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A question about normal ordering (regarding Weinberg's QFT Vol 1. p. 200)

  1. Aug 25, 2011 #1
    In the lower part of page 200 in his book, Weinberg says that any normal-ordered function of the fields can be expressed as a sum of ordinary products of the fields with c-number coefficients.

    I don't quite see this.

    A field can be decomposed in terms of parts that contain either only creation operators or only annihilation operators.
    [tex] \phi(x) = \phi_{+} (x) + \phi_{-} (x)[/tex]
    [tex] \phi^{\dagger}(x) = \phi_{+}^{\dagger} (x) + \phi_{-}^{\dagger} (x)[/tex]

    If we consider a function of fields
    [tex]F(\phi(x), \phi^{\dagger}(x))[/tex]
    and its normal ordered version
    [tex]:F(\phi(x), \phi^{\dagger}(x)):[/tex],
    the difference between the two seems to involve all of the following quantities.
    [tex] \phi_{+} (x), \,\phi_{-} (x),\,\phi_{+}^{\dagger} (x),\, \phi_{-}^{\dagger} (x)[/tex]

    However, Weinberg says that we can express it in terms of only
    [tex] \phi(x),\, \phi^{\dagger}(x)[/tex].

    Is there any simple argument to justify this? My impression is that Weinberg would provide some arguments in the book unless it is too obvious.
     
    Last edited: Aug 25, 2011
  2. jcsd
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