Local Operators and Short-Distance Structure
Products of fields at nearby points can become singular when the points coincide. This chapter explains how local insertions, contact terms, and free Wick products manage that short-distance structure, and why identities for integrated observables do not automatically hold for local operators.
Enter this chapter
Section titled “Enter this chapter”Begin with Local and Composite Operator Insertions to define an insertion through its source. Then examine coincidence before applying point splitting or operator-product expansions.
Topics
Section titled “Topics”- Local and Composite Operator Insertions — Defines composite insertions using independent local sources.
- Coincident Products and Contact Terms — Locates the singularities and delta-supported terms produced by collisions.
- Free Wick Products and Point Splitting — Constructs selected free composites with a specified subtraction.
- Local versus Integrated Operator Redundancies — Tests when equations of motion or total derivatives can be removed.
- Free-Field OPE Preview and Further Developments — Uses a free-field example to introduce short-distance expansion and its limits.
How the ideas fit together
Section titled “How the ideas fit together”A separated product of distributions can be meaningful even when its restriction to coincidence is not. Subtracting a prescribed singular part can define a composite, but finite local terms may remain ambiguous. Scaling-degree extension makes the origin of such local freedom precise; see Brunetti and Fredenhagen 2000, § 5.
The same locality matters when simplifying operators. A total derivative integrates to a boundary term only after the domain and boundary data are chosen. An equation-of-motion insertion inside a time-ordered correlator can produce contact terms at other insertions. Neither expression is simply zero in every local measurement.
Review the chapter
Section titled “Review the chapter”Why can an equation-of-motion operator vanish at separated insertions yet contribute after integration?
Solution
Its correlator may contain delta distributions supported where insertion points coincide. At separated points those terms vanish, but integration samples their support and produces finite contributions. Boundary terms can provide an additional contribution when integrations by parts are used.
Where to go next
Section titled “Where to go next”Continue to Correlators, Sources, and Effective Actions for the source identities behind contact terms, then use the free OPE page’s links for interacting operator renormalization.