In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy, momentum, and angles of particles in a scattering process in a Lorentzinvariant 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 , the Mandelstam variables are then defined by
where p_{1} and p_{2} are the fourmomenta of the incoming particles and p_{3} and p_{4} are the fourmomenta of the outgoing particles.
is also known as the square of the centerofmass energy (invariant mass) and as the square of the fourmomentum transfer.
The letters s,t,u are also used in the terms schannel (timelike channel), tchannel, and uchannel (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 fourmomentum equals s,t,u, respectively.
schannel  tchannel  uchannel 
For example, the schannel corresponds to the particles 1,2 joining into an intermediate particle that eventually splits into 3,4: the schannel is the only way that resonances and new unstable particles may be discovered provided their lifetimes are long enough that they are directly detectable.^{[citation needed]} The tchannel 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 uchannel is the tchannel 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:
Where is the mass of the particle with corresponding momentum .
Note that
where m_{i} is the mass of particle i.^{[1]}
Proof


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 fourmomentum, So finally,

In the relativistic limit, the momentum (speed) is large, so using the relativistic energymomentum equation, the energy becomes essentially the momentum norm (e.g. becomes ). The rest mass can also be neglected.
So for example,
because and .
Thus,