A Covariant Canonical Quantization of General Relativity (published in Advances in High Energy Physics-Hindawi) (original) (raw)
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A Covariant Canonical Quantization of General Relativity
Advances in High Energy Physics, 2018
A Hamiltonian formulation of General Relativity within the context of the Nexus Paradigm of quantum gravity is presented. We show that the Ricci flow in a compact matter free manifold serves as the Hamiltonian density of the vacuum as well as a time evolution operator for the vacuum energy density. The metric tensor of GR is expressed in terms of the Bloch energy eigenstate functions of the quantum vacuum allowing an interpretation of GR in terms of the fundamental concepts of quantum mechanics.
Hamiltonian approach to GR - Part 2: covariant theory of quantum gravity
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. As shown here the connection with CQG-theory is achieved via the classical GR Hamilton-Jacobi equation, leading to the realization of the CQG-wave equation in 4-scalar form for the corresponding CQG-state and wave-function. The new quantum wave equation exhibits well-known formal properties characteristic of quantum mechanics, including the validity of quantum hydrodynamic equations and suitably-generalized Heisenberg inequalities. In addition, it recovers the classical GR equations in the semiclassical limit, while admitting the possibility of developing further perturbative approximation schemes. Applications of the theory are pointed out with particular reference to the construction of the stationary vacuum CQG-wave equation. The existence of a corresponding discrete energy spectrum is pointed out, which provides a possible estimate for the graviton mass together with a new interpretation about the quantum origin of the cosmological constant. PACS numbers: 02.30.Xx, 04.20.Cv, 04.20.Fy, 04.60.Bc, 04.60.Ds, 04.60.Gw, 11.10.Ef
Coupling of quantum gravitational field with Riemann and Ricci curvature tensors
European Physical Journal C, 2021
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 manifestlycovariant 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 spacetime, 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.
Journal of High Energy Physics, Gravitation and Cosmology, 2022
We present a simple way to approach the hard problem of quantization of the gravitational field in four-dimensional space-time, due to non-linearity of the Einstein equations. The difficulty may be overcome when the cosmological constant is non-null. Treating the cosmological contribution as the energy-momentum of vacuum, and representing the metric tensor onto the tetrad of its eigenvectors, the corresponding energy-momentum and, consequently, the Hamiltonian are easily quantized assuming a correspondence rule according to which the eigenvectors are replaced by creation and annihilation operators for the gravitational field. So the geometric Einstein tensor, which is opposite in sign respect to the vacuum energy-momentum (plus the possible known matter one), is also quantized. Physical examples provided by Schwarzschild-De Sitter, Robertson-Walker-De Sitter and Kerr-De Sitter solutions are examined.
Covariant quantization of the gravitational field
Il Nuovo Cimento, 1962
In any quantum theory, in which the metric tensor of Einstein's gravitational theory is also quantized, it becomes meaningless to ask for an initial space-like surface on which to specify the conventional field commutators. The covariant quantum formalism, in which all fields either commute or fail to do so only when the field's points coincide, is proposed as being suitable to quantize gravity. The extension of the covariant quantum formalism to general boson fields that interact in an intrisically nonlinear way with external fields is analysed in some detail. This formalism is applied to the case of the free gravitational field. In a functional representation, the measure en metrics is found to be that proposed by Misner. A basic state of the quantized gravitational theory is proposed, which involves a summation over all permissible metrics in the entire space-time manifohl.
Von Neumann’s quantization of general relativity
Physics of Atomic Nuclei, 2017
Von Neumann's procedure is applied for quantization of General Relativity. We quantize the initial data of dynamical variables at the Planck epoch, where the Hubble parameter coincides with the Planck mass. These initial data are defined via the Fock simplex in the tangent Minkowskian spacetime and the Dirac conformal interval. The Einstein cosmological principle is applied for the average of the spatial metric determinant logarithm over the spatial volume of the visible Universe. We derive the splitting of the general coordinate transformations into the diffeomorphisms (as the object of the second Nöther theorem) and the initial data transformations (as objects of the first Nöther theorem). Following von Neumann, we suppose that the vacuum state is a quantum ensemble. The vacuum state is degenerated with respect to quantum numbers of non-vacuum states with the distribution function that yields the Casimir effect in gravidynamics in analogy to the one in electrodynamics. The generation functional of the perturbation theory in gravidynamics is given as a solution of the quantum energy constraint. We discuss the region of applicability of gravidynamics and its possible predictions for explanation of the modern observational and experimental data.
