Mandelstam variables
In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy, momentum, and angles of particles in a scattering process in a Lorentz-invariant fashion. They are used for scattering processes of two particles to two particles. The Mandelstam variables were first introduced by physicist Stanley Mandelstam in 1958. If the Minkowski metric is chosen to be formula_1, the Mandelstam variables formula_2 are then defined by where "p" and "p" are the four-momenta of the incoming particles and "p" and "p" are the four-momenta of the outgoing particles, and we are using relativistic units  (c=1). s is also known as the square of the center-of-mass energy (invariant mass) and t is also known as the square of the four-momentum transfer. Feynman diagrams. The letters "s,t,u" are also used in the terms s-channel (timelike channel), t-channel, and u-channel (both spacelike channels). These channels represent different Feynman diagrams or different possible scattering events where the interaction involves the exchange of an intermediate particle whose squared four-momentum equals "s,t,u", respectively. For example, the s-channel corresponds to the particles 1,2 joining into an intermediate particle that  eventually splits into 3,4: The t-channel represents the process in which the particle 1 emits the intermediate particle and becomes the final particle 3, while the particle 2 absorbs the intermediate particle and becomes 4. The u-channel is the t-channel with the role of the particles 3,4 interchanged. When evaluating a Feynman amplitude one often finds scalar products of the external four momenta. One can use the Mandelstam variables to simplify these: formula_6 formula_7 formula_8 Where formula_9 is the mass of the particle with corresponding momentum formula_10. Sum. Note that where "m" is the mass of particle  i. To prove this, we need to use two facts: So, to begin, Then adding the three while inserting squared masses leads to, Then note that the last four terms add up to zero using conservation of four-momentum, So finally, Relativistic limit. In the relativistic limit, the momentum (speed) is large, so using the relativistic energy-momentum equation, the energy becomes essentially the momentum norm (e.g. formula_21 becomes formula_22 ). The rest mass can also be neglected. So for example, because formula_24 and formula_25. Thus,