Confusion about interaction in 2nd quantization Interaction terms in second quantization is written as 
$$
\sum_{ijkl}c_{i}^{\dagger}c_{j}^{\dagger}V_{ijkl}c_{k}c_{l}
$$
Now, is spin is there then this term is written like
$$
\sum_{ijkl\sigma\sigma'}c_{i\sigma}^{\dagger}c_{j\sigma'}^{\dagger}V_{ijkl}c_{k\sigma'}c_{l\sigma}
$$
Interaction is spin independent. My question is how to put the indices of $ \sigma, \sigma'$ in the creation and annihilation operators? How, the books write the order of $ \sigma, \sigma'$ it seems to be something nontrivial.
 A: First of all, it's not true for any Hamiltonian / interaction term that interaction would be independent of spin, but let's assume in this particular case it's true (if $V$ is, for example, the Coulomb potential of interacting electrons).
Now for the order of operators. It is indeed something nontrivial. I assume we're talking about fermions here, so the relevant operators obey anticommutation relations, which means that you get a sign changes if you swap them around.
The general idea is that the order of particles in the annihilation operators has to be the reverse of that of the operators in the creation operators. An intuitive reason is that, if we have the total operator $\hat V$ acting on a state, we want all the amplitudes and signs and everything contained inside the matrix elements $V_{ijkl}$ and not in the operators.
For this to work, again very intuitively speaking, we can imagine that, acting on a given state, we first destroy all the particles using the annihilation operators, and then recreate them in their final states with the creation operators, and to not get the signs wrong the orders of the particles have to mirror each other.
The proof for that isn't hard, but very long and tedious. It is discussed in general terms in these lecture notes I found: https://www.phas.ubc.ca/~berciu/TEACHING/PHYS502/NOTES/2ndQ.pdf
