String Theory and M-Theory

2
The bosonic string
This chapter introduces the simplest string theory, called the bosonic string.
Even though this theory is unrealistic and not suitable for phenomenology,
it is the natural place to start. The reason is that the same structures
and techniques, together with a number of additional ones, are required for
the analysis of more realistic superstring theories. This chapter describes
the free (noninteracting) theory both at the classical and quantum levels.
The next chapter discusses various techniques for introducing and analyzing
interactions.
A string can be regarded as a special case of a p-brane, a p-dimensional
extended object moving through space-time. In this notation a point particle
corresponds to the p = 0 case, in other words to a zero-brane. Strings
(whether fundamental or solitonic) correspond to the p = 1 case, so that they
can also be called one-branes. Two-dimensional extended objects or twobranes
are often called membranes. In fact, the name p-brane was chosen
to suggest a generalization of a membrane. Even though strings share some
properties with higher-dimensional extended objects at the classical level,
they are very special in the sense that their two-dimensional world-volume
quantum theories are renormalizable, something that is not the case for
branes of higher dimension. This is a crucial property that makes it possible
to base quantum theories on them. In this chapter we describe the string as
a special case of p-branes and describe the properties that hold only for the
special case p = 1.
2.1 p-brane actions
This section describes the free motion of p-branes in space-time using the
principle of minimal action. Let us begin with a point particle or zero-brane.
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