2016
Developing Planck scale physics requires addressing problem of time, quantum reduction, determinism and continuum limit. In this article on the already known foundations of quantum mechanics, a set of proposals of dynamics is built on fully constrained discrete models: 1) Self-Evolution - Flow of time in the phase space in a single point system, 2) Local Measurement by Local Reduction through quantum diffusion theory, quantum diffusion equation is rederived with different assumptions, 3) Quantum Evolution of a MultiPoint Discrete Manifold of systems through a foliation chosen dynamically, and 4) Continuum Limit, and Determinism are enforced by adding terms and averaging to the action. The proposals are applied to the various physical scenarios such as: 1) Minisuperspace reduced cosmology of isotropic and homogenous universe with scalar field, 2) Expanding universe with perturbation, and 3) Newtonian universe. Ways to experimentally test the theory is discussed. This article is a fur...
Manifest Covariant Hamiltonian Theory of General Relativity
Applied physics research, 2016
The problem of formulating a manifest covariant Hamiltonian theory of General Relativity in the presence of source fields is addressed, by extending the so-called "DeDonder-Weyl" formalism to the treatment of classical fields in curved space-time. The theory is based on a synchronous variational principle for the Einstein equation, formulated in terms of superabundant variables. The technique permits one to determine the continuum covariant Hamiltonian structure associated with the Einstein equation. The corresponding continuum Poisson bracket representation is also determined. The theory relies on first-principles, in the sense that the conclusions are reached in the framework of a non-perturbative covariant approach, which allows one to preserve both the 4-scalar nature of Lagrangian and Hamiltonian densities as well as the gauge invariance property of the theory.
Hamilton–Jacobi Wave Theory in Manifestly-Covariant Classical and Quantum Gravity
Symmetry, 2019
The axiomatic geometric structure which lays at the basis of Covariant Classical and Quantum Gravity Theory is investigated. This refers specifically to fundamental aspects of the manifestly-covariant Hamiltonian representation of General Relativity which has recently been developed in the framework of a synchronous deDonder-Weyl variational formulation (2015-2019). In such a setting, the canonical variables defining the canonical state acquire different tensorial orders, with the momentum conjugate to the field variable g µν being realized by the third-order 4-tensor Π α µν. It is shown that this generates a corresponding Hamilton-Jacobi theory in which the Hamilton principal function is a 4-tensor S α. However, in order to express the Hamilton equations as evolution equations and apply standard quantization methods, the canonical variables must have the same tensorial dimension. This can be achieved by projection of the canonical momentum field along prescribed tensorial directions associated with geodesic trajectories defined with respect to the background space-time for either classical test particles or raylights. It is proved that this permits to recover a Hamilton principal function in the appropriate form of 4-scalar type. The corresponding Hamilton-Jacobi wave theory is studied and implications for the manifestly-covariant quantum gravity theory are discussed. This concerns in particular the possibility of achieving at quantum level physical solutions describing massive or massless quanta of the gravitational field.
A note on the foundation of relativistic mechanics. II: Covariant hamiltonian general relativity
I illustrate a simple hamiltonian formulation of general relativity, derived from the work of Esposito, Gionti and Stornaiolo, which is manifestly 4d generally covariant and is defined over a finite dimensional space. The spacetime coordinates drop out of the formalism, reflecting the fact that they are not related to observability. The formulation can be interpreted in terms of Toller's reference system transformations, and provides a physical interpretation for the spinnetwork to spinnetwork transition amplitudes computable in principle in loop quantum gravity and in the spin foam models.