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.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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

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