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- What is the correct way to write the S-matrix?
$$S = T\left\{e^{-i\int \mathscr{H}_I d^4 x}\right\}$$
or
$$S = T\left\{e^{-i\int :\mathscr{H}_I: d^4 x}\right\}$$
?
Where :: refers to the Normal-ordering.
My question arises when we expand the S-matrix using Wick's theorem, there we need to compute time-ordered products, but is not the same to compute
$$T\{\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1)\}$$ or
$$T\{:\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1):\}$$
While the second one is simply
$$:\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1):$$
the first one is
$$T\{\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1)\} = :\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1): + i \text{Tr}\{{S_F(0)\gamma^\mu\}}A_\mu(x_1)$$
So there's a difference between both, and similar diferences appear in higher order calculations, now I know that this extra terms give tadpole diagrams and that this vanish in QED. But in other theories we need to introduce this terms?
Thanks
$$T\{\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1)\}$$ or
$$T\{:\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1):\}$$
While the second one is simply
$$:\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1):$$
the first one is
$$T\{\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1)\} = :\bar{\psi}(x_1)\gamma^\mu \psi(x_1) A_\mu(x_1): + i \text{Tr}\{{S_F(0)\gamma^\mu\}}A_\mu(x_1)$$
So there's a difference between both, and similar diferences appear in higher order calculations, now I know that this extra terms give tadpole diagrams and that this vanish in QED. But in other theories we need to introduce this terms?
Thanks