One of the key numbers we can extract from mass and momentum is the invariant mass, a number independent of the Lorentz frame we are in
(10.5) |
This quantity takes it most transparent form in the centre-of-mass, where . In that case
(10.6) |
and is thus, apart from the factor , nothing but the energy in the CM frame. For a single particle , the rest mass.
Most considerations about processes in high energy physics are greatly simplified by concentrating on the invariant mass. This removes the Lorentz-frame dependence of writing four momenta. I
As an example we look at the collision of a proton and an antiproton at rest, where we produce two quanta of electromagnetic radiation ( ’s), see fig. 10.1 , where the anti proton has three-momentum , and the proton is at rest.
The four-momenta are
From this we find the invariant mass
(10.8) |
If the initial momentum is much larger than , more accurately
(10.9) |
we find that
(10.10) |
which energy needs to be shared between the two photons, in equal parts. We could also have chosen to work in the CM frame, where the calculations get a lot easier.