量子引力中的离散 $Z_4$ 对称性
Discrete $Z_4$ symmetry in quantum gravity
AI总结：
本文研究了量子引力中引力四重标作为序参量涌现时的离散 $Z_4$ 对称性，借助拓扑超流体 $^3$He-B 的类比论证了该对称性及其自发破缺机制，并探讨了改变标量曲率符号的替代破缺场景。
摘要：
我们考虑离散 $Z_4$ 对称性 ${\hat {\bf i}}$，它出现在引力四重标作为序参量涌现的量子引力场景中。在该对称操作 $\hat {\bf i}$ 下，涌现的四重标乘以虚数单位，即 $\hat {\bf i}\\,e^a_μ= - i e^a_μ$。通过对拓扑超流体 $^3$He-B 中对称性破缺方案的考量，也支持了这种对称性的存在及其自发破缺。序参量 $^3$He-B 是四重标场的类比，但它具有复数值。$\hat {\bf i}$ 对称性操作使复数序参量的相位改变 $π/2$，这对应于量子引力中的 $Z_4$ 离散对称性。我们还考虑了该 $Z_4$ 对称性破缺的替代场景，其中 $\hat {\bf i}$ 操作改变标量曲率的符号，即 $\hat {\bf i} \\,{\cal R} = - {\cal R}$，从而 Einstein-Hilbert 作用量违反了 $\hat {\bf i}$ 对称性。在对称性破缺的替代场景中，引力耦合 $K=1/16πG$ 充当序参量，它在 $\hat {\bf i}$ 变换下改变符号。
英文摘要：
We consider the discrete $Z_4$ symmetry ${\hat {\bf i}}$, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter. Under this symmetry operation $\hat {\bf i}$, the emerging tetrads are multiplied by the imaginary unit, $\hat {\bf i}\,e^a_μ= - i e^a_μ$. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid $^3$He-B. The order parameter $^3$He-B is the analogue of the tetrad field, but it has complex values. The $\hat {\bf i}$-symmetry operation changes the phase of the complex order parameter by $π/2$, which corresponds to the $Z_4$ discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this $Z_4$ symmetry, in which the $\hat {\bf i}$-operation changes sign of the scalar curvature, $\hat {\bf i} \,{\cal R} = - {\cal R}$, and thus the Einstein-Hilbert action violates the $\hat {\bf i}$-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling $K=1/16πG$ plays the role of the order parameter, which changes sign under $\hat {\bf i}$-transformation.
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