---
title: Non-Invertible Time-Reversal Symmetry
url: https://www.emergentmind.com/topics/non-invertible-time-reversal-symmetry
type: topic
---

# Non-Invertible Time-Reversal Symmetry

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Non-invertible time-reversal symmetry is a generalization of ordinary time-reversal in which the symmetry operation reverses time or implements an anti-linear transformation, yet fails to admit an inverse in the usual group-theoretic sense. In the recent literature, the term refers to several related but distinct structures. In gauge theory, a non-invertible time-reversal symmetry is a conserved anti-linear topological operator whose fusion with its adjoint yields a nontrivial condensation defect rather than the identity, and which exists at rational values of the $\theta$-angle such as $\theta=\pi p/N$ [2208.04331]. In defect and categorical settings, anti-unitary symmetries may satisfy algebras such as $\mathsf T^2=C$ or $\mathsf T^2=(-1)^F C$, so that fusion with the orientation-reversed defect produces a sum of sectors rather than a single inverse [2308.11706]. In open quantum systems, a distinct notion of non-invertibility appears when a time-reversal-symmetric Lindblad-type evolution acquires a factor $\mathrm{sgn}(t)$: the resulting dynamical map is non-invertible through $t=0$, even though the equation remains fully time-reversal invariant [2311.08486]. These developments place non-invertible time-reversal at the intersection of higher-form symmetry, topological defects, anomaly inflow, generalized Wigner structures, and nonequilibrium quantum dynamics.

## 1. Definition and basic algebra

In a theory with a $2\pi$-periodic $\theta$-angle, ordinary time-reversal is usually represented only at $\theta=0$ or $\theta=\pi$ by an anti-linear, invertible operator $\mathcal T$ satisfying $\mathcal T^\dagger \mathcal T=1$ and an algebra such as $\mathcal T^2=\pm1$ or $(-1)^F$ [2208.04331]. The non-invertible case departs from this by replacing the anti-unitary operator with an anti-linear topological operator that still commutes with the Hamiltonian and flips the sign of the time coordinate, but is not invertible. At rational $\theta=\pi p/N$, one may construct an anti-linear topological operator $\mathcal T^{\theta=\pi p/N}$ that is conserved and time-reversing, yet obeys
$$(\mathcal T^{\theta=\pi p/N})^\dagger \times \mathcal T^{\theta=\pi p/N}=\mathcal C^{(N)}\neq 1,$$
where $\mathcal C^{(N)}$ is a condensation defect [2208.04331]. This failure of $\mathcal T^\dagger \mathcal T$ to equal the identity is the defining hallmark of non-invertibility in the gauge-theory setting.

A related formulation appears in defect categories. In $2+1$-dimensional topological orders equipped with charge conjugation $C$, one can adjoin an antiunitary defect $\mathsf T$ with algebra
$$\mathsf T^2=C \quad (\text{bosonic}), \qquad \mathsf T^2=(-1)^F C \quad (\text{fermionic}),$$
so that
$$\mathsf T\times\overline{\mathsf T}=1+C \quad (\text{bosonic}), \qquad \mathsf T\times\overline{\mathsf T}=1+(-1)^F C \quad (\text{fermionic}),$$
rather than a single identity sector [2308.11706]. Here non-invertibility is encoded directly in fusion.

A further formulation arises from a generalized Wigner perspective. If the requirement of bijectivity is dropped while preserving transition probabilities, then a non-invertible symmetry may be realized as
$$D=\mathcal U\,P,$$
where $\mathcal U$ is unitary or antiunitary and $P$ is a positive semi-definite, non-invertible Hermitian operator, with $P$ acting as the identity on the physical subspace [2509.25327]. In operator-theoretic terms, such a $D$ is a partial isometry satisfying
$$D^\dagger D=P,\qquad D D^\dagger = Q,$$
with $P$ and $Q$ orthogonal projections [2509.25327]. Specializing to time-reversal gives a non-invertible antiunitary transformation of the form $\Theta=U K\,P$ with $P^2=P$, $P^\dagger=P$, and $[P,H]=0$ [2509.25327]. This formulation is conceptually distinct from topological-defect constructions, but it gives a quantum-mechanical template for non-invertible anti-linear symmetry.

