Invariant Connections in Loop Quantum Gravity
Invariant Connections in Loop Quantum Gravity
复制标题
环量子引力中的不变连接
DOI：
--
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Maximilian Hanusch
中科院分区：
文献类型：
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作者：
Maximilian Hanusch
给定一组 $${G}$$G和一个阿贝尔 $${C^*}$$C *代数 $${mathfrak{A}}$$A，反同态 $${Thetacolon G
ightarrow {mathrm{Aut}}(mathfrak{A})}$$Θ：G→Aut(A)与左边的动作是一一对应的 $${Phicolon G imes {mathrm{Spec}}(mathfrak{A})
ightarrow {mathrm{Spec}}(mathfrak{A})}$$Φ：G×Spec(A)→Spec(A)，其翻译映射 $${Phi_g}$$Φg是连续的；的连续性 $${Theta}$$Θ和 $${Phi}$$Φ等于if $${mathfrak{A}}$$A是单位的。特别是左动作 $${phicolon G imes X
ightarrow X}$$ϕ：G×X→X可以唯一地扩展到a的频谱 $${C^*}$$C * -子代数 $${mathfrak{A}}$$A的有界函数 $${X}$$X如果 $${phi_g^*(mathfrak{A})subseteq mathfrak{A}}$$ϕg∗(A)挨个成立 $${gin G}$$g∈g。在本文中，我们将此应用到环量子引力的框架中。我们证明，在构型空间的水平上，量子化与约简一般不交换，即对称约简的量子构型空间（严格地）大于约简经典理论的量子化构型空间。在这里，量子约简空间具有完全由简单代数关系表征的优点，而量子约简经典空间通常难以计算。
Given a group $${G}$$G, and an abelian $${C^*}$$C∗-algebra $${mathfrak{A}}$$A, the antihomomorphisms $${Thetacolon G
ightarrow {mathrm{Aut}}(mathfrak{A})}$$Θ:G→Aut(A) are in one-to-one with those left actions $${Phicolon G imes {mathrm{Spec}}(mathfrak{A})
ightarrow {mathrm{Spec}}(mathfrak{A})}$$Φ:G×Spec(A)→Spec(A) whose translation maps $${Phi_g}$$Φg are continuous; whereby continuities of $${Theta}$$Θ and $${Phi}$$Φ turn out to be equivalent if $${mathfrak{A}}$$A is unital. In particular, a left action $${phicolon G imes X
ightarrow X}$$ϕ:G×X→X can be uniquely extended to the spectrum of a $${C^*}$$C∗-subalgebra $${mathfrak{A}}$$A of the bounded functions on $${X}$$X if $${phi_g^*(mathfrak{A})subseteq mathfrak{A}}$$ϕg∗(A)⊆A holds for each $${gin G}$$g∈G. In the present paper, we apply this to the framework of loop quantum gravity. We show that, on the level of the configuration spaces, quantization and reduction in general do not commute, i.e., that the symmetry-reduced quantum configuration space is (strictly) larger than the quantized configuration space of the reduced classical theory. Here, the quantum-reduced space has the advantage to be completely characterized by a simple algebraic relation, whereby the quantized reduced classical space is usually hard to compute.
DOI：
10.3842/sigma.2014.025
发表时间：
2014
影响因子：
0.9
作者：
Maximilian Hanusch
通讯作者：
Maximilian Hanusch