Equal time contractions in Wick contractions

[CAF123]

One usually ignores equal time contraction terms when computing time ordered products of interaction hamiltonians arising in the Dyson expansion. E.g it is often said $$T(N(AB\dots)_{x_1} \dots N(AB \dots)_{x_n}) = T((AB \dots)_{x_1} \dots (AB \dots)_{x_n})_{\text{no E.T.C}}$$

Few questions

a)I think Wick's theorem applied to these expressions will not generate any tadpole contributions since there the loop starts at some x and finishes at the same x. Is that correct?

I've read that the S-matrix can be defined as the time ordered product of an exponentiated normal ordered interaction hamiltonian, the normal ordering a means of renormalisation eliminating infinities arising from such self-interactions.

b) Why is this statement completely true? Wick's theorem applied onto above will certainly give e.g a bubble contribution (comprising two propagators, one of which starts at some x and finishes at y and the other vice versa) which is divergent in four dimensions so it seems there is not a complete elimination of infinities.

c) Is there a more physical reasoning why we normal order the hamiltonian in the S-matrix? I've read it gives unphysical results such as non conservation of 4 momentum if we don't normal order but didn't understand this completely. Moreover, as far as I can see, the tadpole diagrams are not generated by normal ordering so it seems we lost contributions (or perhaps the whole process is for 1PI /amputated diagrams?)

Thanks!

 

[CAF123]

Basically I'd just like to understand the motivation behind using a normal ordered hamiltonian in the definition of the S-matrix. I don't see how this would give tadpole diagrams in any Wick expansion given the equation I cite in my OP (tadpole diagrams involve contractions between field at the same space time point). Anyone?