## 2. Gauge-theory construction at rational $\theta$

The canonical gauge-theoretic construction appears in free Maxwell theory and massive QED at rational $\theta$-angle [2208.04331]. For free Maxwell theory,
$$L=-\frac{1}{2e^2}F\wedge \star F+\frac{\theta}{8\pi^2}F\wedge F,\qquad F=dA,\qquad \theta=\pi p/N.$$
The naive time reversal $K$ acts anti-linearly by
$$K:\ t\to -t,\quad A_0(t,x)\to -A_0(-t,x),\quad A_i(t,x)\to A_i(-t,x),\quad \theta\to -\theta.$$
Because this flips $\theta$, it fails to preserve the theory at generic rational $\theta$. The remedy is a codimension-one topological interface $\mathcal I_{2\pi p/N}$ that shifts $\theta\to\theta-2\pi p/N$, realized by a $\nu=p/N$ fractional quantum Hall state on the interface $M$:
$$\mathcal I_{2\pi p/N}=\exp\!\Bigl[i\oint_M \mathcal L_{2\pi p/N}\Bigr],$$
with
$$\mathcal L_{2\pi p/N}[A,a]=\frac{pN}{4\pi}a\wedge da+\frac{1}{2\pi}a\wedge dA,$$
where $a$ is a dynamical $U(1)$ one-form on $M$ [2208.04331].

The resulting time-reversal defect is the composition
$$\mathcal T^{\theta=\pi p/N}\equiv K\circ \mathcal I_{2\pi p/N}.$$
This operator is topological and anti-linear, flips $t\to -t$, and inserts the FQH layer. Its non-invertibility is expressed by the fusion relation
$$(\mathcal T^{\theta=\pi p/N})^\dagger\times \mathcal T^{\theta=\pi p/N}=\mathcal C^{(N)}\neq 1,$$
with $\mathcal C^{(N)}$ the condensation defect obtained by summing over magnetic one-form symmetry lines on $M$ [2208.04331].

The same work extends the construction beyond abelian Maxwell theory. In massive QED at $\theta=\pi p/N$, the composition $K\circ \mathcal D_{p/N}$ remains unbroken and flows in the infrared to the Maxwell non-invertible $\mathcal T$ [2208.04331]. In non-Abelian gauge theory, including $\mathcal N=4$ $SU(2)$ super Yang-Mills along $|\tau|=1$, the non-invertible time-reversal takes the form
$$\mathcal T^{|\tau|=1}=S\circ \mathbb S \circ K,$$
with fusion
$$(\mathcal T)^\dagger\times \mathcal T=\mathcal C^{(2)}_0,$$
the $\mathbb Z_2$ condensation defect [2208.04331]. At special points $\tau=i$ or $e^{2\pi i/3}$, this operator factorizes into known non-invertible duality or triality defects composed with the invertible $\mathcal T$ at $\theta=0,\pi$ [2208.04331].

This framework modifies the standard statement that time-reversal exists only at $\theta=0$ or $\pi$. More precisely, that statement applies to invertible anti-unitary time-reversal. At rational $\theta=\pi p/N$, there is instead a conserved anti-linear topological operator without an inverse [2208.04331].

## 3. Ward identities, selection rules, and conserved defects

Because the non-invertible time-reversal defect extends along a constant-time slice, it commutes with the Hamiltonian:
$$\mathcal T\,H=H\,\mathcal T.$$
For any local operator $O(t)$,
$$\mathcal T\,O(t)=O(-t)\,\mathcal T.$$
These relations imply $\partial_t(\mathcal T)=0$, so the defect is conserved, and they yield Ward identities in correlation functions [2208.04331]. In particular,
$$\langle \cdots O(t)\cdots \rangle
= \langle \cdots \mathcal T\,O(t)\,\mathcal T^\dagger \cdots \rangle
= \langle \cdots O(-t)\,(\mathcal T\mathcal T^\dagger)\cdots \rangle,$$
from which certain time-odd correlators vanish or are related to time-reversed correlators up to insertions of the condensation defect $\mathcal C^{(N)}$ [2208.04331].

The same logic persists in massive QED, where correlators of charged fields satisfy analogous time-reversal Ward identities up to $\mathcal C^{(N)}$ [2208.04331]. These identities show that non-invertible time-reversal does not simply mimic ordinary $\mathbb Z_2$ time reversal with a missing inverse; rather, the missing inverse is replaced by a defect sector that modifies operator selection rules.

