标题： 陈-西蒙斯态在量子引力中的三元组表示 显示英文标题

标题： Triad representation of the Chern-Simons state in quantum gravity

摘要： 我们研究了具有非零宇宙常数的量子引力的Chern-Simons态的三胞胎表示。结果显示，作为Ashtekar理论中一个著名的精确波函数，在适当推广的傅里叶变换作用下，Chern-Simons态可以转换为实三胞胎表示，从而在三胞胎变量中得到相应的态的复积分表示。发现对于复积分路径的不同拓扑不等价选择，在三胞胎表示中会产生线性无关的波函数，这些波函数都来自于Ashtekar变量中的一个Chern-Simons态。对于合适的归一化因子选择，这些态在任意甚至拓扑非平凡的规范变换下都是规范不变的。在几个有趣的渐进行为参数区域中，可以获得三胞胎表示中波函数的显式解析表达式，并讨论了相关的半经典4-几何。在限制到Bianchi型齐次3-度规的情况下，我们将结果与早期对齐次宇宙模型的讨论进行了比较。此外，我们在量子引力的希尔伯特空间上定义了一个内积，并选择了一个自然的规范条件来固定时间规范。根据这个特定的内积，量子引力的Chern-Simons态被证明是一个不可归一化的波函数。 显示英文摘要

摘要： We investigate a triad representation of the Chern-Simons state of quantum gravity with a non-vanishing cosmological constant. It is shown that the Chern-Simons state, which is a well-known exact wavefunctional within the Ashtekar theory, can be transformed to the real triad representation by means of a suitably generalized Fourier transformation, yielding a complex integral representation for the corresponding state in the triad variables. It is found that topologically inequivalent choices for the complex integration contour give rise to linearly independent wavefunctionals in the triad representation, which all arise from the one Chern-Simons state in the Ashtekar variables. For a suitable choice of the normalization factor, these states turn out to be gauge-invariant under arbitrary, even topologically non-trivial gauge-transformations. Explicit analytical expressions for the wavefunctionals in the triad representation can be obtained in several interesting asymptotic parameter regimes, and the associated semiclassical 4-geometries are discussed. In restriction to Bianchi-type homogeneous 3-metrics, we compare our results with earlier discussions of homogeneous cosmological models. Moreover, we define an inner product on the Hilbert space of quantum gravity, and choose a natural gauge-condition fixing the time-gauge. With respect to this particular inner product, the Chern-Simons state of quantum gravity turns out to be a non-normalizable wavefunctional.