In an attempt to attach a PDF file the original of this was bounced by the
server causing me to segment this post into parts before I realized the bounce
was due to the attachment. So here is the post again un segmented and with
one missing equation included. The attachment was a PDF file of Sakharov's
original 1967 paper on his induced gravity proposal.
Vacuum Quantum Fluctuations in Curved Space and the Theory
of Gravitation
I will make this available to anyone upon request. I apologize for these
multiple posts.
****************
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
E(w) = -(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 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 th
e 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
In addition to the two standard solutions of the quantum field equations
having the form e^{+/-(iwt-ikx)}, there exist two additional solutions of the
form e^{+/-(iwt+ikx). By incorporating these latter solutions, deemed
"supplemental solutions", into the development of quantum field theory, one finds a
simple and natural cancellation of terms that results in an energy VEV, and a
cosmological constant, of zero. This fundamental, and previously unrecognized,
inherent symmetry in quantum field theory shows promise for providing a
resolution of the large vacuum energy problem, simply and directly, with little
modification or extension to the extant mathematics of the theory. In certain
scenarios, slight asymmetries could give rise to dark energy.
We study a symmetry, schematically Energy -> - Energy, which suppresses
matter contributions to the cosmological constant. The requisite negative energy
fluctuations are identified with a "ghost" copy of the Standard Model.
Gravity explicitly, but weakly, violates the symmetry, and naturalness requires
General Relativity to break down at short distances with testable consequences.
If this breakdown is accompanied by gravitational Lorentz-violation, the
decay of flat spacetime by ghost production is acceptably slow. We show that
inflation works in our scenario and can lead to the initial conditions required
for standard Big Bang cosmology.
We propose a method of field quantization which uses an indefinite metric in
a Hilbert space of state vectors. The action for gravity and the standard
model includes, as well as the positive energy fermion and boson fields,
negative energy fields. The Hamiltonian for the action leads through charge
conjugation invariance symmetry of the vacuum to a cancellation of the zero-point
vacuum energy and a vanishing cosmological constant in the presence of a
gravitational field. To guarantee the stability of the vacuum, we introduce a Dirac
sea `hole' theory of quantization for gravity as well as the standard model.
The vacuum is defined to be fully occupied by negative energy particles with
a hole in the Dirac sea, corresponding to an anti-particle. We postulate
that the negative energy bosons in the vacuum satisfy a para-statistics that
leads to a para-Pauli exclusion principle for the negative energy bosons in the
vacuum, while the positive energy bosons in the Hilbert space obey the usual
Bose-Einstein statistics. This assures that the vacuum is stable for both
fermions and bosons. Restrictions on the para-operator Hamiltonian density lead
to selection rules that prohibit positive energy para-bosons from being
observable. The problem of deriving a positive energy spectrum and a consistent
unitary field theory from a pseudo-Hermitian Hamiltonian is investigated.
I discuss possible implications a symmetry relating gravity with
antigravity might have for smoothing out of the cosmological constant puzzle. For this
purpose, a very simple model with spontaneous symmetry breaking is explored,
that is based on Einstein-Hilbert gravity with two self-interacting scalar
fields. The second (exotic) scalar particle with negative energy density, could
be interpreted, alternatively, as an antigravitating particle with positive
energy.
We outline the evaluation of the cosmological constant in the framework of
the standard field-theoretical treatment of vacuum energy and discuss the
relation between the vacuum energy problem and the gauge-group spontaneous
symmetry breaking. We suggest possible extensions of the 't Hooft-Nobbenhuis
symmetry, in particular, its complexification till duality symmetry and discuss the
compatible implementation on gravity. We propose to use the discrete
time-reflection transform to formulate a framework in which one can eliminate the
huge contributions of vacuum energy into the effective cosmological constant
and suggest that the breaking of time--reflection symmetry could be responsible
for a small observable value of this constant
The action for gravity and the standard model includes, as well as the
positive energy fermion and boson fields, negative energy fields. The Hamiltonian
for the action leads through a positive and negative energy symmetry of the
vacuum to a cancellation of the zero-point vacuum energy and a vanishing
cosmological constant in the presence of a gravitational field solving the
cosmological constant problem. To guarantee the quasi-stability of the vacuum, we
postulate a positive energy sector and a negative energy sector in the universe
which are identical copies of the standard model. They interact only weakly
through gravity. As in the case of antimatter, the negative energy matter is
not found naturally on Earth or in the universe. A positive energy spectrum
and a consistent unitary field theory for a pseudo-Hermitian Hamiltonian is
obtained by demanding that the pseudo-Hamiltonian is ${\cal P}{\cal T}$
symmetric. The quadratic divergences in the two-point vacuum fluctuations and the
self-energy of a scalar field are removed. The finite scalar field self-energy
can avoid the Higgs hierarchy problem in the standard model.
Abstract: Energy-parity has been introduced by Kaplan and Sundrum as a
protective symmetry that suppresses matter contributions to the cosmological
constant [KS05]. It is shown here that this symmetry, schematically Energy --> -
Energy, arises in the Hilbert space representation of the classical phase
space dynamics of matter. Consistently with energy-parity and gauge symmetry, we
generalize the Liouville operator and allow a varying gauge coupling, as in
"varying alpha" or dilaton models. In this model, classical matter fields can
dynamically turn into quantum fields (Schroedinger picture), accompanied by a
gauge symmetry change -- presently, U(1) --> U(1) x U(1). The transition
between classical ensemble theory and quantum field theory is governed by the
varying coupling, in terms of a one-parameter deformation of either limit.
These corrections introduce diffusion and dissipation, leading to decoherence.
In this paper we study a new symmetry argument that results in a vacuum
state with strictly vanishing vacuum energy. This argument exploits the
well-known feature that de Sitter and Anti- de Sitter space are related by analytic
continuation. When we drop boundary and hermiticity conditions on quantum
fields, we get as many negative as positive energy states, which are related by
transformations to complex space. The paper does not directly solve the
cosmological constant problem, but explores a new direction that appears
worthwhile.
We outline the evaluation of the cosmological constant in the framework of
the standard field-theoretical treatment of vacuum energy and discuss the
relation between the vacuum energy problem and the gauge-group spontaneous
symmetry breaking. We suggest possible extensions of the 't Hooft-Nobbenhuis
symmetry, in particular, its complexification till duality symmetry and discuss the
compatible implementation on gravity. We propose to use the discrete
time-reflection transform to formulate a framework in which one can eliminate the
huge contributions of vacuum energy into the effective cosmological constant
and suggest that the breaking of time--reflection symmetry could be responsible
for a small observable value of this constant.