Defect-worldvolume constructions furnish an analogous structure. In $(3+1)$-dimensional self-dual theories with $\mathbb Z_N^{(1)}$ one-form symmetry, the duality defect $\mathcal D$ carries a hidden anti-unitary time-reversal $\mathsf T$ on its worldvolume [2212.14605]. Geometrically, this is constructed as $\mathsf T=S\circ R_\pi$, where a $\pi$-rotation around an axis in the defect swaps $\mathcal Q\leftrightarrow S\mathcal Q$, and composition with gauging $S$ reverses defect orientation while preserving the boundary condition [2212.14605]. On defect anyons,
$$\mathsf T(a)=a^{(p)^{-1}_N},$$
and sequential application yields
$$\mathsf T^2=\frac{1}{N}\sum_{\gamma\in H_1(M,\mathbb Z_N)} a^{\,1+(p)^{-1}_N}(\gamma)\cong C,$$
where $C$ is charge conjugation [2212.14605]. This is a non-invertible anti-unitary action internal to the defect worldvolume TQFT.

A different but related phenomenon occurs in the $1+1$-dimensional lattice Kramers–Wannier setting. There the exact non-invertible duality operator $R$ obeys
$$R^2=(1+\eta)T^{-1},\qquad R^\dagger R=1+\eta,\qquad R R^\dagger=T^{-1}(1+\eta),$$
with $\eta=\prod_{j=1}^L X_j$ the on-site $\mathbb Z_2$ spin-flip and $T$ the lattice translation [2401.12281]. In the presence of the defect $\mathcal D$ implementing $R$, parity and other invertible symmetries act projectively; for example,
$$\eta_{\mathcal D}P_{\mathcal D}=-P_{\mathcal D}\eta_{\mathcal D}.$$
The source describes this as the $1+1$-dimensional manifestation of a mixed anomaly between parity and the non-invertible duality symmetry [2401.12281]. This is not a literal non-invertible time-reversal operator, but it is closely related because the symmetry algebra involving parity/time-reversal is realized projectively in the presence of the defect.

## 4. Anomalies, indicators, and hidden anti-unitary structure

In $(3+1)$ dimensions, anomalies of non-invertible symmetries can be studied by $4+1$-dimensional bulk topological quantum field theories built from Abelian two-form gauge theories with a $0$-form permutation symmetry [2308.11706]. Gauging the $0$-form symmetry produces the inflow theory for the non-invertible symmetry. Two levels of anomalies appear: the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and the Frobenius–Schur indicator of the non-invertible symmetry may be incompatible with trivial boundary dynamics [2308.11706].

Within this framework, defects associated with ordinary $\mathbb Z_4$ symmetry host worldvolume theories with time-reversal symmetry $\mathsf T$ obeying
$$\mathsf T^2=C \quad \text{or} \quad \mathsf T^2=(-1)^F C,$$
with $C$ a unitary charge-conjugation symmetry [2308.11706]. Since $C\neq 1$, one may equivalently write
$$\mathsf T^{-1}=C\,\mathsf T \quad \text{(boson)}, \qquad \mathsf T^{-1}=(-1)^F C\,\mathsf T \quad \text{(fermion)}.$$
Fusion with the orientation-reversed defect gives
$$\mathsf T\times\overline{\mathsf T}=1+C \quad \text{(bosonic)}, \qquad
\mathsf T\times\overline{\mathsf T}=1+(-1)^F C \quad \text{(fermionic)},$$
which makes non-invertibility explicit [2308.11706].

The anomalies of this algebra in $2+1$ dimensions are classified by $\mathbb Z_4\times \mathbb Z_4$ in the bosonic case and by $\mathbb Z_4$ in the fermionic case [2308.11706]. After gauging charge conjugation $C$, one can define two indicators,
$$\eta_1=\frac1D\sum_{a\in \mathcal C} d_a^2 e^{2\pi i h[a]}
=e^{2\pi i\,\frac{c_-}{8}},$$
and
$$\eta_2=\frac1D\sum_{a\,\text{fixed by }\mathsf T} d_a\,\mathsf T^2_a\,e^{2\pi i h[a]},$$
with $h[a]$ the topological spin and $d_a$ the quantum dimension [2308.11706]. According to the source, if $(\eta_1,\eta_2)=(1,1)$, $\mathsf T$ is an ordinary invertible symmetry with no ’t Hooft anomaly; if $\eta_1=1$ but $\eta_2=-1$, gauging $C$ forces $\mathsf T$ into a $2$-group extension; and if $\eta_1=-1$, gauging $C$ makes $\mathsf T$ genuinely non-invertible [2308.11706]. The paper describes this as a higher-dimensional analogue of the $1+1$-dimensional Frobenius–Schur obstruction.

These anomaly discussions connect directly to the hidden time-reversal symmetry on defect worldvolumes studied in self-dual $\mathbb Z_N^{(1)}$ theories [2212.14605]. There, the worldvolume TQFT $\mathcal A^{N,p}$ has anyon spins
$$h(a^r)=\frac{p\,r^2}{2N}\ \mathrm{mod}\ 1,$$
and the anti-unitary action $\mathsf T(a)=a^{(p)_N^{-1}}$ flips spins $h\mapsto -h$ precisely when $p^2=-1\mod N$ [2212.14605]. This provides an explicit realization of anti-unitary structure tied to self-duality and condensation, rather than to a group-like involution.

