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:
Definition of momenta for four 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.
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