# Daily Archives: May 22, 2008

## Mandelstam Variables and Veneziano Amplitudes

Quantum mechanics is a probabilistic theory. The probabilities are the squared magnitudes of (probability) amplitudes. Amplitudes are computed through perturbation theory, or whatever method one can find. The state of the art method is to use Feynman diagrams to find the amplitudes.

When one considers interactions between two particles, there are two Feynman diagrams that are of particular interest, the “t-channel” and the “s-channel”. The “s” and “t” are Mandelstam variables. These variables define the interaction in a way that automatically preserves Lorentz invariance.

Special Relativity Momenta

This is a relativistic problem, so all momenta are 4-momenta, that is, they are vectors with four components. The time or zero portion of the vector gives the negative energy of the particle, while the space or 1-3 portion of the vector gives its momentum. A 2-particle interaction involves 4 momenta, two for the incoming particles, and two more for the outgoing particles. That’s a total of 4×4 = 16 variables.

Let $p_1, p_2, p_3, p_4$ be the four momenta. The first two are for the incoming particles, that is, the ones before the interaction, the 3rd and 4th are for the outgoing particles:

Now each of these $p_i$ are a 4-vector. I will write them as $p_i^\mu$ where $p_i^0$ is the energy of the particle, $p_i^1$ is the momentum in the x-direction, etc.