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

[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.

 

[eljose79]

I can offer some insights on this topic. Weinberg's statement is based on the concept of normal ordering, which is a mathematical tool used in quantum field theory to simplify calculations. Normal ordering involves rearranging the operators in a product so that all creation operators are to the left of all annihilation operators.

In the case of a function of fields, F(\phi(x), \phi^{\dagger}(x)), the normal ordered version can be expressed as a sum of ordinary products of the fields with c-number coefficients. This can be seen by considering the expansion of the function in terms of creation and annihilation operators. Since normal ordering involves rearranging these operators, it can be rewritten in terms of just the fields \phi(x) and \phi^{\dagger}(x).

To understand this concept better, let's consider a simpler example. Suppose we have a function of two variables, f(x,y), and we want to normal order it. The normal ordered version would be written as :f(x,y): = f(x,y) - f(y,x). Here, we can see that the difference between the normal ordered version and the original function involves both x and y variables, but the normal ordered version can be expressed in terms of just x and y.

Similarly, in the case of a function of fields, the normal ordered version can be expressed in terms of just the fields \phi(x) and \phi^{\dagger}(x), even though the difference between the two involves all four components \phi_{+} (x), \,\phi_{-} (x),\,\phi_{+}^{\dagger} (x),\, \phi_{-}^{\dagger} (x). This is because the normal ordering process involves rearranging these components in a specific way, resulting in a simplification of the expression.

In conclusion, Weinberg's statement is justified by the concept of normal ordering, which allows us to express a function of fields as a sum of ordinary products of the fields with c-number coefficients. This simplification is possible due to the specific rearrangement of creation and annihilation operators involved in the normal ordering process.