量子约束、Dirac观测量和演化：LQC中的群平均与薛定谔图像
Quantum constraints, Dirac observables and evolution: group averaging versus Schroedinger picture in LQC
摘要：
研究了$C= - \partial_T^2 \otimes B - I\otimes H$形式的一般量子约束（特别是在圈量子宇宙学模型中实现的）。群平均被应用于定义解的希尔伯特空间和相关的狄拉克观测量。考虑两种情况。在第一种情况下，假设算子$(1/2)π^2 B - H$的谱是离散的。由约束定义的量子理论采用具有广义哈密顿量$\sqrt{B^{-1} H}$的类薛定谔量子力学的形式。在第二种情况下，谱是绝对连续的，并假设了本征函数的一些特殊的渐近性质。由此产生的希尔伯特空间和动力学的特点是一个连续的家庭薛定谔类量子理论。然而，关系可观察量混合了家庭的不同成员。我们的假设的动机是新的圈量子宇宙学模型的量子FRW时空。这两种情况下考虑的文件分别对应于负和正的宇宙学常数。我们的结果应该也适用于许多其他广义相对论的背景下。
英文摘要：
A general quantum constraint of the form $C= - \partial_T^2 \otimes B - I\otimes H$ (realized in particular in Loop Quantum Cosmology models) is studied. Group Averaging is applied to define the Hilbert space of solutions and the relational Dirac observables. Two cases are considered. In the first case, the spectrum of the operator $(1/2)π^2 B - H$ is assumed to be discrete. The quantum theory defined by the constraint takes the form of a Schroedinger-like quantum mechanics with a generalized Hamiltonian $\sqrt{B^{-1} H}$. In the second case, the spectrum is absolutely continuous and some peculiar asymptotic properties of the eigenfunctions are assumed. The resulting Hilbert space and the dynamics are characterized by a continuous family of the Schroedinger-like quantum theories. However, the relational observables mix different members of the family. Our assumptions are motivated by new Loop Quantum Cosmology models of quantum FRW spacetime. The two cases considered in the paper correspond to the negative and, respectively, positive cosmological constant. Our results should be also applicable in many other general relativistic contexts.
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