The highly successful Feynman approach to elementary particle interactions emanates from the physics of quantum field theory. Representing the particles of a system by appropriate fields, a Hamiltonian for the system can be constructed out of the various quantum field operators. The interaction term in this Hamiltonian can be treated as a perturbation, and the dynamical equations of the system can be solved in terms of a perturbation expansion. Higher orders in the expansion denote the multiplicity of intermediate virtual states. Illustratively, for a fermion field with interactions mediated by a scalar boson, we might have the first order amplitude, or single boson exchange approximation, with one possible Feynman diagram,
.
The labels indicate initial and final spin and momenta, 
and 
is the squared momentum transfer effected by the virtual boson. By application of the momentum space Feynman rules the contribution from this diagram to the corresponding S-matrix [28, 12] element is [26],
where 
and 
are the fermionic and bosonic field operators respectively,
iM is the reduced S-matrix element (S-I) and g represents a coupling constant. Evaluating the bosonic propagator
and applying Wick's theorem [29] yields the scattering amplitude contributing to the reduced S-matrix element for this
diagram, and for these momentum and spin states, represented by 
where u(p,s) is a 4-component spinor,
The factor 
may be dropped by exchanging the dummy variables x and y. In relativistic scattering 
denotes 
and similarily for 
.
In addition, for identical particle scattering there is a second possible Feynman diagram due to the indistinguishability of the two particles. This diagram is constructed by exchanging the labels on the outgoing particles, ie
where u denotes the quantity 
.
This adds an exchange term
to the form of the amplitude, carrying a minus sign with respect to the direct term. Hence the antisymmetry requirement of the amplitude with respect to exchange of the two fermions is satisfied. Inspecting the denominators of the two terms it is clear that the second term can be neglected for scattering in the forward direction where the momentum transfer (P - P') is small.