Group Classification of Many-Body Interactions

For an $n$-fermion system in which each particle can occupy any one of $k$ states, an "irreducible $b$-body operator" is defined as an operator belonging to [$2^b$, $1^{k-2b}$] of ${\mathrm{SU}}_k$. It is shown that such an operator can always be written as a $b$-body operator multiplied by a function of $n$. A measure of the failure of a basis to diagonalize the Hamiltonian is constructed. If the Hamiltonian is analyzed into irreducible $b$-body parts, the measure of error for $n$ particles can be calculated from two-particle parameters. An upper limit on the error of a group theoretically defined basis can be found without having to calculate any $n$-particle interaction matrix elements. A measure of the magnitude of an interaction is defined, and shown to depend differently on $n$ for irreducible 0-body, 1-body, and 2-body interactions. The effect of the Pauli principle on the formation of shell-model potentials is discussed from this point of view.