UNITS AND FOUR VECTORS
Before going into much detail about kinematics, it is important to remember
a number of basic identities. First of all, it must be clear that in particle
physics most things move with large velocities - in other words with a
considerable fraction of the speed of light. Since particle physics also
is the world of quantum physics, it is common to set
This allows to write four-vectors without any disturbing c's or Planck's
Four-vectors have to satisfy the following on-shell relation
where, as above E is the energy, the pi are the components of the
three-momentum, and m is the mass of the particle in question.
FOUR MOMENTUM CONSERVATION: EXAMPLE
Of course, in physical processes, both energy and momentum must be conserved.
This can easily be interpreted in a relativistically invariant fashion as
four-momentum conservation. In particular, if, for instance, a particle with
four-momentum p decays into two particles with four-momenta q and r, respectively,
a(p) → b(q) + c(r), the momenta must satisfy
or, in components
From this, it is clear that in the centre-of mass frame of the decaying particle
p the spatial components of the outgoing momenta r and q must balance each other.
Also, for massless q and r it is obvious that both particles have the same amount
of energy in the very same frame, namely half the (virtual) mass of the decaying
particle. The relative amounts of energy are shifted if the two outgoing particles
acquire different masses.
The determination of the exact energies of the two particles q and r in the rest
frame of the decaying particle p is left as an exercise.
In order to integrate over four-momenta, as needed in the calculation of cross
sections from the S-matrix (see above),
a corresponding covariant volume element dP in momentum space needs to be
introduced. Apart from the integration measure and some factors of 2π, there
are two more ingredients, namely a δ-function incorporating the on-shell
condition, and the requirement that all physical particle energies are positive.
Taking together, this yields