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Vacuum Energy

Vacuum energy is an underlying background energy
that exists in space throughout the entire Universe.
In Quantum Field Theory, the fields are comprised
of harmonic oscillators with energy E = (n + 1/2)hω.
Each oscillator has a zero point energy of E = (1/2)hω.
That is to say that at 0 displacement, an oscillator
still has some energy. The 0 point energy is a
consequence of the Uncertainty Principle. Consider
the analysis for the photon field in 2 dimensional
momentum (k) space.
k_{1}

 + 2,2
  
 +
 1,1
 k_{2}
k = 2π/λ = 2πn/L
Let n = 1 so the box has sides of length, k = 2π/L.
Consider a volume in this space, V_{k}.
The number of modes, N, in a cube with sides, L is
given by:
N = V_{k}/k^{3}
= L^{3}V_{k}/(2π)^{3}
The volume density of modes, n, is given by:
n = N/Total volume of space = N/L^{3} = V_{k}/(2π)^{3}
This is effectively the density of nspace
The number of modes between k and k + dk is the
volume of k space in a thin spherical shell. Thus,
dn = (1/2π)^{3}4πk^{2}dk
Now setting, h = c = 1 we get E = k = ω. Now, each
mode has an energy ω/2 associated with it. Therefore,
dρ = (1/2π)^{3}4πω^{3}/2dω
or,
ρ = (1/2π)^{3}∫dω 2πω^{3}
= ω^{4}/8π^{2}
= hω^{4}/8π^{2}c^{3} after restoring h and c.
There is an extra factor of 2 here to account for the
fact that the photon has 2 polarization states.
Evidence for the existence of vacuum energy can be
found in the CASIMIR EFFECT.
The Casimir Effect

The Casimir effect is a small attractive force that
acts between two close parallel, uncharged, grounded,
conducting plates. On the interior surface of the
plates the electric field must be zero since a nonzero
value would dissipate energy and violate the condition
of equilibrium. This imposes a constraint on the nodes
of the field between the plates. The distances between
the plates can only fit 1/2 period sine waves. The
vacuum energy contains contributions from all wavelengths,
except those excluded by the spacing between plates. As
the plates draw together, more wavelengths are excluded
and the vacuum energy decreases. The decrease in energy
means there must be a force doing work on the plates as
they move.
.. ..
 
> < P
 
.. ..
< x >
The lack of low frequency modes between the plates
lowers the vacuum energy between them. This causes
there to be a small correction term:
Δρ = π^{2}A/720x^{4}
Where A is the area of the plates.
Thus, the modified ρ is:
ρ = ω^{4}/8π^{2}  π^{2}A/720x^{4}
The difference in energy results in a pressure, P,
acting on the plates. If the spacing between the
plates is increased by dx, the work done is:
W = PdV
= PAdx
P = (1/A)dU/dx
= π^{2}240x^{4}
The force acting on the plates is the CASIMIR FORCE.
One may view the force as arising due to the radiation
pressure of the vacuum.
The Casimir Effect has been observed in the laboratory
and is considered to be evidence of the existance of
vacuum energy. The Casimir effect does not measure
the value of the vacuum energy, but only identifies
that it exists.
Other Fields and the Vacuum Catastrophe

All fields have a zeropoint energy. Therefore, the
vacuum energy can be viewed as the combination of all
zeropoint fields. The above equation is also a fair
approximation for these other quantum fields. Therefore,
ρ = Σω^{4}/4π^{3} over all Bosonic and Fermionic fields.
This is clearly divergent as ω > ∞.
This divergence can also be interpreted in terms of
Feynman loop diagrams with virtual particle pairs that
blink into existence and then annihilate in a timespan
too short to observe.
Now, Fermionic fields contribute a negative amount ω/2
for each degree of freedom to the vacuum energy. While
Supersymmetry might provide some cancellations at higher
energies it does not work for lower energies. This is
due to the fact that Supersymmetry must be spontaneously
broken and the masses of the superpartners are not equal
(if the masses were equal physicists would have discovered
them by now). Thus, even if Supersymmetry turns out to be
real, there remains a considerable vacuum energy governed
by equations of the above form.
This zeropoint energy density of the vacuum, due to all
quantum fields, is enormously large, even when we cut off
the largest allowable frequencies based on plausible
physical arguments.
QFT is at the heart of modern day physics and has been
spectacularly successful. Such an enormous vacuum energy
is generally not important in these field theories because
we are only interested in energy differences and can use
normal ordering procedures to eliminate the contributions
from the zeropoint field energies. This has the effect
of subtracting out the vacuum energy regardless of its size.
However, a large vacuum energy presents a huge problem for
gravity. To see this, consider the following classical
argument involving the Sun and Pluto. Using Newton's and
Gauss's Law of Universal Gravity we can write:
F_{S} = GMm/r^{2}
and,
F_{V} = M(r)m/r^{2}
Now, M(r) = 4πr^{3}ρ/3
Therefore,
F_{V} = 4πρGmr/3
Newtonian mechanics does an excellent job of describing
planetary motion therefore we must conclude F_{V} << F_{S}
4πρGmr/3 << GMm/r^{2}
∴ 4πρr/3 << M/r^{2}
ρ < 3M/4πr^{3}
For M = 1.99 x 10^{30}kg (Sun's mass) and r = 5.9 x 10^{12}m
(orbital radius of Pluto), we get ρ < 2.3 x 10^{9} kg/m^{3}.
Cosmological Constant Problem

The vacuum energy is one possible explanation for the
Cosmological constant and the source of dark energy.
The estimated mass density of the Universe is (including
Dark matter) thought to be roughly equal to 10^{26} kg/m^{3}.
If the energy density of the vacuum is not close to
this value, the behavior of the Universe would be
seriously affected and the curvature and expansion
of space would not agree with astronomical observation.
This can be seen from the equation:
..
(a/a) = 4πG(ρ + 3p)/3 + Λ/3
Which describes the acceleration of the expansion of the
Universe (see the note on "Geometries of the Universe").
So, a large vacuum energy presents a huge problem for
General Relativity because the absolute amount of vacuum
energy has a real physical meaning. In fact, the
Cosmological constant and the vacuum energy differ by
about an astonishing 120 orders of magnitude! This
is the infamous "Cosmological constant problem" which
remains one of the greatest unsolved mysteries of physics
in the modern era.