Symplectic Quantization II: Dynamics of Space\u2013Time Quantum Fluctuations and the Cosmological Constant

Abstract

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.