String Theory Demystified

22 String Theory Demystifi ed
the actual universe while force-carrying particles are indeed bosons, fundamental
matter particles (like electrons) have half-integral spin, that is, they are fermions.
So a theory that describes a world consisting entirely of bosons does not describe
the real universe.
Nonetheless, we start here because it is an easier way to approach string theory
and we can learn the nuts and bolts in a slightly simpler context. We are going to
approach bosonic strings in three steps. In this chapter, we will develop the theory
of classical, relativistic strings starting with the action principle and deriving the
equations of motion. In Chap. 3, we will learn about the stress-energy tensor and
conserved currents, specifi cally conserved worldsheet currents. Finally, in the last
chapter of this part of the book, we will quantize the strings using a procedure of
fi rst quantization (i.e., fi rst quantization of point particles gives single-particle
states). In the end you have a quantized relativistic theory.
To this end, we begin our journey into the world of classical relativistic point
particle moving in space-time to illustrate the techniques used.
The Relativistic Point Particle
The task at hand is to describe the motion of a free (relativistic) point particle in
space-time. One way to approach the problem is by using an action principle.
Before we do that, let’s set up the arena in which the particle moves. Let its motion
be defi ned with respect to space-time coordinates X µ where X 0 is the timelike
0 i
coordinate (i.e., X = ct)
and X where i ≠ 0 are the spacelike coordinates (say x, y,
and z). While you are probably used to lowercase letters like x µ to represent
coordinates, in string theory uppercase letters are used, so we will stick to that
convention.
Anticipating the fact that string theory takes place in a higher-dimensional arena,
rather than the usual one time dimension and three spatial dimensions we are used
to, we consider motion in a D-dimensional space-time. There is one time dimension
but now we allow for the possibility of d = D − 1 spatial dimensions. We reserve 0
to index the time dimension hence our coordinates range over µ = 0, ... , d .
Now, the motion or trajectory of a particle is described such that the coordinates
are parameterized by τ , which parameterizes the world-line of the particle. That is,
this is the time given by a clock that is moving or carried along with the particle
itself. We can emphasize this parameterization by writing the coordinates as
functions of the proper time:
µ µ
X = X ( τ)
(2.1)