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

[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?

 

[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).

 

[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.

 

[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)?

 

[vanhees71]

Only the S-matrix is invariant under gauge transformations. Off-shell Green's functions are not!

See my QFT notes,

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

Section 7.2.2

 

[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)

 

[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?