辛量子化II：时空量子涨落动力学和宇宙学常数
Symplectic quantization II: dynamics of space-time quantum fluctuations and the cosmological constant
摘要：
本文将“辛量子化Ⅰ”中提出的物质标量场的辛量子化方案推广到时空量子涨落的情形。辛量子化考虑了度量张量$g_{μν}$对一个额外的时间变量的显式依赖性，该时间变量名为“固有时间”，与相对论的坐标时间不同。真时的物理意义是在四维时空连续体的给定点上标记$g_{μν}$量子涨落的序列。由于这个原因，辛量子化必然包含一个新的自由度，即度规场对真时的导数$\dot{g}_{μν}$，对应于共轭动量$π_{μν}$。辛量子化通过由广义作用量泛函$\mathcal{A}[g_{μν},π_{μν}] = \mathcal{K}[g_{μν},π_{μν}] - S[g_{μν}]$产生的辛动力学描述引力的量子涨落，该作用量泛函在形式上扮演Hamilton函数的角色，其中$S[g_{μν}]$是爱因斯坦-希尔伯特作用量，$\mathcal{K}[g_{μν},π_{μν}]$是包含场的动力学自由度的新项。这样的行动使我们能够定义一个伪微正则系综的量子涨落的$g_{μν}$，建立在广义作用$\mathcal{A}[g_{μν},π_{μν}]$，而不是能量的守恒。$S[g_{μν}]$在保作用辛动力学中扮演着势项的角色：它的涨落就是$g_{μν}$的量子涨落。它示出了如何辛量子化映射到路径积分引力的方法。通过这样做，我们解释了如何在共轭动量场$π_{μν}$的积分引起的路径积分中的宇宙学常数项。
英文摘要：
The symplectic quantization scheme proposed for matter scalar fields in the companion paper "Symplectic quantization I" is generalized here to the case of space-time quantum fluctuations. Symplectic quantization considers an explicit dependence of the metric tensor $g_{μν}$ on an additional time variable, named "proper time" at variance with the coordinate time of relativity. The physical meaning of proper time is to label the sequence of $g_{μν}$ 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 $\dot{g}_{μν}$ of the metric field with respect to proper time, corresponding to the conjugated momentum $π_{μν}$. Symplectic quantization describes the quantum fluctuations of gravity by means of the symplectic dynamics generated by a generalized action functional $\mathcal{A}[g_{μν},π_{μν}] = \mathcal{K}[g_{μν},π_{μν}] - S[g_{μν}]$, playing formally the role of a Hamilton function, where $S[g_{μν}]$ is the Einstein-Hilbert action and $\mathcal{K}[g_{μν},π_{μν}]$ is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define a pseudo-microcanonical ensemble for the quantum fluctuations of $g_{μν}$, built on the conservation of the generalized action $\mathcal{A}[g_{μν},π_{μν}]$ rather than of energy. $S[g_{μν}]$ plays the role of a potential term along the symplectic action-preserving dynamics: its fluctuations are the quantum fluctuations of $g_{μν}$. It is shown how symplectic quantization maps to the path-integral approach to gravity. By doing so we explain how the integration over the conjugated momentum field $π_{μν}$ gives rise to a cosmological constant term in the path-integral.
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