Commutate my hamiltonian H with a fermionic anihillation operator

[aaaa202]

Homework Statement

I have problem where I need to commutate my hamiltonian H with a fermionic anihillation operator. Had H been written in terms of fermionic operators I would know how to do this, but the problem is that it describes phonon oscillations, i.e. is written in terms of bosonic operators:
H = ∑_kb_k(dagger)b_k
How do I commute this with a fermionic operator?

 

[vanhees71]

If you have different kinds of (quasi) particles the corresponding annihilation and creation operators all commute.

 

[aaaa202]

So in terms of symmetrization of wave functions, should I understand different quasiparticles as distinguishable particles such that the complete wavefunction is a simple product of the fermion and boson wavefunction?
edit: wait.. phonon oscillations are even real particles so wave function symmetrization etc. does not really make sense to talk about.
How should I understand, starting from the basic wave function symmetrization (from which the creation and anihillation operators came), that the operators commute?

 

[vanhees71]

Phonons are elementary excitations of the normal modes of lattice vibrations of solid. These you can describe either by the position and momentum variables of the molecules making up the lattice or, equivalently, in terms of annihilation and creation operators as for one simple oscillator. These annihilation and creation operators fulfill the same commutation relations as the analogous operators for bosonic particles. That leads to the picture of various collective modes (here lattice vibrations) of a many-body system as "quasi-particle excitations". As long as the coupling of the quasiparticles to other degrees of freedom is weak, they have sharply peaked spectral functions. For non-interacting phonons the spectral function is given by a sum of $\delta$ distributions, and thus they behave mathematically like stable particles. If you include interactions, e.g., with electrons, single atoms or impurities of the lattice leads to scattering and thus collisional broadening of the phonon's spectral function. The classical analogue of this is that the lattice vibrations are damped, i.e., energy is dissipated from the vibrations to other degrees of freedom.

 

[paranormal]

To commutate a Hamiltonian with a fermionic annihilation operator, you would first need to express the Hamiltonian in terms of fermionic operators. In this case, since the Hamiltonian is written in terms of bosonic operators, you would need to use the boson-fermion transformation to rewrite it in terms of fermionic operators. This transformation involves replacing the bosonic operators with linear combinations of fermionic operators. Once you have expressed the Hamiltonian in terms of fermionic operators, you can then commute it with the fermionic annihilation operator as you normally would. It is important to note that the result of this commutation will depend on the specific form of the Hamiltonian and the fermionic operator, so it is difficult to give a general answer without more information.