Credit-address for the header photo. In this post, I will discuss and use non-linear multigravity theory to model quantum foam and probe solutions to the cosmological constant cosmic/Planck-scales ‘discrepancy paradox’, related to the hierarchy problem: namely, the 10-47GeV 4/EZP E ≈ 1071GeV cut-off one. Note first that spacetime/quantum randomness-foamy-chaos can be interpreted as a large N composition of Schwarzschild wormholes with a scalar curvature in n-dimensions being

with the Regge-Wheeler hypersurface and let me begin with the action involving N massless gravitons without matter fields

with  and  being the cosmological constant and the Newton constant corresponding to the i-th universe, respectively, and the total action takes the following form

In this way, the action  describes a Bose-Einstein condensates of gravitons. Start with the N = 1 Einstein field equations

with  the Einstein tensor and injecting a time-like unit vector  such that yields

which is the Hamiltonian constraint expressed in terms of the equation of motion. So far, we are at the classical ‘level’. The discrepancy between the observed classically-regimed cosmological constant and the quantum-numerical result is in its quantum version: that is, one can derive the expectation value

and given that

dimensional analysis for 3-D gives us

with the r.h.s. equal to

A

and we integrated over the Regge-Wheeler hypersurface and divided by its volume, and it can be derived starting with the Wheeler-De Witt equation which represents invariance under time reparametrization: that is the Sturm-Liouville cosmological constant problem

The boundary conditions are given by the choice of the quantum fluctuational Gaussian wavefunctionals. Extracting the TT tensor, second order in perturbation, contribution from A, of the spatial part of the metric into a background term, and a perturbation,  , yields

B

with  the inverse DeWitt metric and with the following definition

for the propagator

with  the eigenfunctions of .

Now, the expectation value of  is obtained by inserting the form of the propagator into A and minimizing with respect to the variational function . So, the total one loop energy density for TT tensors is

and its contribution to the spin-two operator for the Schwarzschild metric is

and  is the scalar curved Laplacian, given by

with

the mixed Ricci tensor

hence, the scalar curvature is traceless

Thus, we must analyse the eigenvalue equation

the eigenvalue of the corresponding equation. Following Regge-Wheeler, the 3-D gravitational perturbation is represented by its even-parity form

Hence, the system

from the throat of the bridge, becomes

Hence, we have, for  and

and

So, one can write

where we have

We can now explicitly evaluate:

in terms of the effective mass. Via the ‘t Hooft BWM method, one can derive

by using the zeta function regularization method to numerically analyse the energy densities  and by introducing the mass parameter so as to restore the correct dimension for the regularized quantities. So the energy density is renormalized due to the absorption divergences, yielding the classical constant

removing the dependence on the mass scale , it is appropriate to use the renormalization group equation, which means imposing:

After solving, one realizes that the renormalized constant  should be treated as a running one

and the cosmological constant takes the form

with a minimum

and the condition

We are …