## 5. Constraints on phases and dynamics

One of the central uses of non-invertible time-reversal and related defect symmetries is to constrain infrared dynamics. In $(3+1)$-dimensional self-dual theories with $\mathbb Z_N^{(1)}$ one-form symmetry, a symmetry-preserving vacuum state with a gapped spectrum is often forbidden [2212.14605]. The source states the obstruction theorem as follows: a self-dual theory with $\mathbb Z_N^{(1)}$ one-form symmetry is gapless or spontaneously breaks the self-duality symmetry unless
$$N=k^2\ell \qquad \text{where } -1 \text{ is a quadratic residue modulo } \ell,$$
and a symmetry-preserving gapped phase exists iff this condition holds [2212.14605]. When it does fail, the combined non-invertible symmetry enforces either gaplessness or spontaneous symmetry breaking.

The same paper gives explicit examples. For $N=4$, a $\mathbb Z_2$ TQFT preserving duality is allowed. For $N=5$, a duality-invariant SPT exists but no non-trivial TQFT. For $N=3$, no suitable factorization exists, so at self-duality the lattice model spontaneously breaks duality in a first-order transition [2212.14605]. The source also notes that for small $N$ the duality or triality defect is spontaneously broken at the self-dual point, whereas for large $N$ the system remains symmetric but gapless in the Coulomb phase [2212.14605]. This places non-invertible anti-unitary structure within the general program of symmetry-enforced gaplessness beyond ordinary ’t Hooft anomalies.

A related constraint appears in $1+1$ dimensions for the exact lattice non-invertible Kramers–Wannier symmetry $R$ on a tensor-product Hilbert space. Any finite-range Hamiltonian commuting with $R$ must satisfy either: the system is gapless, or it is gapped and its global symmetry $(R,\eta)$ is spontaneously broken [2401.12281]. In the gapped case, the number of superselection sectors is a multiple of three [2401.12281]. The source explicitly compares this with Lieb–Schultz–Mattis-type obstructions. Although the operator $R$ is not itself labeled as time reversal, the work states that the symmetry algebra involving parity/time-reversal is realized projectively in the presence of the defect [2401.12281]. This suggests a broader pattern in which non-invertible defects obstruct trivially gapped symmetric phases and force either criticality or symmetry breaking.

In gauge theory, the rational-$\theta$ non-invertible time-reversal similarly constrains RG flows. Massive QED at $\theta=\pi p/N$ cannot trivially gap without saturating the non-invertible constraints [2208.04331]. A plausible implication is that the anti-linear topological defect should be regarded as an RG-invariant datum that organizes admissible infrared phases, much as ordinary anomalies do.

## 6. Open quantum systems and non-invertibility through $t=0$

A distinct notion of non-invertible time-reversal appears in the study of open quantum systems [2311.08486]. There the starting point is a system $S$ coupled to a bath $B$ with total Hamiltonian
$$H_{\mathrm{tot}}=H_S+H_B+H_{SB},$$
and an anti-unitary time-reversal operator $\theta$ acting on $\mathcal H_S\otimes \mathcal H_B$ such that
$$\theta\,i\,\theta^{-1}=-i,\qquad \theta\,Q\,\theta^{-1}=Q,\qquad \theta\,P\,\theta^{-1}=-P,\qquad \theta H_{\mathrm{tot}}\theta^{-1}=H_{\mathrm{tot}}.$$
Thus the microscopic dynamics is time-reversal symmetric [2311.08486]. Correlation functions of bath forces obey
$$\langle f(t)f(t')\rangle =K(t-t'),\qquad K(\tau)=K(-\tau),$$
so the bath correlations are stationary and even in time difference [2311.08486].

The key claim is that the Markov approximation does not imply a violation of time-reversal symmetry. Rather, if one keeps track of the sign of time, the secularized Markovian master equation takes the form
$$
\frac{d\rho_S}{dt}
= -\frac{i}{\hbar}[H_S+H_{\rm LS},\rho_S]
+\mathrm{sgn}(t)\sum_{\alpha\beta\omega}\gamma_{\alpha\beta}(\omega)
\left[
A_\beta(\omega)\rho_S A_\alpha^\dagger(\omega)
-\frac12\{A_\alpha^\dagger(\omega)A_\beta(\omega),\rho_S\}
\right],
$$
valid for all $t\neq 0$ [2311.08486]. The factor $\mathrm{sgn}(t)$ renders the dynamics non-invertible at $t=0$: one cannot run the Lindblad-type evolution continuously through the origin. Yet every term is chosen so as to commute appropriately with time reversal, and the equation satisfies
$$T[d\rho_S(t)/dt]T^{-1}=-d\rho_S(-t)/dt,$$
that is, full time-reversal invariance [2311.08486].

