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[Phys-l] More On Induced Gravity and The Ghost Sector.




I have made several posts on the possible connection between A.D. Sakarov's
1967 "Induced Gravity" proposal (" Vacuum Quantum Fluctuations in Curved Space
and the Theory of Gravitation.") and the new proposal by several physicists
on the existence of a negative energy "ghost" sector which is hoped will
solve the Zero Point Energy problem, the non existence of a huge vacuum energy
density which would generate a huge cosmological constant, so large in fact
the structure we see in our Universe would be totally impossible.

I think some direct quotes from Sakarov's original 1967 paper may be
helpful.

*******************

I will provide this paper in Pdf format for any interested party. Please
email me at

_RBZannelli@aol.com_ (mailto:RBZannelli@aol.com)

to request a copy
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Sakharov writes


" Here we consider the hypothesis which identifies the action (
Einstein-Hilbert action) with the change in the action of quantum fluctuations of the
vacuum if space is curved. Thus we consider the metrical elasticity of space
as a sort of level displacement effect.) ( see E.M Lifshitz zH ekap .Teor.
Fiz.29,94 ( 1954) [ Sov. Phys-JETP2, 73 (19540]

" In present-day quantum field theory it is assumed that the energy
momentum tensor of the quantum fluctuations of the vacuum T^i_k(0) and the
corresponding action S(0) , formally proportional to a divergent integral of the
fourth power over the momenta of the virtual particles of the form - Integral
k^2 dk - are actually equal to zero."

" Recently Ya B. Zel'dovitch [ ZhETF pis.Red. 6 922 ( 1967) [ JETP Lett.6
245 (1976)] suggested that gravitational interactions could lead to a
"small" disturbance of this equilibrium and thus to a finite value of Einstein's
cosmological constant , in agreement with the recent interpretation of the
astrophysical data. Here we are interested in dependence of the action of the
quantum fluctuations on the curvature of space."

End Quote

Sakharov then goes to derive the gravitational constant from the second term
of a Lagrange power series function of curvature terms.

The quantum fluctuations in question are the result of applying the
uncertainty principle to the harmonic Hamiltonian which results in


H= w[ a^dag*a +(1/2)] ( for boson fields)

Note I will ignore the fermion states because mass spitting make the
predominate contribution due to virtual boson particles, especially the
electromagnetic field.


Giving us a energy eigenvalue of


E= (N+(1/2)8hbar*w

So given the vacuum state [ 0> we have

E(w)= hbar*w/2


So we have to integrate over the whole energy spectrum of the virtual
particles


< rho> = (1/4*pi^2)*Integral { 0 to k_c} w*k^2 dk

Which is the integral Sakharov mentions in his paper.

This gives a huge divergent vacuum energy density. Even at the expected
cut off , the Planck scale, the vacuum energy density prediction is off from
the value constrained by observation by a factor of 1E120.

The solution to this dilemma is to invoke a "ghost" sector for virtual
states. In this Ghost sector we have



a_gh= -i*a^dag a_gh^dag= -i*a


Given us the energy eigen value

E= -(N+(1/2))*hbar*w


So that for the vacuum state [0> we have




E(w)= -hbar*w/2

Which gives us

< rho> =- (1/4*pi^2)*Integral { 0 to k_c} w*k^2 dk

Thereby giving us a overall zero vev for the vacuum.


However, based on the Zel'dovitch-Sakharov proposal on the effect of
gravitational interactions on the "equilibrium" of vacuum state we can postulate
that this proposed balance between the normal particle sector and the
"ghost" sector holds only in a space time metric with zero curvature. In GR we
have the relationship

R_mu,nu-(1/2)*R*g_mu,nu= -( 8*pi*G/c^4)*<rho>*g_mu,nu


So given a positive vacuum energy density we have negative curvature and
given a negative energy density we have positive curvature.


This suggests that a condensed physics analog might be appropriate in
modeling the vacuum state. And since we can equate the vacuum energy state with
space time curvature , we can equate the gravitational field with the energy
density of the vacuum relative to an observer in a gravitational field.

We can view a space time with positive curvature as related to a local
vacuum state which is "depleted" in the action density of the normal virtual
states or equivalently an increase of the action density of the virtual particles
in the ghost sector.

