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arXiv 2005.00988gr-qchep-th
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约化相空间圈量子引力新路径积分公式的半经典极限

Semiclassical Limit of New Path Integral Formulation from Reduced Phase Space Loop Quantum Gravity

Muxin Han, Hongguang Liu

备注：
25+3 pages, no figure. ILQGS talk: http://relativity.phys.lsu.edu/ilqgs/han041420.mp4. Mathematica files can be downloaded from https://github.com/LQG-Florida-Atlantic-University/Classical-EOM
发表信息：
Phys. Rev. D 102, 024083 (2020)
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摘要：

最近，arXiv：1910.03763从正则圈量子引力（LQG）的约化相空间公式导出了一个新的LQG路径积分公式.本文着重于这种路径积分公式的半经典分析。我们发现，占主导地位的贡献的路径积分来自半经典运动方程（EOMs）的解决方案，这减少到汉密尔顿的方程的完整性和通量${h}(e),{p}^a(e)$在减少相空间$\mathcal{P}_γ$的立方晶格$γ$：$\dot{h}(e)=\{{h}(e),\, \mathbf{H}\}$，$\dot{p}^a(e)=\{{p}^a(e),\, \mathbf{H}\}$，其中$\mathbf{H}$是离散的物理哈密顿量。由路径积分得到的半经典动力学问题转化为$\mathcal{P}_γ$中哈密顿时间演化的初值问题。此外，当我们采取的连续极限的格$γ$，这些汉密尔顿方程正确地再现经典的约化相空间EOM的重力耦合到尘埃场的连续，只要初始和最终状态是半经典的。我们的结果证明了新的路径积分公式具有正确的半经典极限，并表明LQG中的约化相空间量子化是半经典相容的。在此基础上，我们将该路径积分公式与自旋泡沫公式进行了比较，结果表明该公式具有有限性、与正则LQG的关系、不受余弦和平坦性问题的限制等优点.

英文摘要：

Recently, a new path integral formulation of Loop Quantum Gravity (LQG) has been derived in arXiv:1910.03763 from the reduced phase space formulation of the canonical LQG. This paper focuses on the semiclassical analysis of this path integral formulation. We show that dominant contributions of the path integral come from solutions of semiclassical equations of motion (EOMs), which reduces to Hamilton's equations of holonomies and fluxes ${h}(e),{p}^a(e)$ in the reduced phase space $\mathcal{P}_γ$ of the cubic lattice $γ$: $\dot{h}(e)=\{{h}(e),\, \mathbf{H}\}$, $\dot{p}^a(e)=\{{p}^a(e),\, \mathbf{H}\}$, where $\mathbf{H}$ is the discrete physical Hamiltonian. The semiclassical dynamics from the path integral becomes an initial value problem of Hamiltonian time evolution in $\mathcal{P}_γ$. Moreover when we take the continuum limit of the lattice $γ$, these Hamilton's equations reproduce correctly classical reduced phase space EOMs of gravity coupled to dust fields in the continuum, as far as initial and final states are semiclassical. Our result proves that the new path integral formulation has the correct semiclassical limit, and indicates that the reduced phase space quantization in LQG is semiclassically consistent. Based on these results, we compare this path integral formulation and the spin foam formulation, and show that this formulation has several advantages including the finiteness, the relation with canonical LQG, and being free of cosine and flatness problems.

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