<Normal ordering> defines a composite field by subtracting all singular self-contractions before bringing its constituent points together. For example,
$$
{}:\!\partial X\partial X\!:(z)
=\lim_{w\to z}\left[
\partial X(z)\partial X(w)-
\langle\partial X(z)\partial X(w)\rangle\right].
$$
This makes $T$ and $\bar T$ well-defined local operators and is equivalent in oscillator language to placing creation operators to the left of annihilation operators.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 3 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-306/3/ii/solution