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

TL;DR

This paper extends symplectic quantization to space-time quantum fluctuations, introducing a new degree of freedom and linking quantum gravity fluctuations to the cosmological constant via path-integral methods.

Contribution

It generalizes symplectic quantization to include space-time fluctuations, incorporating a conjugate momentum for the metric and connecting to the cosmological constant.

Findings

Symplectic quantization describes quantum gravity fluctuations with a generalized action.

The approach maps to the path-integral formulation of gravity.

Integration over conjugate momentum yields a cosmological constant.

Abstract

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.