# Problem with Wick's theorem at first order

by GQuinta
Tags: order, theorem, wick
 P: 1 Hey there, first post here! I've been struggling with a detail in Second Quantization which I really need to clear out of my head. If I expand the S-matrix of a theory with an interaction Hamiltonian $H_I(x)$ then I have $S - 1= \int^{+\infty}_{-\infty} d^4 x H_I(x) + \int^{+\infty}_{-\infty} \int^{+\infty}_{-\infty} d^4 x d^4 y T[ H_I(x) H_I(y) ] + ...$ where the T operator is unnecessary in the first term. Now, if I choose a $\overline{\psi}(x)\psi(x)$ theory for example, the first term gives some contributions which I can calculate most easily by doing the expansion $\psi(x) = \psi^+(x) + \psi^-(x)$, which is the essence of Wick's theorem. I know the contributions will be trivial, but the point is Wick's theorem is not defined for the same spacetime points, so I can't understand why everywhere I look people assume implicitly that $\overline{\psi}(x)\psi(x) = T[ \overline{\psi}(x)\psi(x) ]$ in the first term of $S-1$ if the time ordering operators aren't even the same, since this one has a minus sign in its definition. As far as I can see, what everyone says is that since the T operator in the first term can be there, then when substituting $H_I(x)$ we simply retain the operator and use Wick's theorem like the spacetime points were different and impose x=y at the end, but this doesn't make any sense since the operators T are different. Sorry for the long text... Thanks!