the correlation function: Using the inverse transform to substitute in the LSZ reduction formula, with some effort, the following result can be obtained: Leaving aside normalization factors, this formula asserts that S-matrix elements are the residues of the poles that arise in the Fourier transform of the correlation functions as four-momenta are put on-shell. Reduction formula for fermions Recall that solutions to the quantized free-field Dirac equation may be written as where the metric signature is mostly plus, is an annihilation operator for b-type particles of momentum and spin , is a creation operator for d-type particles of spin , and the spinors and satisfy and . The Lorentz-invariant measure is written as , with . Consider now a scattering event consisting of an in state of non-interacting particles approaching an interaction region at the origin, where scattering occurs, followed by an out state of outgoing non-interacting particles. The probability amplitude for this process is given by where no extra time-ordered product of field operators has been inserted, for simplicity. The situation considered will be the scattering of b-type particles to b-type particles. Suppose that the in state consists of particles with momenta and spins , while the out state contains particles of momenta and spins . The in and out states are then given by Extracting an in particle from yields a free-field creation operator acting on the state with one less particle. Assuming that no outgoing particle has that same momentum, we then can write where the prime on denotes that one particle has been taken out. Now recall that in the free theory, the b-type particle operators can be written in terms of the field using the inverse relation where . Denoting the asymptotic free fields by and , we find The weak asymptotic condition needed for a Dirac field, analogous to that for scalar fields, reads and likewise for the out field. The scattering amplitude is then where now the interacting field appears in the inner product. Rewriting the limits in terms of the integral of a time derivative, we have where the row vector of matrix elements of the barred Dirac field is written as . Now, recall that is a solution to the Dirac equation: Solving for , substituting it into the first term in the integral, and performing an integration by parts, yields Switching to Dirac index notation (with sums over repeated indices)