Pythagoras
could be called the first known string theorist. Pythagoras, an excellent
lyre player, figured out the first known string physics  the harmonic
relationship. Pythagoras realized that vibrating Lyre strings of equal
tensions but different lengths would produce harmonious
notes (i.e. middle C and high C) if the ratio
of the lengths of the two strings were a whole
number.
Pythagoras
discovered this by looking and listening. Today that information is more
precisely encoded into mathematics, namely the wave equation for a string
with a tension T and a mass per unit length m.
If the string is described in coordinates as in the drawing below, where
x is the distance along the string and y is the height of the string,
as the string oscillates in time t,
then the equation of motion is the onedimensional wave equation
where
v_{w} is the wave velocity along the string.
When solving the equations of motion, we need to know
the "boundary conditions" of the string. Let's suppose that
the string is fixed at each end and has an unstretched length L. The
general solution to this equation can be written as a sum of "normal
modes", here labeled by the integer n, such that
The condition for a normal mode is that the wavelength be some integral
fraction of twice the string length, or
The frequency of the normal mode is then
The
normal modes are what we hear as notes. Notice that the string wave
velocity v_{w} increases as the tension of the string is increased,
and so the normal frequency of the string increases as well. This is
why a guitar string makes a higher note when it is tightened.
But
that's for a nonrelativistic string, one with a wave velocity much smaller
than the speed of light. How do we write the equation for a relativistic
string?
According
to Einstein's theory, a relativistic equation has to use coordinates
that have the proper Lorentz transformation properties. But then we
have a problem, because a string oscillates in space and time, and as
it oscillates, it sweeps out a twodimensional surface in spacetime
that we call a world sheet (compared
with the world line of a particle).
In
the nonrelativistic string, there was a clear difference between the
space coordinate along the string, and the time coordinate. But in a
relativistic string theory, we wind up having to consider the world
sheet of the string as a twodimensional
spacetime of its own, where the division between space and time
depends upon the observer.
The
classical equation can be written as
where s and t
are coordinates on the string world sheet representing space and time
along the string, and the parameter c^{2} is the ratio of the
string tension to the string mass per unit length.
These
equations of motion can be derived from EulerLagrange equations from
an action based on the string world sheet
The spacetime coordinates X^{m}
of the string in this picture are also fields X^{m}
in a twodimension field theory defined on the surface that a string
sweeps out as it travels in space. The partial derivatives are with
respect to the coordinates s
and t on the world sheet and
h^{mn} is the twodimensional metric defined on the string world
sheet.
The
general solution to the relativistic string equations of motion looks
very similar to the classical nonrelativistic case above. The transverse
space coordinates can be expanded in normal modes as
The string solution above is unlike a guitar string in that it isn't
tied down at either end and so travels freely through spacetime as it
oscillates. The string above is an open
string, with ends that are floppy.
For
a closed string, the boundary conditions
are periodic, and the resulting oscillating solution looks like two
open string oscillations moving in the opposite direction around the
string. These two types of closed string modes are called rightmovers
and leftmovers, and this difference
will be important later in the supersymmetric heterotic
string theory.
This
is classical string. When we add quantum mechanics by making the string
momentum and position obey quantum commutation relations, the oscillator
mode coefficients have the commutation relations
The quantized string oscillator modes wind up giving representations
of the Poincaré group, through
which quantum states of mass and spin
are classified in a relativistic quantum field theory.
So this is where the elementary particle arise in string theory. Particles
in a string theory are like the harmonic notes played on a string with
a fixed tension
The parameter a' is called the string parameter and the square root
of this number represents the approximate distance scale at which string
effects should become observable.
In
the generic quantum string theory, there are quantum states with negative
norm, also known as ghosts. This happens
because of the minus sign in the spacetime metric, which implies that
So there ends up being extra unphysical states in the string spectrum.
In
26 spacetime dimensions, these extra unphysical states wind up disappearing
from the spectrum. Therefore. bosonic string quantum mechanics is only
consistent if the dimension of spacetime is
26.
By
looking at the quantum mechanics of the relativistic string normal modes,
one can deduce that the quantum modes of the string look just like the
particles we see in spacetime, with
mass that depends on the spin according to the formula
Remember
that boundary conditions are important for string behavior. Strings
can be open, with ends that travel at the speed of light, or closed,
with their ends joined in a ring.
One
of the particle states of a closed string
has zero mass and two
units of spin, the same mass and spin as a graviton,
the particle that is supposed to be the carrier of the gravitational
force.