Projecting onto the energy basis yields a time-symmetric Pauli master equation,
$$
\frac{dp_n}{dt}
=\mathrm{sgn}(t)\sum_{m\neq n}
\big[W_{n\leftarrow m}\,p_m(t)-W_{m\leftarrow n}\,p_n(t)\big],
$$
with Golden-Rule rates
$$
W_{n\leftarrow m}
=\frac{2\pi}{\hbar}\sum_{\alpha\beta}\gamma_{\alpha\beta}(\epsilon_m-\epsilon_n)
\langle n|A_\alpha|m\rangle \langle m|A_\beta^\dagger|n\rangle.
$$
Detailed balance holds in each temporal direction:
$$
\frac{W_{m\leftarrow n}}{W_{n\leftarrow m}}
=e^{-(\epsilon_m-\epsilon_n)/k_B T}.
$$
The paper concludes that both the Lindblad and Pauli equations predict approach to Gibbs equilibrium for $t\to +\infty$ and $t\to -\infty$, producing two opposing arrows of thermodynamic time pointing away from $t=0$ [2311.08486].

The associated dynamical map,
$$E(t)=\exp[\mathrm{sgn}(t)\mathcal L\,|t|],$$
has the semigroup property on each temporal half-axis but not across $t=0$ [2311.08486]. The source therefore proposes a time-symmetric definition of Markovianity based on divisibility for pairs of times with the same sign [2311.08486]. This use of “non-invertible time-reversal symmetry” differs from the topological-defect notion: the non-invertibility is not the failure of a topological defect to have an inverse, but the impossibility of continuously extending the dissipative evolution through the time origin while preserving the time-symmetric form.

## 7. Conceptual synthesis and common distinctions

The literature now supports several technically precise meanings of non-invertible time-reversal symmetry. In gauge theory and topological-defect language, it denotes a conserved anti-linear codimension-one operator whose fusion is not group-like. At rational $\theta=\pi p/N$, this operator is $\mathcal T^{\theta=\pi p/N}=K\circ \mathcal I_{2\pi p/N}$ and satisfies $\mathcal T^\dagger \mathcal T=\mathcal C^{(N)}\neq 1$ [2208.04331]. In defect-worldvolume settings, anti-unitary symmetry can obey $\mathsf T^2=C$ or $\mathsf T^2=(-1)^F C$, so non-invertibility is encoded by fusion into multiple sectors [2212.14605; 2308.11706]. In generalized Wigner form, non-invertible antiunitary symmetry appears as a partial isometry $\Theta=U K\,P$ on an extended gauged Hilbert space [2509.25327]. In open-system dynamics, non-invertibility refers to the inability to continue a time-symmetric Markovian semigroup through $t=0$ despite full microscopic $T$-invariance [2311.08486].

A common misconception is that non-invertible time-reversal is merely ordinary time reversal at a special parameter value with some anomaly attached. The rational-$\theta$ construction shows something sharper: the operator is conserved and anti-linear, but it cannot be implemented by an anti-unitary operator because its adjoint times itself is a condensation defect rather than the identity [2208.04331]. Another misconception is that anti-unitary symmetry must always be invertible by Wigner’s theorem. The generalized result states that once bijectivity is relinquished, probability-preserving non-invertible symmetries are realized as partial isometries on an extended gauged Hilbert space [2509.25327]. Conversely, in open quantum systems, the presence of a sign function in the generator does not by itself signal broken time-reversal symmetry; the cited work argues that the Markov approximation itself does not break microscopic time-reversal symmetry, and that the apparent arrow of time arises only when one restricts to $t>0$ or $t<0$ alone [2311.08486].

Taken together, these results indicate that non-invertible time-reversal symmetry is not a single universal algebraic object but a family of related anti-linear structures appearing in QFT, topological order, lattice models, quantum mechanics, and open-system dynamics. What unifies them is the replacement of group-like inversion by condensation, projection, or half-axis semigroup structure, together with new Ward identities, anomaly indicators, and infrared constraints that are absent for ordinary invertible time reversal.

Source: https://www.emergentmind.com/topics/non-invertible-time-reversal-symmetry