A Quantized Hamiltonian Formulation of General

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A Quantized Hamiltonian Formulation of General Relativity Stuart Marongwe

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Abstract 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 closed matter free manifold serves as the Hamiltonian density of the vacuum as well as a time evolution operator for the vacuum energy density. The closed matter free manifold in which this formalism occurs is a geodesic ball mapped by a four vector of Minkowski space. The four vector is represented as a expanding or contracting spherically symmetric wave pact of space-time called the Nexus graviton. Matter fields are then introduced into the Nexus graviton to derive the Hamiltonian for a quantized vacuum perturbed by matter fields. Keywords Quantum Gravity; Quantization; General Relativity; Dark Matter; Dark Energy; Voids.

1 Introduction 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 [1], named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner first published in 1959. This formalism starts from the assumption that space is foliated into a family of time slices Σt , labeled by their time coordinate t, and with space coordinates on each slice given by xk . The dynamic variables of Department of Physics and Astronomy Botswana International University of Science and Technology P. Bag 16, Palapye Botswana [email protected]

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this theory are then taken to be the metric tensor of three dimensional spatial slices γij (t, xk ) and their conjugate momenta nij (t, xk ). Using these variables it is possible to define a Hamiltonian, and thereby write the equations of motion for GR in Hamilton’s form. The time slices are then welded together using four Lagrange multipliers and components of a shift vector field. An extensive review of this formalism can be found in the literature notably in [2-5]. The ADM formalism was first applied by Bryce De Witt in 1967 [6-8] to quantize gravity which resulted in the Wheeler De Witt equation of quantum gravity. It is a functional differential equation in which the three dimensional spatial metrics have the form of an operator acting on a wave function. This wave function contains all of the information about the geometry and matter content of the universe of each time slice. However, the Hamiltonian no longer determines the evolution of the system and leads to the problem of timelessness. Hawking rightly points out that the very act of splitting space-time into space and time destroys the spirit of GR and therefore not much can be gained from this approach to quantization of gravity. In this letter we report a successful 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 a closed manifold mapped by a four vector of Minkowski space called the Nexus Graviton. Though the Nexus Paradigm has been introduced in the following papers [9-11] the aim of this letter is to explicitly express the Hamiltonian formulation of the theory using the wave function of the quantum vacuum. The wave function of this formalism contains information about the state of the quantum vacuum which in turn dictates the geometry of spacetime. 2 The Nexus Graviton The Nexus graviton in the n-th quantum state as described in Refs:[9-11]is a four displacement vector in Minkowski space whose components are Z ∞ i(kµ xµ ) µ 2∆rHS e sin(kµ xµ )dkµ γµ (1) ∆xµn (kµ ) = nπ kµ xµ −∞ and are associated with a four-momentum Z ∞ i(kµ xµ ) 2n∆pµ1 e sin(kµ xµ )dkµ µ ∆pn = γµ π kµ xµ −∞

(2)

where ∆pµ1 is the four-momentum of the ground state graviton and γµ are the Dirac gamma matrices. The minimum length of the displacement vector is the µ Planck four length and the maximum being the Hubble four radius rHS . Thus the Nexus graviton can exist in 1060 eigenstates and can also be intepreted as a pulse of four-space ∆xµ associated with a four-momentum ∆pµ such that they satisfy the Uncertainty relation ¯h ∆pµn ∆x(n)µ ≥ (3) 2

A Quantized Hamiltonian Formulation of General Relativity

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Hence a displacement four vector in Minkowski space can be considered as a Gaussian wave packet consisting of a linear superposition of waves characterized by the following eigen four wave vectors. kµ,n =

nπ where n = ±1, ±2...1060 aµ0

(4)

µ where aµ0 = rHS The wave packet is essentially a particle of 4-space in which the four vector traces out a geodesic ball. The geodesic ball is the closed manifold. The spin of this particle can be determined from the fact that each component of the displacement four vector transforms according to the law 0 1 ∆a0µ = exp( ωµν [γµ , γν ])∆aµ0 8

(5)

Where ωµν is an antisymmetric 4x4 matrix parameterizing the transformation Therefore, each component of the four vector has a spin half. A summation of all the four half spins yields a total spin of 2 for the Nexus Graviton. The four momentum of the n-th state graviton is 2 En2 nhH0 = 3 (6) c2 c where H0 is the Hubble constant (2.2 × 10−18 s−1 ) . This four-momentum is expressed in terms of the cosmological constant, Λ as 2 2 kn En = = n2 Λ (7) Λn = hc 2π The curvature of space-time in the n-th quantum state is thus expressed as G(n)µν = n2 Λgµν

(8)

where G(n)µν is the Einstein tensor of space-time in the n-th state. Eqn.(8) depicts a contracting geodesic ball. The DE which results in an expanding geodesic ball arises from the emission of a ground state graviton such that Eqn.(8) becomes G(n)µν = (n2 − 1)Λgµν (9) In the presence of baryonic matter Eqn.(9) becomes G(n)µν = kTµν + (n2 − 1)Λgµν

(10)

The Schwarzchild solution to Eqn.(8) as found in Ref:[10] shows that the curvature of space-time is a function of its quantum state as follows ds2 = −(1 −

2 2 2 2 )c dt + (1 − 2 )−1 dr2 + r2 (dθ2 + sin2 θdφ2 ) n2 n

There are no singularities in Eqn.(11).

