In quantum mechanics if (Fermionic or Bosonic) operators do not commute with each other, one cannot swap position of two operators easily. For example, let $(c^\dagger, c)$ are Fermionic operators, then to change the position of operators in expression $c_1^\dagger c_3 c_2^\dagger c_1$ one has to take care of anti-commutation relations for these operators.

My question is about the expressions which have both Fermionic and Bosonic operators. Let we have electron-phonon interaction that is written as $$c_k^\dagger c_{k'} a_{-q}^\dagger + c_k^\dagger c_{k'}a_q$$

$(a^\dagger, a)$ are Bosonic operators.

Are there any rules that have to be obeyed while changing the position of Bosonic operators in this expression? Can I simply write it as $(a_{-q}^\dagger c_k^\dagger c_{k'} + c_k^\dagger a_q c_{k'})$? Does the position of Bosonic operators in this expression have any physical meaning?


1 Answer 1


The composite system of two distinguishable objects is described by the tensor product of the individual Hilbert spaces. Hence operators acting on the respective Hilbert spaces commute.

Now replace system one with 'Fermions' and system two with 'Bosons'.

The short answer is, yes they commute.


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