Why can we apply the symmetries of S-Matrix to part of Feynman diagram

1. Nov 9, 2013

ndung200790

How can we demonstrate that the symmetries of S-Matrix can be applyed to parts of Feynman diagrams?The S-Matrix is the sum of infinite diagrams,why we know each or part of each diagram has the same symmetries as the symmetries of S-Matrix?

2. Nov 9, 2013

Chopin

They don't always--sometimes you need to sum up multiple diagrams in order to produce a symmetry of the S-matrix. For instance, the Ward-Takahashi identity is not satisfied per-diagram--you need to sum across all insertion points of the photon into the fermion lines in order to show it.

However, the converse is true--if you can show that a relation is true for all diagrams, then it is automatically true for the S-matrix as well (assuming that perturbation theory converges).

3. Nov 9, 2013

vanhees71

Ward identities (or Ward-Takahashi, Slavnov-Taylor identities) hold order by order in the $\hbar$ expansion if they hold for the full expression, because $\hbar$ enters the theory as an overall factor in the path-integral for generating functions.

4. Nov 9, 2013

ndung200790

Can Ward Identity ensure that the correspondent sum of diagrams(the sum satisfies the Ward Identity) is invariant under the symmetry transformation(the symmetry of S-Matrix or of Lagrangian)?

5. Nov 9, 2013

6. Nov 9, 2013

ndung200790

So I do not understand what does author mean in Weinberg's QFT &10.1 when writing:
''One obvious but important use of the theorem quote above(the theorem saying:the sum of all diagrams for a process α--->β with extra vertices inserted corresponding to operators Oa(x),Ob(x),etc is given by the matrix element of the time ordered product of the corresponding Heisenberg-picture operators...) is to extend the application of symmetry principles from S-matrix elements,where all external lines have four-momenta on the mass-shell,to part of Feynman diagrams,with some or all external lines off the mass-shell''
(At the end of the section,he leads to Furry's theorem as an example of charge-conjugate symmetry of sum of diagrams)

7. Nov 10, 2013

ndung200790

I do not understand why the matrix element of the time ordered product of corresponding Heisenberg-picture operators:

($\Psi$$^{-}_{\beta}$,T{-iO$_{a}$(x),O$_{b}$(y)...}$\Psi$$^{+}_{\alpha}$)

is invariant under the symmetries if S-matrix is invariant?