# Tag Info

Hints to the question (v1): Recall that the operator time ordering is symmetric $$\tag{1}T[A_1(t_1)\ldots A_n(t_n)]~=~T[A_{\pi(1)}(t_{\pi(1)})\ldots A_{\pi(n)}(t_{\pi(n)})],$$ where $\pi\in S_n$ is a permutation. (Here we assume for simplicity that all operators are Grassmann-even. Else there will be additional sign factors.) Recall that if $t_1> ... 2 Hint$\int_0^t \int_0^{t_1} dt_1 dt_2 \, a(t_1) a(t_2) = \frac{1}{2!} \int_0^t\int_0^t dt_1 dt_2\, \mathcal{T}\{ a(t_1) a(t_2) \}$and so forth. You can see this by noting that the (square) integration region in the second integral can be split up into two triangular integration regions like in the first integral. This is one way to define the ... 2 If you can calculate vacuum-to-vacuum transition amplitudes, you can calculate S-matrix elements, because the two are related by the LSZ reduction formula. The LSZ will in any case chop off the propagators for external lines that the generating functional inserts, so you will end up only needing to compute amputated diagrams. 3 The primary utility in introducing the generating functional is in using it to compute correlation functions of the given quantum field theory. Let's restrict the discussion to that of a theory of a single, real scalar field on Minkowski space, and let$x_1, \dots, x_n\$ denote spacetime points. Of central importance are time-ordered vacuum expectation ...