Symmetrized Faddeev Equations

We define a full symmetrizer (antisymmetrizer) $$S_N$$ for a system of N identical bosons (fermions), which can be built recursively as a product of $$(N-1)$$ simple permutation operators $$ (1 + k \epsilon \tau _k)$$ , $$1 \le k < N$$ , and the operator $$S_{N-1}$$ : $$S_N= (1+ \epsilon \tau _1)(1+2\epsilon \tau _2)\cdots (1+(N-1)\epsilon \tau _{N-1}) S_{N-1} /N! $$ , where $$\tau _{k}$$ denotes the transposition $$P_{k, k + 1}$$ and $$\epsilon = 1 (-1) $$ for identical bosons (fermions). Using this form of $$S_N$$ we could rewrite the N-body Faddeev equation as $$\psi = \left( {\begin{array}{c}N\\ 2\end{array}}\right) \, G_0 V S_N \psi $$ , where $$\psi $$ , $$\left( {\begin{array}{c}n\\ k\end{array}}\right) $$ , $$G_0$$ and V are the Faddeev component, the standard binomial coefficient, the N-body Green’s function and the two-body potential, respectively. Our formulation of the N-body Faddeev equation can be naturally extended to include three- and other many-body forces. We demonstrate that the proposed form of $$S_N$$ significantly simplifies the process of building the N-body wave function.