hep-th)
Mathematical Physics (math-ph)
Quantum Physics (quant-ph)
Statistical Mechanics (cond-mat.stat-mech)
Strongly Correlated Electrons (cond-mat.str-el)