What's the definition of the time ordering operator for more than two particles?

Question

For two particles, $\langle {\mathcal T} a(t_1) a^\dagger (t_2) \rangle = \langle a(t_1) a^\dagger (t_2)\rangle \theta (t_1-t_2) + \xi \langle a^\dagger (t_2)a(t_1) \rangle \theta (t_2-t_1)$ with $\xi$ is a plus sign for bosons and a minus sign for fermions.

How would I write, for example, $\langle {\mathcal T} a(t_1) a^\dagger (t_2) a(t_3) a^\dagger (t_4) \rangle$ ?

Answer 1

Just sum over each permutation of [1,2,3,4], for each permutation $[I_1,I_2,I_3,I_4]$you would have a factor of $\theta(t_{I_1}-t_{I_2})\theta(t_{I_2}-t_{I_3})\theta(t_{I_3}-t_{I_4})$ times the corresponding operator product, etc.