标题： 在非平凡几何中描述幺正演化面临的挑战：图像和表示 显示英文标题

标题： Challenges for describing unitary evolution in nontrivial geometries: pictures and representations

摘要： 在一般时空中的空间切片之间的演化描述面临一个显著的困难：在给定基下，切片上的状态不是通过幺正变换相关联的。 这一问题出现在大于二维的时空维度中，直接与规范对易子的无限多个不等价表示有关，并且特别是在时变时空中的相互作用理论中会出现。 我们联系了这个问题的不同方面，并讨论其可能的解决方法。 它直接与标准薛定谔图像演化失败的讨论以及通过“多指时间”进行演化的讨论相关。 需要一个指定物理幺正等价类状态的条件；一般来说，这个等价类随时间演化，一个重要的问题是它是如何确定的。 在自由理论中，一种方法是通过对两点函数施加Hadamard条件。 我们探讨了一种不同的方法，这可能对相互作用理论也有帮助，分析了状态在局部极限中的结构，并将这些方法进行了关联。 我们还阐明了非物理真空的非Hadamard行为，并讨论了涉及宇宙学和黑洞演化的这些方法的具体例子。 在量子动力几何的背景下，这些问题得到了扩展，并对宇宙波函数的正确描述以及Wheeler-DeWitt方程的作用提出了重要问题。 显示英文摘要

摘要： Description of evolution between spatial slices in a general spacetime suffers from a significant difficulty: the states on the slices, in a given basis, are not related by a unitary transformation. This problem, which occurs in spacetime dimensions above two, is directly related to the infinite number of inequivalent representations of the canonical commutators, and in particular will arise for interacting theories in time-dependent spacetimes. We connect different facets of this issue, and discuss its possible resolution. It is directly related to discussions of failure of a standard Schr\"odinger picture of evolution, and of evolution via "many-fingered time." One requires a condition specifying a physical unitary equivalence class of states; in general this equivalence class evolves with time, and an important question is how it is determined. One approach to this in free theories is by imposing a Hadamard condition on the two point function. We explore a different approach, which also may be helpful for interacting theories, analyzing the structure of the state in a local limit, and relate these approaches. We also elucidate the non-Hadamard behavior of unphysical vacua, and discuss concrete examples of these approaches involving cosmological and black hole evolution. The issues are extended in the context of quantum dynamical geometry, and raise important questions for the proper description of the wavefunction of the universe and for the role of the Wheeler-DeWitt equation.