Knowing how pure particle helicity states may be transformed from one frame to another, a formalism may now be devised to connect the scattering matrix, and the superposition of incoming and outing particle states describing a scattering experiment. The natural method when working with helicity states is to use the helicity density matrix.
A single particle with momentum 
is described by a sum in the helicity basis of , ie
For the spin- 
case to be dealt with here, 
may of course take only two values.
For a beam of uniform , made up of individual particle helicity states 
, of statistical
weight 
the helicity density matrix may be defined,
This density matrix describes all the physically observable quantities for the specified particles with
momentum . There are various ways of disseminating this information from the density matrix.
Given that the range of defines a basis for a particle i in its helicity rest frame in terms of the
vectors 
, the expectation value of an operator 
in that rest frame will be
where 
are the matrix elements of in the basis of rest states.
Summing this over all i particles,
and rearranging the indices
Referring back to the definition of the helicity density matrix, clearly
So the expectation value of in the rest frame of the beam is the trace of 
For spin half particles, the helicity density matrix must be a 
Hermitian matrix of unit trace.
This reduces its parameters to 3. It is therefore possible to express the density matrix in the form,
where the identity I, and the three Pauli matrices [Appendix 1 ] 
provide four independent matrices, and P defines the Polarization vector for the collection of particles in question.
The inverse of the above being,
illustrating that 
is the expectation value of 
or that of 
.
For a system containing two types of distinct particles, which may mean identical particles of different momentum, a joint density matrix is defined. An observable of this interacting system, is given by the sum,
where the operator 
, with 
and 
acting in the helicity rest
frames of A and B respectively. 
is therefore a joint expectation value for observations made on A in
its helicity rest frame and likewise for B.
In the case where a beam and target have been prepared separately the systems are uncorrelated
and the expectation value of an operator for the particles A and B together can be factorized into the
expectation values for A and B separately such that
This is only possible if the joint density matrix will factorize also.
