# Imaginary time dependent operators: (anti)commutation relations

If I know trivial (anti)commutation relation for some operators (let's say Fermi operators), I can only use it if they are in the same moment of time. If I have their time dependence and they don't have to be observed in the same moment of time, then those (anti)commutation relations are not that trivial and so I can't use them that way.

But what about imaginary time dependence (via the exponential factor)? For example, let's consider Fermi operators in Heisenberg picture, or Dirac (interaction) picture. Do they have some trivial (anti)commutation relations that can be used or not?

• You can use them in a time-ordered operator $T_\tau$.
– Jack
Aug 7, 2021 at 23:57

$$\{c_{k_1}(t_1),c^\dagger_{k_2}(t_2)\}=\delta_{k_1k_2}\delta(t_1-t_2)\tag{1}$$
$$\{c_{k_1}(\tau_1),c^\dagger_{k_2}(\tau_2)\}=\delta_{k_1k_2}\delta(\tau_1-\tau_2)\tag{2}$$