Symplectic quantization II: dynamics of space-time quantum fluctuations and the cosmological constant

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 $g_{\mu\nu}$ 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 $g_{\mu\nu}$ 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 proper time, corresponding to the conjugated momentum $\pi_{\mu\nu}$. Symplectic quantization describes the quantum fluctuations of gravity by means of the 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 Einstein-Hilbert action and $\mathcal{K}[g_{\mu\nu},\pi_{\mu\nu}]$ is a new term including the kinetic degrees of freedom of the field. Such an action allows us to define a pseudo-microcanonical ensemble for the quantum fluctuations of $g_{\mu\nu}$, built on the conservation of the generalized action $\mathcal{A}[g_{\mu\nu},\pi_{\mu\nu}]$ rather than of energy. $S[g_{\mu\nu}]$ plays the role of a potential term along the symplectic action-preserving dynamics: its fluctuations are the quantum fluctuations of $g_{\mu\nu}$. 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 $\pi_{\mu\nu}$ gives rise to a cosmological constant term in the path-integral.