In gauge theory, it is commonly stated that time-reversal symmetry only exists at $\theta=0$ or $\pi$ for a $2\pi$-periodic $\theta$-angle. In this paper, we point out that in both the free Maxwell theory and massive QED, there is a non-invertible time-reversal symmetry at every rational $\theta$-angle, i.e., $\theta= \pi p/N$. The non-invertible time-reversal symmetry is implemented by a conserved, anti-linear operator without an inverse. It is a composition of the naive time-reversal transformation and a fractional quantum Hall state. We also find similar non-invertible time-reversal symmetries in non-Abelian gauge theories, including the $\mathcal{N}=4$ $SU(2)$ super Yang-Mills theory along the locus $|\tau|=1$ on the conformal manifold.

在规范理论中，人们通常认为只有时间反转对称性 存在于$2\pi$-周期$\theta$-角的$\theta=0$或$\pi$处。在这 文中指出，在自由麦克斯韦理论和质量量子电动力学中，都存在 在每一个有理的角内都是一个不可逆的时间反转对称， 即$\theta=\pi p/N$。不可逆时反对称是 由不带逆的守恒的反线性运算符实现。这是一个 朴素时间反转变换与分数量子的合成 霍尔州。我们也发现了类似的不可逆时反对称。 非阿贝尔规范理论，包括$\数学{N}=4$$SU(2)$SUPER 共形流形上沿轨迹$|\tau|=1$的Yang-Mills理论。