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

  1. Nov 9, 2013 #1
    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. jcsd
  3. Nov 9, 2013 #2
    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).
     
  4. Nov 9, 2013 #3

    vanhees71

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    Ward identities (or Ward-Takahashi, Slavnov-Taylor identities) hold order by order in the [itex]\hbar[/itex] expansion if they hold for the full expression, because [itex]\hbar[/itex] enters the theory as an overall factor in the path-integral for generating functions.
     
  5. Nov 9, 2013 #4
    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)?
     
  6. Nov 9, 2013 #5

    vanhees71

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  7. Nov 9, 2013 #6
    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)
     
  8. Nov 10, 2013 #7
    I do not understand why the matrix element of the time ordered product of corresponding Heisenberg-picture operators:

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

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