Hindawi Advances in High Energy Physics Volume 2018, Article ID 4537058, 7 pages https://doi.org/10.1155/2018/4537058 Research Article A Covariant Canonical Quantization of General Relativity Stuart Marongwe Department of Physics and Astronomy, Botswana International University of Science and Technology, P. Bag , Palapye, Botswana Correspondence should be addressed to Stuart Marongwe; [email protected] Received 13 June 2018; Accepted 8 November 2018; Published 19 December 2018 Guest Editor: Farook Rahaman Copyright © Stuart Marongwe. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e publication of this article was funded by SCOAP. 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. e 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. 1. Introduction the geometry and matter content of the universe of each time slice. However, the Hamiltonian no longer determines e gravitational field which is elegantly described by Ein- the evolution of the system and leads to the problem of stein’s field equations has so far eluded a quantum descrip- timelessness [, ]. Hawking rightly points out that the very tion.MuchefforthasbeenplacedintoformulatingGeneral act of splitting space-time into space and time destroys the Relativity (GR) in terms of Hamilton’s equations since a spirit of GR (general covariance) and therefore not much Hamiltonian formulation of a classical field theory leads can be gained from this approach to quantization of gravity. naturally to its quantization. e earliest such attempt is the Perturbative covariant approaches to the problem of quantum ADM formalism [], named for its authors Richard Arnowitt, gravity have an inherent weakness in that they depend on Stanley Deser, and Charles W. Misner first published in a fine classical background. It is therefore difficult to obtain . is formalism starts from the assumption that space a self-consistent quantum theory of gravity with a classical Σ is foliated into a family of time slices �,labeledbytheir background space-time. Steven Carlip [ , ] and Claus time coordinate t and with space coordinates on each slice � Kiefer [, ] have made excellent and extensive reviews of given by � . e dynamic variables of this theory are then the problems faced by current approaches to the problem taken to be the metric tensor of three-dimensional spatial of quantum gravity. Carlip [ ] in particular singles out the � �� � slices ��� (�.� ) andtheirconjugatemomenta� (�.� ).Using lack of a firm conceptual understanding of the foundational these variables it is possible to define a Hamiltonian and concepts of quantum gravity as the source of much of the thereby write the equations of motion for GR in Hamilton's difficulty in understanding quantum gravity. ese deep form.etimeslicesarethenweldedtogetherusingfour conceptual issues result in technical problems in the attempt Lagrange multipliers and components of a shi vector field. to develop a consistent quantum theory of gravity. An extensive review of this formalism can be found in the In this paper we report a successful covariant canonical literature notably in [– ]. quantization of the gravitational field which preserves the e ADM formalism was first applied by Bryce De Witt in success of GR while simultaneously explaining Dark Energy [] to quantize gravity which resulted in the Wheeler–De (DE) and Dark Matter (DM). is approach to quantization Witt equation of quantum gravity. It is a functional differen- takes place in -space of metric signature (-,,,) in which tial equation in which the three-dimensional spatial metrics the quanta are excitations of the quantum vacuum called havetheformofanoperatoractingonawavefunction. Nexus gravitons. ough the Nexus Paradigm has been is wave function contains all of the information about introduced in the following papers [– ], the aim of this Advances in High Energy Physics study is to explicitly express the Hamiltonian formulation where � is the amplitude of a displacement vector in of the theory using the Bloch eigenstate functions of the Minkowski space. If we consider the Hubble diameter as quantum vacuum. ese wave functions contain information the maximum dimension of the local patch of space then about the energy state of the quantum vacuum which in turn �=��� where ��� istheHubbleradius.ischoiceofa dictates the geometry of space-time. maximum amplitude is justified from the fact that we cannot physically interact with objects beyond the Hubble -radius. 