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Suppose you have an interaction term in your hamiltonian that looks like

\begin{equation} H=\sum_{ijkl}U_{ijkl}c^\dagger_ic^\dagger_jc_kc_l \end{equation}

where $U$ is the coupling and $c$, $c^\dagger$ are fermion operators. The questions is: What can you do to figure out the properties of the tensor $U_{ijkl}$ under index exchange? I first thought that I can demand that $H$ is hermitian, but then I run into the problem of not knowing what is the adjoint of a tensor with more than two indeces. If it is what I think so, then I'd get

\begin{equation} H^\dagger=\sum_{ijkl}U^*_{lkji}c^\dagger_lc^\dagger_kc_jc_i \end{equation}

But then only demanding that $U$ is real seems enough. However I've seen in many places that $U$ needs to have a specific symmetry under index exchange. My other approach was to use the commutation rules, For example switching the first two operators we get

\begin{equation} \begin{aligned} H&=\sum_{ijkl}U_{ijkl}c^\dagger_ic^\dagger_jc_kc_l \\ &=-\sum_{ijkl}U_{ijkl}c^\dagger_jc^\dagger_ic_kc_l \\ &=-\sum_{ijkl}U_{jikl}c^\dagger_ic^\dagger_jc_kc_l \end{aligned} \end{equation} which would mean that the matrix is antisymmetric under exchange of $i$ and $j$. Is this reasoning correct?

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