It was long ago shown by L.D. Faddeev [Fad61] that the eigenfunctions of the three-body scattering problem with a general potential could be easily written down as the sum of three parts which would obey a set of three coupled integral equations. The mathematical formalism worked out by Faddeev was an important improvement on earlier techniques (i.e. Lippmann-Schwinger type equations) for two main reasons. First of all, the equations have a unique solution. Secondly, the kernel of the integral equations for short range pairwise forces, such as nuclear forces, is compact. This means that the numerically discretized form will converge [Glo96] which enables a solution to be found using modern computer resources.
The basic Faddeev equations can be understood, using a derivation discussed in depth in [Glo96], by modeling the three-body interaction with a infinite Born series of two body forces:
where denotes the initial state, a pair interaction between the and particles, and the free propagator between consecutive interactions. The superscript (1) on the three-body interaction means that particle 1 is the free particle in the initial state.
Next it is possible to split the series into three subseries based on which interaction, , , or , immediately precedes the final state.
where the second superscript indicates the particle not participating in the final interaction. Inspection of the series above shows that each of the subseries can be represented in a recursive, compact manner.
defines the subseries . The interactions and can similarily be defined as the sum of the infinite subseries and a final pair interaction.
Mathematically the expression for would be
The two-nucleon t-matrix, which is defined as
and is dependent only on the two-nucleon potential, can be substituted into equation to get
Similarly the other two infinite subseries would give
Equations , , and form a set of coupled Faddeev equations which can, in principle be solved with enough computer power using standard two-body nn (nucleon-nucleon) potentials, such as the Argonne, Bonn or Nijmegan potentials.
To calculate scattering observables it is necessary to find the transition amplitudes, . The determination of these quantities is usually done by working in momentum space and using a partial wave representation of the states. Therefore, a given 3-nucleon state, , can be expressed as where and are the typical Jacobi momenta describing the motion of the three particles [Glo96]
where ijk equal cyclic combinations of 123, and the 's are the individual momentum of the particles. indicates the partial wave state. In other words, it describes the spin and angular momentum of the quantum mechanical state. Typically determining the transition amplitudes would require an integration over the continuous variables and , and a summation over all partial waves. This represents an infinite set of coupled integral equations in and . In actually obtaining a solution, it is usually acceptable to truncate the partial wave expansion after a certain number of terms (usually 5). This can be justified because the short range nature of the nuclear force, and since the and two-nucleon forces are much stronger than p-wave or higher l-wave forces [Glo82].
The second step is to come up with a discretization of the momentum variables, and , and a scheme for interpolation, such as a spline interpolation. After all this is done what is left is the homogeneous eigenvalue problem
where represent all the indices for , , and , which can be solved given enough calculational power to invert the kernel, .
The Faddeev calculational method has been relatively successful in reproducing three-nucleon scattering observables. With recent advances into the inclusion of three-body forces and the removal of on-shell discrepancies it is proving an accurate and useful tool for studying nuclear dynamics within the three-body system.