Quantum Kinetic Theory by M. Bonitz: Traces over Commutators

[Jezuz]

Hi!
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.

 

[Civilized]

I have never encountered that book, it looks interesting. Most of my non-equilibrium stat mech books concentrate on classical phenomena.

As for your question, I agree that the trace of a commutator of two matrices with complex number entries is always zero. But if the perators are matrices whose entries are not c-numbers but are themselves non-commutative operators, we can have situations where operators fail to commute with themselves, and it seems we can also have situations where the trace of such a commutator is not zero.

 

[DrFaustus]

From a mathematical point of view, the cyclic invariance of the trace only holds if all the operators are bounded, which most physically relevant operators are not (position operator, Hamiltonian operator for the harmonic oscillator...).

I don't know the book, but chances are it's a (mathematically) more rigorous treatment of the subject than what physicists are used to.