String Theory Demystified

10 String Theory Demystifi ed
.
Old quantum field theory:
A particle is a mathematical
point, with no extension
interactions (zero distance) imply infi nite momentum. This leads to divergent loop
integrals, and infi nities in calculations.
So in string theory, we replace a point particle by a one-dimensional string. This
is illustrated in Fig. 1.4.
Some Basic Analysis in String Theory
In string theory, we don’t go all the way to ∆x→0but instead cut it off at some
small, but nonzero value. This means that there will be an upper limit to momentum
and hence ∆ p /→ ∞ . Instead momentum goes to a large, but fi nite value and the loop
integral divergences can be gotten rid of.
If we have a cutoff defi ned by the length of a string, then the uncertainty relations
must be modifi ed. It is found that in a string theory uncertainty in position ∆x is
approximately given by
∆ = + ′
∆ ∆ p
x α (1.10)
p
A new term has been introduced into the uncertainty relation, α ′ ( ∆ p/ ) which can
serve to fi x a minimum distance that exists in the theory. The parameter α ′ is related
to the string tension T as S
α ′ =
String theory: Particles are strings,
with extension in one dimension.
This gets rid of infinities
Figure 1.4 In string theory, particles are replaced by strings, spreading out interactions
over space-time so that infi nities don’t result.
1
2πTS The minimum distance scale we can see in string theory is given by
(1.11)
xmin ∼ 2 α ′
(1.12)