arXiv 1908.108722019-08-28gr-qchep-th

量子路径、引力和时空微结构的一种测度

A Measure for Quantum Paths, Gravity and Spacetime Microstructure

T. Padmanabhan

备注：: Extended version of the essay which received an Honorable Mention in the Gravity Research Foundation Essay Competition, 2019; publication in the special issue of IJMPD; 23 pages; no figures
发表信息：: Int. Jour. Mod. Phys D, (2019), 1944009
DOI：: 10.1142/S0218271819440097

摘要：

在（伪）黎曼时空中，连接任意两个事件的给定长度的经典路径的数目当然是无限的。然而，对于任意时空中[连接两个事件$(x_1,x_2)$的长度为$σ$]的量子路径的有效数目，可以定义一个有用的有限测度$N(x_2,x_1;σ)$。当为$x_2=x_1$时，这简化为$C(x,σ)$，给出了包含事件$x$的长度为$σ$的闭合量子环的度量。$N(x_2,x_1;σ)$和$C(x,σ)$都有明确的定义，并且只依赖于时空的几何。各种其他物理量，例如，有效拉格朗日量可以用$N(x_2,x_1;σ)$表示。在时空区域$\mathcal{V}$中，由闭合环贡献的总路径长度的相应度量由$L(σ;x) \equivσC(σ;x)$在$\mathcal{V}$上的积分给出。值得注意的是，$L(0;x) \propto R(x)$，即里奇标量，即对时空区域中无穷小闭合环所贡献的总长度的度量，给出了爱因斯坦-希尔伯特作用量。它的变化，当我们改变度量，可以提供一个新的路线诱导/紧急重力描述。在存在背景电磁场的情况下，$N(x_2,x_1;σ)$和$C(x,σ)$的相应表达式可以与场的完整性相关联。测度$N(x_2,x_1;σ)$也可以用来计算一类作用量和测度是路径长度的任意函数的路径积分。作为一个例子，我计算了一个修改的路径积分，其中包括零点的时空长度。我还描述了$N(x_2,x_1;σ)$的其他几个属性，并概述了几个简单的应用。

英文摘要：

The number of classical paths of a given length, connecting any two events in a (pseudo) Riemannian spacetime is, of course, infinite. It is, however, possible to define a useful, finite, measure $N(x_2,x_1;σ)$ for the effective number of quantum paths [of length $σ$ connecting two events $(x_1,x_2)$] in an arbitrary spacetime. When $x_2=x_1$, this reduces to $C(x,σ)$ giving the measure for closed quantum loops of length $σ$ containing an event $x$. Both $N(x_2,x_1;σ)$ and $C(x,σ)$ are well-defined and depend only on the geometry of the spacetime. Various other physical quantities like, for e.g., the effective Lagrangian, can be expressed in terms of $N(x_2,x_1;σ)$. The corresponding measure for the total path length contributed by the closed loops, in a spacetime region $\mathcal{V}$, is given by the integral of $L(σ;x) \equivσC(σ;x)$ over $\mathcal{V}$. Remarkably enough $L(0;x) \propto R(x)$, the Ricci scalar; i.e, the measure for the total length contributed by infinitesimal closed loops in a region of spacetime gives us the Einstein-Hilbert action. Its variation, when we vary the metric, can provide a new route towards induced/emergent gravity descriptions. In the presence of a background electromagnetic field, the corresponding expressions for $N(x_2,x_1;σ)$ and $C(x,σ)$ can be related to the holonomies of the field. The measure $N(x_2,x_1;σ)$ can also be used to evaluate a wide class of path integrals for which the action and the measure are arbitrary functions of the path length. As an example, I compute a modified path integral which incorporates the zero-point-length in the spacetime. I also describe several other properties of $N(x_2,x_1;σ)$ and outline a few simple applications.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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