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 $$g_{\mu \nu }$$ g μ ν 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 $$g_{\mu \nu }$$ g μ ν 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 $${\dot{g}}_{\mu \nu }$$ g ˙ μ ν of the metric field with respect to intrinsic time, corresponding to the conjugated momentum $$\pi _{\mu \nu }$$ π μ ν . Our proposal is to describe the quantum fluctuations of gravity by means of a symplectic dynamics generated by a generalized action functional $${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }] = {\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }] - S[g_{\mu \nu }]$$ A [ g μ ν , π μ ν ] = K [ g μ ν , π μ ν ] - S [ g μ ν ] , playing formally the role of a Hamilton function, where $$S[g_{\mu \nu }]$$ S [ g μ ν ] is the standard Einstein–Hilbert action while $${\mathcal {K}}[g_{\mu \nu },\pi _{\mu \nu }]$$ K [ g μ ν , π μ ν ] 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 $$g_{\mu \nu }$$ g μ ν analogous to the microcanonical one in statistical mechanics, with the only difference that in the present case one has conservation of the generalized action $${\mathcal {A}}[g_{\mu \nu },\pi _{\mu \nu }]$$ A [ g μ ν , π μ ν ] and not of energy. Since the Einstein–Hilbert action $$S[g_{\mu \nu }]$$ S [ g μ ν ] 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 $$g_{\mu \nu }$$ g μ ν . 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 $$\pi _{\mu \nu }$$ π μ ν gives rise to a cosmological constant term in the path-integral approach.