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

1. Aug 25, 2011

### weejee

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.
$$\phi(x) = \phi_{+} (x) + \phi_{-} (x)$$
$$\phi^{\dagger}(x) = \phi_{+}^{\dagger} (x) + \phi_{-}^{\dagger} (x)$$

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

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

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