2. Methods Itisimportanttonotethatthelineelementisthesquare of the amplitude of the displacement -vector. us if we Our first step towards a covariant canonical quantization are to express GR in terms of the language of QM we must begins with defining a quantized space-time and its quanta. make the radical assumption that the displacement vectors We then modify Einstein’s vacuum equations to be consistent in Minkowski space are pulses of -space which can be with the quantized space-time followed by the defining of expressed in terms of Fourier functions as follows: Hamilton’s equations of the quantized space-time. is step ∞ is then followed by the Poisson brackets which provide the � 2��� � ��� Δ�� = � ∫ sinc (��) � �� bridge between classical and quantum mechanics (QM). e �� −∞ ∞ ( ) covariant canonical quantization procedure is carried out � =� ∫ ����(���)�� within the context of the Nexus Paradigm of quantum gravity. −∞ 2� �=+∞ .. Quantization of -Space and the Nexus Graviton. e pri- �� = ∑ � Where �� �� () mary objective of physics is the study of functional relation- �=−∞ ships amongst measurable physical quantities. In particular, a unifying paradigm of physical phenomena should reveal ��� Here �(���) = sinc(��)� are Bloch energy eigenstate the functional relationship between the fundamental physical functions. e Bloch functions can only allow the four wave quantities of -space and -momentum. Currently GR and vector to assume the following quantized values: QM offer the best predictions of the results of measurement �� of physical phenomena in their respective domains using �� = �=±1,±2...1060 � () different languages. GR describes gravitation in the language ��� of geometry and thus far, it has been difficult to apply thethelanguageofwavefunctionsusedinQMtogive e minimum -radius in Minkowski space is the Planck - it a QM description. e problem of quantum gravity is length since it is impossible to measure this length without 60 therefore to interpret GR in terms of the wavefunctions of forming a black hole. e statesarisefromtheratioof QM.TranslatingthelanguageofmeasurementofGRintothat Hubble -radius to the Planck -length. e displacement - of QM becomes the primary objective of this present attempt vectors in each eigenstate of space-time generate an infinite to resolve the problem of quantum gravity. Bravais -lattice. Also, condition () transforms ( ) to Measurements in GR take place in a local patch of a ��1 Reimannian manifold. is local patch can be considered � � Δ�� =� ∫ ����(�,�,�)�� () asaflatMinkowskispace.elineelementinMinkowski −��1 space which is the subject of measurement can be computed through the inner product of the local coordinates as e second assumption we make is that each displacement - � 2 2 2 2 2 vector is associated with a conjugate pulse of four-momentum Δ� Δ�� =Δ� +Δ� +Δ� −� Δ� which can also be expressed as a Fourier integral =(�Δ�+�Δ�+�Δ�+���Δ�) () �� 2��1 1 ⋅ (�Δ� + �Δ� + �Δ� + ���Δ�) Δ�(�)� = �� ∫ �(�,�,�)�� � −�� 1 ( ) On multiplying the right hand side we see that to get all the ��1 cross terms such as Δ�Δ� to cancel out we must assume =�� ∫ ����(�,�,�)�� −��1 �� + �� = 0 () where �(1)� is the four-momentum of the ground state. �2 =�2 =⋅⋅⋅=1 A displacement -vector and its conjugate -momentum e above conditions therefore imply that the coefficients satisfy the Heisenberg uncertainty relation (�,�,�,�) generate a Clifford algebra and therefore must ℎ be matrices. We rewrite these coefficients in the -tuple Δ� Δ� ≥ () 1 2 3 0 � � 2 form as (� ,� ,� ,� ) which may be summarized using the Minkowski metric on space-time as follows: e Uncertainty Principle plays the important role of gener- � ] �] {� ,� }=2� () ating a vector bundle, out of the total uncertainty space E of trivial displacement -vectors from which a closed compact egammasareofcoursetheDiracmatrices.usinorder manifold X is formed, i.e., (� : � �→ �). Each point on the to satisfy () we can express a displacement -vector as manifold is associated with a vector which is along a normal � � Δ� =�� () to the manifold. Advances in High Energy Physics 2 2 −1 e wave packet described by () is essentially a particle of ��2 =−(1−( )) �2��2 +(1−( )) ��2 2 2 four-space. e spin of this particle can be determined from � � () 2 2 2 2 the fact that each component of the four-displacement vector +� (�� + sin ��� ) transforms according to the law ere are no singularities in (). At high energies, character- �� 1 � ized by microcosmic scale wavelengths of the Nexus graviton Δ� = exp ( ��� [��,��]) Δ� () � 8 � and high values of n,space-timeisflatandhighlycompact.