CHAPTER 9
Quantum Strings and Random Surfaces
We have seen in the previous chapters that while chiral fields in the
large N limit describe free particles, gauge theories describe, or are
described by, noninteracting strings. The notion of “free string” is far
from being trivial, and actually, we have used this term before rather
loosely. It is the purpose of this chapter to present the theory of strings.
Unfortunately this theory is not yet completed. So, we shall give a
review of existing results and indicate the possible directions of future
development.
We begin our discussion with a very simple special case of infinitely
short strings or, which is the same, point-like particles. After this warmup we will proceed to our main case of interest—the general theory of
strings.
9.1 Mathematical Preliminaries: Summation of Random Paths
The position of a point-like particle is described by a four-vector x^.
The meaningful question is: what is the amplitude G(x, x') for a particle
to go from the point x to a point x'. As usual, this amplitude is given by
the sum over all possible trajectories connecting the points x and x'. In
Euclidean space the amplitude is given by:
G(x, x') = X
-
(Px,x')
(9.1)
Here
are paths connecting x and x', and
is the classical
action for the given path. As such, one takes the simplest invariant
characteristic of the path—its length. So, we have the following
expression for the action:
] = moL(P,,0
151
(9.2)
DOI: 10.1201/9780203755082-9
Quantum Strings and Random Surfaces
We have seen in the previous chapters that while chiral fields in the
large N limit describe free particles, gauge theories describe, or are
described by, noninteracting strings. The notion of “free string” is far
from being trivial, and actually, we have used this term before rather
loosely. It is the purpose of this chapter to present the theory of strings.
Unfortunately this theory is not yet completed. So, we shall give a
review of existing results and indicate the possible directions of future
development.
We begin our discussion with a very simple special case of infinitely
short strings or, which is the same, point-like particles. After this warmup we will proceed to our main case of interest—the general theory of
strings.
9.1 Mathematical Preliminaries: Summation of Random Paths
The position of a point-like particle is described by a four-vector x^.
The meaningful question is: what is the amplitude G(x, x') for a particle
to go from the point x to a point x'. As usual, this amplitude is given by
the sum over all possible trajectories connecting the points x and x'. In
Euclidean space the amplitude is given by:
G(x, x') = X
-
(Px,x')
(9.1)
Here
are paths connecting x and x', and
is the classical
action for the given path. As such, one takes the simplest invariant
characteristic of the path—its length. So, we have the following
expression for the action:
] = moL(P,,0
151
(9.2)
DOI: 10.1201/9780203755082-9