Re: Tapping the Vacuum

From: Russell E. Rierson (analog57_at_yahoo.com)
Date: 08/11/04 

Date: 11 Aug 2004 02:05:59 -0700

Still researching the tapping of the vacuum

The energy in a section of vacuum can be altered by the material
around it, giving the "Casimir Effect". The effect has been
experimentally verified by positioning two uncharged parallel metal
plates, creating a symmetric - attractive force, pressing them
together; the force only becomes measurable when the distance between
the two plates is extremely small. If the two opposing metal plates
move rapidly, some of the vacuum waves can become real waves.

Some people propose that a form of "dynamical Casimir effect" may be
responsible for the mysterious phenomenon known as sonoluminescence,
which is the emission of short bursts of light from imploding bubbles
in a liquid when excited by sound waves.

Alas, the enigmatic vacuum energy appears to be completely symmetric
though.

Casimir Force:

F_c / A = hbar*c*pi^2 / 240d^2

F_c is the force on the plates

A is the area of the plates

hbar is [Planck's constant] / 2pi

c is the speed of light,

pi is the ratio of the circumference of a circle to its diameter

d is the distance between the two plates.

In bosonic string theory, using the operator formalism, an infinite
looking zero point energy is an infinite sum:

1+2+3+4+5...

representing the frequencies of standing waves between the plates;
each possible standing wave acts like a quantum harmonic oscillator
whose ground state energy is equal to hbar*omega/2 contributing to the
total potential energy. Subtract off the zero point energy... There
arise complications, however. Thus, infinity must be replaced with
-1/12 to keep things consistent.

There is a mathematical mechanism that allows one to see 1+2+3+4+...
as equalling -1/12.

Let the series be seen as:

1+2^-n +3^-n +...

The Riemann Zeta function!

Reimann Zeta converges for large n but can be analytically continued
to n = -1, even though the series doesn't converge there. Zeta(-1) is
-1/12. So in a bizarre string theoretic sense 1+2+3+4+5... becomes
-1/12.

The -1/12 can be obtained via a different route altogether - using the
Feynman path integral, rather than the operator formalism. The -1/12
is tied up in a profound way with the geometry of string theory, so
there appears to be a lot more to it than just simple mathematical
trickery.

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