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 (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 spacetime continuum. For this reason symplectic quantization necessarily incorporates a new degree of freedom, the derivative 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 A[g_{mu,nu}, pi_{mu,nu}] = 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 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 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.