I'm trying to solve this problem, I am not sure how to go about it. Some help would be highly appreciated.
Let $\mathcal{H}$ be a (one-body) Hilbert space and let $\{u_\alpha\}^\infty_{\alpha=1}$ be an orthonormal basis for $\mathcal{H}$. Let W be a two-body potential for identical particles, i.e. an operator on $\mathcal{H} \otimes\mathcal{H}$ such that $ExWEx = W$ (Recall that $Ex(u_1 \otimes u_2) = u_2 \otimes u_1$). Assume that $u_\alpha \otimes u_\beta \in D(W)$ (The domain of $W$) for all $\alpha, \beta = 1, 2, ...$ show that $$ \bigotimes^\infty_{N=2} \sum_{1<i<j \leq N} W_{i,j} = \sum^\infty_{\alpha, \beta, \mu, \nu = 1} (u_\alpha \otimes u_\beta, Wu_\mu \otimes u_\nu)a^*_\pm(u_\alpha)a^*_\pm(u_\beta)a_\pm(u_\nu)a_\pm(u_\mu) $$
as quadratic forms on finite linear combinations of pure symmetric (+) or antisymmetric (-) tensor products of basis vectors from $\{u_\alpha\}^\infty_{\alpha=1}$.
