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[Phys-l] POSITIVE AND NEGATIVE ENERGY SYMMTERY AND INDUCED GRAVITY PT1





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 attach this paper)
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*)*hbar*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
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 an 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
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 we can that only positive
fluctuation can create the Hawking radiation.
[+ + > + [- - >
So we end up with (as one of two possibilities)
[+ +> 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 being emitted
from a black hole event horizon. Therefore we have;
Rho<0
Can we model this for negative curvature? We can use 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 we see that only negative fluctuation
can create the anti- Hawking radiation. Given
[+ - > + [+ - >
We end up with (as one of two possibilities.)
[+ -> 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
Below is listing of papers which have proposed the energy symmetry as a
solution to the zero point energy problem related above.
Bob Zannelli






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