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 one-dimensional wave equation
where vw 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 vw 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 two-dimensional 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 two-dimensional 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 c2 is the ratio of the string tension to the string mass per unit length.
These equations of motion can be derived from Euler-Lagrange equations from an action based on the string world sheet
The spacetime coordinates Xm of the string in this picture are also fields Xm in a two-dimension 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 hmn is the two-dimensional 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 right-movers and left-movers, 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.
