My book has a chapter on the general mean field approach. I have attached a page of it I don't understand very well. I do understand the general idea of meanfield theory: Expand operators around their average and assume small fluctuations and neglect the second order term. But in the attached page they use a form of the Hamiltonian I am only just getting used to. And for mean field theory to work I need to be able to separate the product of the a-operators and b-operators (in the given hamiltonian they appear as abba with daggers on some of them of course). Can I assume that a and b operators commute? They describe different particles but on the other hand it is interaction between a and b particles that is relevant. So what do I do?