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Casimir
Effect and an Analysis of Image Method performed by Brown and Maclay
to find the Vacuum Expectation of Stress-Energy Tensor
It
is first predicted by a Dutch physicist, Hendrick Casimir,
in 1948 that there will be an attractive force between two close
parallel-uncharged conducting plates because of the quantum vacuum fluctuations
of the electromagnetic field. According to QED, the total vacuum expectation of zero point energy
is infinite when summed over all the possible photon modes.
The
stress tensor for the electromagnetic case is given by
Eq.1
where
.
Eq.2
Brown and
Maclay
used the image method to find the vacuum expectation of stress-energy tensor,
. The stress-energy tensor they found at finite temperature and plate
separation, a, is
Eq.3
where
is the zero point vacuum expectation of stress tensor,
is the finite temperature vacuum expectation of stress tensor at a large
separation, and
is the finite temperature vacuum expectation of stress tensor at a distance a.
They looked at the solution of
, sometimes called by some physicists as the pressure on one of the plates. So,
the right hand side of Eq.3 found by Brown and Maclay is,
Eq.4
So, the vacuum expectation of stress-energy tensor will be,
Eq.5
where
, according to Brown and Maclay.
Brown
and Maclay connect their result with thermodynamics. According to their
approach,
can be written in terms of the
infinite sum of spatial images functions,
Eq.6
such
that
Eq.7
where
Eq.8
Here,
Eq.9
The
reflected coordinate
is defined as
.
REFERENCES:
1.
Gunter Plunien, Berndt Muller and Walter Greiner, The Casimir Effect
(Physics Reports 134, Nos. 2 & 3, 1986, pp.87-193)
2.
Ashok Das, Relativistic Quantum Mechanics Lecture Notes.
3.
Albert D. Wheelon, Tables of Summable Series and Integrals Involving
Bessel Functions (Holden-Day Inc., San Francisco, 1968)
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