Hamiltonian approach to GR – Part 2: covariant theory of quantum gravity

Abstract

A non-perturbative quantum field theory of General Relativity is presented which leads to a new realization of the theory of covariant quantum gravity (CQG-theory). The treatment is founded on the recently identified Hamiltonian structure associated with the classical space-time, i.e., the corresponding manifestly covariant Hamilton equations and the related Hamilton-Jacobi theory. The quantum Hamiltonian operator and the CQG-wave equation for the corresponding CQG-state and wave function are realized in 4-scalar form. The new quantum wave equation is shown to be equivalent to a set of quantum hydrodynamic equations which warrant the consistency with the classical GR Hamilton-Jacobi equation in the semiclassical limit. A perturbative approximation scheme is developed, which permits the adoption of the harmonic oscillator approximation for the treatment of the Hamiltonian potential. As an application of the theory, the stationary vacuum CQG-wave equation is studied, yielding a stationary equation for the CQG-state in terms of the 4-scalar invariant-energy eigenvalue associated with the corresponding approximate quantum Hamiltonian operator. The conditions for the existence of a discrete invariant-energy spectrum are pointed out. This yields a possible estimate for the graviton mass together with a new interpretation about the quantum origin of the cosmological constant.

Generalized Lagrangian Path Approach to Manifestly-Covariant Quantum Gravity Theory

“…First, CQG-theory is intrinsically non-perturbative in character, so that the background metric tensor can be identified with an arbitrary continuum solution of the Einstein equations (not necessarily the flat space-time), while a priori the canonical variable g µν is not required to be necessarily a perturbation field. On the other hand, a decomposition of the type in Equation (5) resembling the one invoked in covariant literature approaches can always be introduced a posteriori for the implementation of appropriate analytical solution methods, like GLP theory proposed here or the analytical evaluation of discrete-spectrum quantum solutions discussed in [10]. Second, the present theory is constructed starting from the DeDonder-Weyl manifestly-covariant approach [12,13].…”

“…A realization of the parameterization ψ ≡ ψ(g, r, s) is provided by the geodetics of the metric field tensor g ≡ g(r), namely the integral curves of the initial-value problem [9,10] . Since each point r µ ≡ r µ (s) can be crossed by infinite arbitrary geodetics having different tangent 4-vectors t µ ≡ t µ (s) it follows that the wave-function parameterization ψ ≡ ψ(g, r, s) may generally depend explicitly on the choice of the geodetics, i.e., on t µ too.…”

“…This task is achieved by means of the CQG-wave equation [10]. Written again in the Eulerian form, the latter is realized by the initial-value problem…”

“…This property will be denoted here as "second-type emergent-gravity paradigm". A basic requirement needed for its verification is manifestly the determination of a suitable class of particular solutions of the quantum wave-function, i.e., the CQG-wave equation for the quantum state ψ(g, r, s) earlier pointed out in [10].…”

“…As shown in Section 5, the adoption of the GLP parameterization for CQG-theory actually leaves unchanged the underlying axioms established in [10], thus providing a Lagrangian representation of CQG-theory which is ontologically equivalent to CQG-theory itself. The remarkable new aspects of the GLP formalism, however, are that it will be shown: First, to determine a solution method for the CGQ-wave equation, to be referred to here as GLP-approach, permitting the explicit construction of physically-relevant particular realizations of the CGQ-quantum state ψ (s).…”