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Extra Dimensions
----------------
In the discussion on open strings we encountered the
Tachyon - a particle with negative mass (m_{0}^{2} = -1).
This is a problem that is overcome by the requiring the
theory to have extra spacial dimensions. To see how
this works consider the following.
The energy of the nth harmonic oscillator is:
E = (1/2)nh where n ≡ ω
Therefore, the total energy is:
E = (1/2)Σnh
= (1/2)(1 + 2 + 3 + 4 + 5 + ...)h
At first sight this would appear to result in an infinite
energy. However, it is possible to perform the following
mathematical 'trick'. We can write (1 + 2 + 3 + 4 + ... )
as:
∞
Σn = exp(-ε) + 2exp(-2ε) + 3exp(-3ε) + 4exp(-4ε) + ...
1
∞
= Σnexp(-nε)
1
∞
= -∂/∂ε{Σexp(-nε)}
1
= -∂/∂ε{exp(-ε) + exp(-2ε) + exp(-3ε) + exp(-4ε) + ...}
= -∂/∂ε{exp(-ε)(1 + exp(-ε) + exp(-2ε) + exp(-3ε) + ...)}
The () term is a Geometric series with the formula
1/(1 - exp(-ε)) (it can also be derived using Taylor
series). Therefore,
= -∂/∂ε{exp(-ε)/(1 - exp(-ε))}
Again using the Taylor series for exp(-ε) we get:
(1 - ε + ε^{2}/2 - ε^{3}/6 ...)
= -∂/∂ε ------------------------------
(1 - (1 - ε + ε^{2}/2 - ε^{3}/6 ...))
= -∂/∂ε(1/ε) ({1 - ε + ε^{2}/2)/(1 - ε/2 + ε^{2}/6))}
Again using 1/(1 - x) = 1 + x + x^{2} + x^{2} + ...
= -∂/∂ε(1/ε){(1 - ε + ε^{2}/2)(1 + ε/2 - ε^{2}/6 + ε^{2}/4)}
= -∂/∂ε(1/ε){(1 - ε + ε^{2}/2)(1 + ε/2 + ε^{2}/12)}
= -∂/∂ε(1/ε){1 + ε/2 + ε^{2}/12 - ε - ε^{2}/2 + ε^{2}/2}
= -∂/∂ε(1/ε){1 - ε/2 + ε^{2}/12}
= 1/ε^{2} - 1/12
We can now write our 0-point oscillator energy as:
E = (1/2)Σnh
= (1/2ε^{2} - 1/24)h
= (1/2)(1/ε^{2} - 1/12)
We get the odd result the the sum to ∞ = -1/12!!
A more rigorous mathematical analysis involves the
RIEMANN ZETA FUNCTION defined as:
∞
ζ(s) = Σ(1/n^{s})
n=1
Where s is any complex number whose real partis > 1.
The ζ function is a fascinating entity that has been
studied extensively and has far reaching applications in
physics and the theory of prime numbers. There is a
technique to extend the domain of such a function and
define a new region where a divergent infinite series
can be made to converge. This technique is referred to
as ANALYTIC CONTINUATION. By applying the technique to
the ζ function it can be shown that:
ζ(-1) = -1/12
Using the ζ function, we can see that the 0-point
energy becomes -h/24. In order to make this term equal
-1 we need 24 additional dimensions! Thus, to get the
desired 0-point energy condition and overcome the tachyon
problems requires a total of 26 dimensions 24 + z + t).
The extra dimensions can be imagined as being tiny and
curled up. A simple analogy is a common garden hose.
From a distance, the hose looks like a straight line
and a bug on the hose could move up and down its length.
But if you move closer to the hose, it becomes apparent
that it has another dimension, its circumference, and
therefore the bug could walk around the hose as well.