Coupling of quantum gravitational field with Riemann and Ricci curvature tensors

The theoretical problem of establishing the coupling properties existing between the classical and quantum gravitational field with the Ricci and Riemann curvature tensors of General Relativity is addressed. The mathematical framework is provided by synchronous Hamilton variational principles and the validity of classical and quantum canonical Hamiltonian structures for the gravitational field dynamics. It is shown that, for the classical variational theory, manifestly-covariant Hamiltonian functions expressed by either the Ricci or Riemann tensors are both admitted, which yield the correct form of Einstein field equations. On the other hand, the corresponding realization of manifestly-covariant quantum gravity theories is not equivalent. The requirement imposed is that the Hamiltonian potential should represent a positive-definite quadratic form when performing a quadratic expansion around the equilibrium solution. This condition in fact warrants the existence of positive eigenvalues of the quantum Hamiltonian in the harmonic-oscillator representation, to be related to the graviton mass. Accordingly, it is shown that in the background of the deSitter space-time, only the Ricci tensor coupling is physically admitted. In contrast, the coupling of quantum gravitational field with the Riemann tensor generally prevents the possibility of achieving a Hamiltonian potential appropriate for the implementation of the quantum harmonic-oscillator solution.

Quantum-Gravity Screening Effect of the Cosmological Constant in the DeSitter Space–Time

Small-amplitude quantum-gravity periodic perturbations of the metric tensor, occurring in sequences of phase-shifted oscillations, are investigated for vacuum conditions and in the context of the manifestly-covariant theory of quantum gravity. The theoretical background is provided by the Hamiltonian representation of the quantum hydrodynamic equations yielding, in turn, quantum modifications of the Einstein field equations. It is shown that in the case of the DeSitter space–time sequences of small-size periodic perturbations with prescribed frequency are actually permitted, each one with its characteristic initial phase. The same perturbations give rise to non-linear modifications of the Einstein field equations in terms of a suitable stochastic-averaged and divergence-free quantum stress-energy tensor. As a result, a quantum-driven screening effect arises which is shown to affect the magnitude of the cosmological constant. Observable features on the DeSitter space–time solution and on the graviton mass estimate are pointed out.