Likewise we can view negative curvature as related to a local vacuum state
which is "depleted" in the action density of the Ghost sector virtual states
or equivalently an increase of the action density of the virtual particles
in the normal sector.


But is there any mechanism which can relate to this "depletion?" I think
here we get a hint from the Davies-Hawking-Unruh effect. Based on the
equivalence principle gravity is an acceleration field. Therefore we see that


T= (hbar/2*pi*c)*dv/dt

In any non inertial frame virtual particles are "promoted" into real
particles. In effect acceleration "measures" the vacuum state. But does such a
process really deplete the appropriate virtual states of the vacuum to
generate the curvature of space time predicted by general relativity. Perhaps a
look at Black holes and the Hawking process in light of this proposal may be
helpful.


We need to invent a notation to make the explanation clearer.
We can define the four possible matter states , two normal, two in the ghost
sector, using the ket notation.
We use the SGN function which is
x>0 SGN[x] = +1
x<0 SGN[x] = -1
x=0 SGN[x] =0
We have the four matter states defined as
Psi= [SGN [E] SGN[t] >
So that
NORMAL SECTOR
Matter ( Positon)
psi_m= [+ + >
Anti matter ( positon)
psi_am = [- - >
GHOST SECTOR
Negative matter (Negaton)
psi_nm = [- + >
Negative anti matter (Negaton)
psi_nam = [+ - >
The Zero point energy fluctuation can be positive or negative in this
proposal.
Positive fluctuation can be modeled as
[0 0 > = [+ + > + [- - >
and a Negative Fluctuation can be modeled as
[0 0 > = [+ - > + [- + >
Now we introduce a black hole horizon and look at the Hawking process. We
can model this as a virtual pair being created and one member of the pair
crossing the horizon. Now negaton particles can not cross the horizon toward the
singularity because they anti gravitate. So only positive fluctuation create
the Hawking radiation.
[+ + > + [- - >
So we end up with say
[+ +> to infinity and [- - > crossing the horizon.
Well this represents a depletion of positon virtual states. In fact the
Hawking process does in fact require that the vacuum state just outside the
horizon violate the weak energy condition. Relative to a particle be emitted from
a black hole event horizon we have;

rho<0


Can we model this for negative curvature? We can using a thought
experiment with a negative mass black hole which would generate an anti-Horizon. Of
course such a black hole does not exist in our Universe.


Here we can say that positon particles can not cross the horizon toward
the singularity because they gravitate. So only negative fluctuation create
the anti- Hawking radiation. So given
[+ - > + [+ - >
So we end up with say
[+ -> to infinity and [+ - > crossing the horizon.
This represents a depletion of negaton virtual states.

So we would get

rho>0


But of course we can never have real negative energy particles. This is
of course true but it must be remembered all measurement events are
manifestations of the decoherence process whose functional has a positivity
requirement. therefore while the negaton is a negative energy particle in its virtual
state it can only be measured as positive energy particle. Of course this
point must be dealt with in much greater rigor than this hand waving explanation.
Hopefully a better understanding of the Decoherence process might allow this
point to be better formulated.


I will close this with another quote from Sakharov's paper.


" Consideration of the density of the vacuum Lagrange function in a
simplified model of the theory for non interacting free fields with particles M
approx k(0) shows that for fixed ratios of the masses of real particles and
"ghost" particles ( i.e. hypothetical particles which give an opposite
contribution to that of real particles to the R dependent action) a finite change of
action arises that is proportional to m^2R which we identify with R/G. Thus
the magnitude of the gravitational interaction is determined by the masses and
equations of motion of free particles , and also , probably , by the momentum
cut-off"

End quote


It occurs to me that all the ideas which have intrigued me such as the
Biverse and the application of negative energy state in the ZPE formulation was
just unknowingly following in the foot steps of A.D Sakharov.


Bob Zannelli




We have got rid of the fetish of the divine right of kings, and that
slavery is of divine origin and authority. But the divine right of
property has taken its place. The tendency plainly is towards .. "a
government of the rich, by the rich, and for the rich."
Rutherford Birchard Hayes (1822-1893), U.S. president.






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