The known elementary particles range in mass from about 0.0004eV for the lightest neutrino to around 170 billion eV = 1.7 x 10^11 for the heaviest quark, the top. This is a ratio of about 400,000,000,000,000 to 1.
On the other hand, the energy scale available from Einstein’s theory of gravitation (which relates mass to energy) suggests that the natural mass for a typical particle should be the Planck mass, about 2.43 × 10^27 eV, 11 orders of magnitude larger than even the top quark and 25 orders of magnitude larger than the lightest neutrino mass.
From the point of view of the Planck energy, all particles known to man have mass very close to zero. Let’s write the Hamiltonian for the system as a first order Hamiltonian , in which the energies (and therefore the masses) of all our usual particles are zero, plus a perturbation , which will provide a correction to the zero energies. For the full Hamiltonian we have:
where is a small number.
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 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 are a 4-vector. I will write them as where is the energy of the particle, is the momentum in the x-direction, etc.