The symplectic quantization scheme proposed for matter scalar fields in the companion paper (Gradenigo and Livi, arXiv:2101.02125, 2021) is generalized here to the case of space–time quantum fluctuations. That is, we present a new formalism to frame the quantum gravity problem. Inspired by the stochastic quantization approach to gravity, symplectic quantization considers an explicit dependence of the metric tensor \(g_{\mu \nu }\) on an additional time variable, named intrinsic time at variance with the coordinate time of relativity, from which it is different. The physical meaning of intrinsic time, which is truly a parameter and not a coordinate, is to label the sequence of \(g_{\mu \nu }\) quantum fluctuations at a given point of the four-dimensional space–time continuum. For this reason symplectic quantization necessarily incorporates a new degree of freedom, the derivative \({\dot{g}}_{\mu \nu }\) of the metric field with respect to intrinsic time, corresponding to the conjugated momentum \(\pi _{\mu \nu }\). Our proposal is to describe the quantum fluctuations of gravity by means of a symplectic dynamics generated by a generalized action functional \({\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }] = {\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }] - S[g_{\mu \nu }]\), playing formally the role of a Hamilton function, where \(S[g_{\mu \nu }]\) is the standard Einstein–Hilbert action while \({\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }]\) is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define an ensemble for the quantum fluctuations of \(g_{\mu \nu }\) analogous to the microcanonical one in statistical mechanics, with the only difference that in the present case one has conservation of the generalized action \({\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }]\) and not of energy. Since the Einstein–Hilbert action \(S[g_{\mu \nu }]\) plays the role of a potential term in the new pseudo-Hamiltonian formalism, it can fluctuate along the symplectic action-preserving dynamics. These fluctuations are the quantum fluctuations of \(g_{\mu \nu }\). Finally, we show how the standard path-integral approach to gravity can be obtained as an approximation of the symplectic quantization approach. By doing so we explain how the integration over the conjugated momentum field \(\pi _{\mu \nu }\) gives rise to a cosmological constant term in the path-integral approach.

随书（Gradenigo and Livi，arXiv：2101.02125，2021）中针对物质标量场提出的辛量化方案在此被推广到时空量子涨落的情况。也就是说，我们提出了一种新的形式主义来描述量子引力问题。受重力随机量化方法的启发，辛量化考虑了度量张量\（g _ {\ mu \ nu} \）对附加时间变量的显式依赖，该时间变量称为内在时间，其方差为相对坐标时间，从中它是不同的。固有时间的物理含义（实际上是一个参数而不是一个坐标）是标记\（g _ {\ mu \ nu} \）的序列二维时空连续体给定点的量子涨落。因此，辛量化必须包含一个新的自由度，即度量域相对于固有时间的导数\（{\ dot {g}} _ {\ mu \ nu} \），对应于共轭动量\ { \ pi _ {\ mu \ nu} \）。我们的建议是通过广义作用函数\（{\ mathcal {A}} [g _ {\ mu \ nu}，\ pi _ {\ mu \ nu}]产生的辛动力学来描述引力的量子涨落。= {\\ mathcal {K}} [g _ {\ mu \ nu}，\ pi _ {\ mu \ nu}]-S [g _ {\ mu \ nu}] \），正式扮演汉密尔顿函数的角色，其中\（S [g _ {\ mu \ nu}] \）是标准的爱因斯坦–希尔伯特动作，而\（{\ mathcal {K}} [g _ {\ mu \ nu}，\ pi _ {\ mu \ nu}] \）是一个新术语，其中包括磁场的自由度。这样的动作使我们能够为\（g _ {\ mu \ nu} \）的量子涨落定义一个集合，类似于统计力学中的微经典动作，唯一的区别是，在当前情况下，该动作具有广义动作的守恒性。\（{{数学{A}} [g _ {\ mu \ nu}，\ pi _ {\ mu \ nu}] \）而不是能量。由于爱因斯坦–希尔伯特动作\（S [g _ {\ mu \ nu}] \）在新的伪哈密​​尔顿形式主义中扮演潜在术语的角色，因此它可以沿辛动保持动力学变化。这些涨落是\（g _ {\ mu \ nu} \）的量子涨落。最后，我们展示了如何获得标准的重力路径积分方法，作为辛量化方法的近似值。通过这样做，我们解释了在共轭动量场\（\ pi _ {\ mu \ nu} \）上的积分如何在路径积分方法中产生宇宙常数项。