(11)

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3 The Hamiltonian formulation for the quantum vacuum The Nexus graviton is a pulse of space-time which can expand or contract but cannot execute translational motion implying that the Hamiltonian density of the system is equal to the Lagrangian density. H=L

(12)

GR is a metric field therefore it is imperative that the metric be expressed in terms of a wave function. To this end we shall express the four displacement vector components in the form Z ∞ µ 2∆rHS µ µ γµ ψn,k dkµ (13) ∆xn (kµ ) = z(kµ ) = nπ −∞ µ where ψn,k are Bloch eigenstate functions of the quantum vacuum in the n-th state. An elementarly four vector can be computed as

drµ =

∂z µ (k µ ) µ dk ∂k µ

(14)

The interval is then computed as ds2 = drµ drν =

∂z µ (k µ ) ∂z ν (k ν ) µ ν µ ν dk dk = cµ cν ψ(n,k) ψ(n,k) dk µ dk ν ∂k µ ∂k ν

(15)

Here the interval is described in terms of the reciprocal lattice. The metric tensor of the Nexus graviton in the n-th quantum state is therefore associated with the Bloch eigenstate functions of the quantum vacuum as follows gµν = cµ cν ψ(n,k)µ ψ(n,k)ν

(16)

The Lagrange density for Eqn. (9) following Einstein and Hilbert is LEH = k(R − 2(n2 − 1)Λ

(17)

Given that the Einstein tensor in a closed manifold is equal to the Ricci flow 1 −∂t gµν = ∆gµν = Rµν − Rgµν = Gµν 2

(18)

The equations of motion of the quantum vacuum obtained from Eqn.(17) yield the following quantized field equations −∂t (ψ(n,k)µ ψ(n,k)ν) = (n2 − 1)Λψ(n,k)µ ψ(n,k)ν

(19)

which can be written as ∂t (ψ(n−1,k)µ ψ(n+1,k)ν = =

−i2 ∇µ ∇ν ψ(n−1,k)µ ψ(n+1,k)ν (2π)2 )

1 ∇µ ∇ν ψ(n−1,k)µ ψ(n+1,k)ν 4π 2

(20) (21)


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where ψ(n−1,k)µ = sinc((n − 1)k1µ xµ )ei(n−1)k1 x

(22)

ψ(n+1,k)ν = sinc((n + 1)k1ν xν )ei(n+1)k1 x

(23)

k12 4π 2

=Λ

(24)

The quantum vacuum can therefore be interpreted as a system in which there is a constant annihilation and creation of quanta as implied by Eqn.(22) and Eqn.(23) which causes the Nexus graviton to either expand or contract. The exchange of quanta within the Nexus graviton generates regions of low and high quantum states. A region of low quantum state is Dark Matter and a region of high quantum state results in a void.

3.1 Hamiltonian formulation in the presence of matter fields We now seek to introduce matter fields into the quantum vacuum. If we compare the quantized metric of Eqn.(11) with the Schwarzschild metric we notice that 2GM (r) 2 = n2 c2 r

(25)

This yields a relationship between the quantum state of space-time and the amount of baryonic matter embedded within it as follows n2 =

c2 r c2 = 2 GM (r) v

(26)

Here v is the intrinsic flow rate of space-time in the n-th quantum state. The result of Eqn.(26) can be added to Eqn.(19) to yield the Hamiltonian formulation of the quantum vacuum in the presence of baryonic matter as 1 ∇µ ∇ν ψ(n−1,k)µ ψ(n+1,k)ν − n2 Λψ(n,k)µ ψ(n,k)ν 4π 2 (27) The first term on the right is the field momentum and the second term is the field momentum in the presence of baryonic matter. The second term is also an 8-cell or 4-cube that operates as a sinc filter with a four-wave cut-off of s c2 r µ µ kc = nk1 = 2π Λ. (28) GM (r) ∂t (ψ(n−1,n,k)µ ψ(n+1,n,k)ν =

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.

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We now introduce a test particle of mass m, into the quantum vacuum perturbed by matter fields. The particle will flow along with the Ricci flow and the Hamiltonian of the system becomes ˆ ˆ ˆ (n−1,n,k)µ ψ(n+1,n,k)ν = Pµ Pν ψ(n−1,k)µ ψ(n+1,k)ν − V ψ(n,k)µ ψ(n,k)ν (29) H(ψ 2m Here V = n2

h2 rc2 ¯h2 k12 Λ= . 2m GM (r) 2m

(30)

Pˆµ = −i¯ h∇µ

(31)

ˆ µ = −i¯ H h∂t

(32)

Eqn.(29) is equivalent to Eqn.(12) in which the Hamiltonian is equal to the Lagrangian. ˆ (n−1,n,k)µ ψ(n+1,n,k)ν = L(ψ(n−1,n,k)µ ψ(n+1,n,k)ν H(ψ

(33)

The Lagrangian is not relativistic because the energy reference frames involved in the transition dynamics n − 1, n and n + 1 are separated by a small energy gap. The exchanged quantum of energy between states is responsible for ’welding’ them together. Eqn.(30) is the gravitational interaction between the states. The weakness of the gravitational interaction is due to the small value of the cosmological constant such that the energy of the exchanged quantum of gravity is √ (34) E = 3.hH0 A measurement of the value of the cosmological constant is therefore a detection of a quantum of gravity.

4 Conclusion A successful canonical quantization of the gravitational field has been presented in which we find that gravity is a flow of space-time and that spacetime 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 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. For instance, the sinc filtter could be shielding the heavy Higgs boson from high energy quantum vacuum fluctuations resulting in the Higgs’ unaturally low expected mass.


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5 Acknowledgements I gratefully appreciate the support from the Department of Physics and Astronomy at the Botswana International University of Science and Technology (BIUST).

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