Abstract
Space-time quantum contributions to the classical Einstein equations of General Relativity are determined. The theoretical background is provided by the non-perturbative theory of manifestly-covariant quantum gravity and the trajectory-based representation of the related quantum wave equation in terms of the Generalized Lagrangian path formalism. To reach the target an extended functional setting is introduced, permitting the treatment of a non-stationary background metric tensor allowed to depend on both space-time coordinates and a suitably-defined invariant proper-time parameter. Based on the Hamiltonian representation of the corresponding quantum hydrodynamic equations occurring in such a context, the quantum-modified Einstein field equations are obtained. As an application, the quantum origin of the cosmological constant is investigated. This is shown to be ascribed to the non-linear Bohm quantum interaction of the gravitational field with itself in vacuum and to depend generally also on the realization of the quantum probability density for the quantum gravitational field tensor. The emerging physical picture predicts a generally non-stationary quantum cosmological constant which originates from fluctuations (i.e., gradients) of vacuum quantum gravitational energy density and is consistent with the existence of quantum massive gravitons.
Highlights
The theory of manifestly-covariant quantum gravity (CQG-theory) recently proposed in a series of papers provides a possible new self-consistent route to Quantum Gravity and the cosmological interpretation of quantum vacuum
Our claim is that CQG-theory gives rise to a well-defined quantum prescription of the cosmological constant, its physical interpretation being ascribed to the action of the non-linear quantum vacuum interaction of the gravitational field with
In the literature such a route is usually regarded as to lead to unphysical predictions. It is worth mentioning in this regard that recent numerical calculations including 28 bosonic fields and based on the estimate of the stochastic fluctuations of the associated total quantum-vacuum energy density, rather than the energy density itself, are claimed to provide lower estimates and a resulting acceleration of the universe comparable to the observed one [72,73]. This type of studies should be regarded as complementary to the present quantum theory, they differ from it for the following main reasons: (1) they are based on numerical calculations, and are subject to the accuracy of the numerical codes implemented, while the theory proposed here is analytical; (2) they assume that the source of the universe expansion is the vacuum populated by bosonic fields, while in the present model the cosmological constant is shown to arise purely from quantum gravitational field with its quantum dynamics being predicted by CQG-theory, without needing to invoke any additional field; (3) they realize non-manifestly covariant solutions in which the coordinate time is singled out with respect to space coordinates, while the investigation based on CQG-theory preserves manifest covariance
Summary
The theory of manifestly-covariant quantum gravity (CQG-theory) recently proposed in a series of papers (see Refs. [1,2,3,4,5,6]) provides a possible new self-consistent route to Quantum Gravity and the cosmological interpretation of quantum vacuum. The analytical estimate for the graviton mass and its quantum discrete invariant energy spectrum, supporting the interpretation of the graviton DeBroglie length as being associated with the quantum ground-state related to the cosmological constant [4] It must be stressed in this connection that the prediction of massive gravitons represents an intrinsic property of CQG-theory which marks an important point of distinction with respect to past literature. [4], the existence of massive gravitons and their mass estimate are found to be associated with a non-vanishing cosmological constant Based on these outcomes, the target of this paper is to show that in the context of the CQG-theory the equation for the background metric tensor gb can self-consistently be determined by the CQG-wave equation itself, together with its relationship with the classical Einstein equations. As we intend to show, besides quantum gravity theory itself, this is relevant in the context of theoretical astrophysics and cosmology to reach a quantum-gravity interpretation/explanation of selected physical evidences emerging from large-scale phenomenology of the universe
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