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Has anyone studied the book Quantum Kinetic Theory by M. Bonitz?

I am reading it on my own and have a hard time understanding some of the things that are done in the book. Specifically, in the second chapter, he keeps terms which are traces over a commutator. Isn't the trace over a commutator always zero due to cyclic invariance of the trace? Perhaps this only holds for some operators. In that case, can someone give me an example for when it does not hold.

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# Quantum Kinetic Theory

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