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 $g_{mu u}$ 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 $g_{mu u}$ 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 $dot{g}_{mu u}$ of the metric field with respect to proper time, corresponding to the conjugated momentum $pi_{mu u}$. Symplectic quantization describes the quantum fluctuations of gravity by means of the symplectic dynamics generated by a generalized action functional $mathcal{A}[g_{mu u},pi_{mu u}] = mathcal{K}[g_{mu u},pi_{mu u}] - S[g_{mu u}]$, playing formally the role of a Hamilton function, where $S[g_{mu u}]$ is the Einstein-Hilbert action and $mathcal{K}[g_{mu u},pi_{mu u}]$ 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 $g_{mu u}$, built on the conservation of the generalized action $mathcal{A}[g_{mu u},pi_{mu u}]$ rather than of energy. $S[g_{mu u}]$ plays the role of a potential term along the symplectic action-preserving dynamics: its fluctuations are the quantum fluctuations of $g_{mu u}$. 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 $pi_{mu u}$ gives rise to a cosmological constant term in the path-integral.