Classical String Theory
Before we consider free strings let us first take a look at the description of a free massless relativistic point-particle:(3.1) |
where
Strings are the one-dimensional generalisation of particles, so they sweep out a two dimensional worldsheet in space-time. The dynamics of the string is descibed by its coordinate
The Action describing a free bosonic string moving in flat space then becomes:
(3.2) |
where
This string action has the following symmetries: it is invariant under reparametrization of the worldsheet (ofcourse):
and it is invariant under conformal rescaling of the internal metric:
this is called Weyl invariance.
We can generalise this to a string moving in curved space-time. The action for a bosonic string propagating in a gravitational background then becomes:
(3.3) |
This bosonic string is the simplest example of string theory, but it is not realy a realistic theory for the desciption of physical phenomina. For instance its spectrum of particles includes a tachyon and no fermions. But there are more complicated string theories that have more realistic features. The supersymmetric version of string theory, superstrings, for instance solves the tachyon problem. There are now many different consistent string theories known, all with different worldsheet symmetries or different worldsheet topologies. Next to that we can change the background in which the string propagates.
For instance the action of a bosonic string moving in a general background is described by the generalised sigma model:
where the backgroundfields
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