^{1}

^{3}.

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.

The gravitational field which is elegantly described by Einstein’s field equations has so far eluded a quantum description. Much effort has been placed into formulating General Relativity (GR) in terms of Hamilton’s equations since a Hamiltonian formulation of a classical field theory leads naturally to its quantization. The earliest such attempt is the ADM formalism [

The ADM formalism was first applied by Bryce De Witt in 1967 [

In this paper we report a successful covariant canonical quantization of the gravitational field which preserves the success of GR while simultaneously explaining Dark Energy (DE) and Dark Matter (DM). This approach to quantization takes place in 4-space of metric signature (-1,1,1,1) in which the quanta are excitations of the quantum vacuum called Nexus gravitons. Though the Nexus Paradigm has been introduced in the following papers [

Our first step towards a covariant canonical quantization begins with defining a quantized space-time and its quanta. We then modify Einstein’s vacuum equations to be consistent with the quantized space-time followed by the defining of Hamilton’s equations of the quantized space-time. This step is then followed by the Poisson brackets which provide the bridge between classical and quantum mechanics (QM). The covariant canonical quantization procedure is carried out within the context of the Nexus Paradigm of quantum gravity.

The primary objective of physics is the study of functional relationships amongst measurable physical quantities. In particular, a unifying paradigm of physical phenomena should reveal the functional relationship between the fundamental physical quantities of 4-space and 4-momentum. Currently GR and QM offer the best predictions of the results of measurement of physical phenomena in their respective domains using different languages. GR describes gravitation in the language of geometry and thus far, it has been difficult to apply the the language of wave functions used in QM to give it a QM description. The problem of quantum gravity is therefore to interpret GR in terms of the wavefunctions of QM. Translating the language of measurement of GR into that of QM becomes the primary objective of this present attempt to resolve the problem of quantum gravity.

Measurements in GR take place in a local patch of a Reimannian manifold. This local patch can be considered as a flat Minkowski space. The line element in Minkowski space which is the subject of measurement can be computed through the inner product of the local coordinates as^{60} states arise from the ratio of Hubble 4-radius to the Planck 4-length. The displacement 4-vectors in each eigenstate of space-time generate an infinite Bravais 4-lattice. Also, condition (

A displacement 4-vector and its conjugate 4-momentum satisfy the Heisenberg uncertainty relation

The wave packet described by (

Therefore, each component of the 4-vector has a spin half. A summation of all the four half spins yields a total spin of

From (_{0} is the Hubble constant (2.2 x 10^{−18} s^{−1}) and can be expressed in terms of the cosmological constant,

From [

The gravitational effects of the Nexus graviton manifest at large scales and for galaxies, these effects begin to manifest when the density of DE is equal to the density of baryonic matter as described by (

The first term on the right is the Newtonian gravitational acceleration, the second term is a radial acceleration due to space-time in the

In classical mechanics, a system is described by

We can also rewrite the Hamiltonian equations in terms of Poisson brackets which are invariant under canonical transformations as

The Nexus graviton is a pulse of 4-space which can only expand or contract and does not execute translational motion implying that the Hamiltonian density of the system is equal to the Lagrangian density.

We initiate the translation procedure of GR into QM by first finding the Lagrange density for the quantized vacuum from (

We now seek to introduce matter fields into the quantum vacuum. If we compare the quantized metric of (

Having obtained the relationship between the quantum state of space-time and the amount of baryonic matter embedded within it, the result of (

The filtration of high frequencies from the vacuum lowers the quantum vacuum state and generates a gravitational field in much the same way as the Casimir Effect is generated. The physical process of filtration occurs as follows: A 4-cube with baryonic matter embedded with in it acquires inertia. The inertia gives the cell inductive impedance and becomes less reponsive to high frequency vibrations of the quantum vacuum. Thus the heavier the cell, the lower the cut-off frequency. Equation (

We now introduce a test particle of mass

Equation (

A close inspection of (^{120}. This huge number represents the maximum number of pixels that can fit on the largest surface in K-space. Hence the expression

Shadow of Sagittarius A

The radii are calculated from (

A successful covariant canonical quantization of the gravitational field has been presented in which we find that gravity is akin to a thermal flow of space-time and that space-time can also be described in terms of a reciprocal lattice. The presence of matter creates an impure lattice and a potential well arises in the region of perturbation through filtration of high frequencies from the quantum vacuum. A test particle flows along with the quantum vacuum and the presence of a potential well will cause it to flow towards the sink. This formulation will find important applications in high energy lattice gauge field theories where it may help expand the standard model of particle physics by eliminating divergent terms in the current theory and predicting hitherto unknown phenomena.

The results of this theoretical research will be confirmed by data from the Event Horizon Collaboration on the size and shape of the shadow of Sgr A

The author declares no conflicts of interest.

The author gratefully appreciates the funding and support from the Department of Physics and Astronomy at the Botswana International University of Science and Technology (BIUST).