Local Operators and Short-Distance Structure

Enter this chapter by identifying which operation you need. Source-coupled insertions at separated points are the starting language; collisions of insertion points raise a distribution-extension problem and can generate contact terms; selected free composites can be defined by Wick subtraction or point splitting; and equations of motion or total derivatives must be tested separately as local insertions and as integrated expressions. The final free-field short-distance preview supplies vocabulary for later operator-product expansions without claiming a general interacting definition or convergence theorem.

The three main routes are therefore: local insertions and contacts, free composite operators, and the boundary from a free OPE example to its developed treatments. Choose a route · Read the chapter synthesis · Review the chapter

Enter this chapter

The durable chain is

$$\text{operator-valued distributions} \longrightarrow \text{insertions} \longrightarrow \text{collisions and contacts} \longrightarrow \text{defined composites}.$$

At a finite regulator $\Lambda$, an auxiliary bosonic source $K^A$ can be coupled to a formal bosonic local expression $\mathcal O_{A,\Lambda}$. With the site’s Lorentzian source sign,

$$\mathcal Z_\Lambda[J,K] =\int\mathcal D\phi\, \exp\!\left( iS_\Lambda[\phi]+iJ\!\cdot\!\phi +i\sum_A\int\mathrm d^d x\, K^A(x)\mathcal O_{A,\Lambda}(x) \right),$$

and

$$\frac{1}{i} \frac{\delta\mathcal Z_\Lambda}{\delta K^A(x)} =\int\mathcal D\phi\, \mathcal O_{A,\Lambda}(x)e^{iS_\Lambda+iJ\cdot\phi+iK\cdot\mathcal O}.$$

Here $\mathcal Z_\Lambda$ is unnormalized. Division by it gives the source-dependent full insertion,

$$\frac{1}{i\mathcal Z_\Lambda[J,K]} \frac{\delta\mathcal Z_\Lambda[J,K]}{\delta K^A(x)} =\langle\mathcal O_{A,\Lambda}(x)\rangle_{J,K}.$$

After the source-independent normalization $Z_\Lambda=\mathcal Z_\Lambda/\mathcal Z_\Lambda[0,0]$, derivatives of $W_\Lambda=-i\log Z_\Lambda$ generate connected insertions with the inherited Lorentzian phases. These are insertion rules, not yet a regulator-independent definition of an interacting composite operator. When two or more insertion points are subsequently brought together—or when functional differentiation of nonlinear local source counterterms produces diagonal-supported distributions—additional local terms can be required. Zinn-Justin develops the source method, mixing with operators of equal or lower dimension, and delta-supported counterterms in a four-dimensional Euclidean convention in Zinn-Justin 2021, §§ 11.1–11.1.3, pp. 240–244; the displayed factors of $i$ above are the translation to the site’s Lorentzian weight.

The free scalar supplies the chapter’s sign checkpoint. With

$$D_F(x-y)=\langle0|\mathrm T\phi(x)\phi(y)|0\rangle,$$

the global conventions give

$$(\Box_x+m^2)D_F(x-y) =-i\delta^{(d)}(x-y).$$

The field equation holds away from $x=y$; differentiating the time ordering produces the contact distribution on the diagonal. This is the simplest model of the distinction the chapter keeps returning to. The Lorentzian sign and its operator/path-integral interpretation are checked in Schwartz 2014, § 14.7, pp. 273–276.

The boundary around this chapter

This chapter develops physical orientation to local insertions, coincidence singularities, contact terms, selected free Wick products and point-split definitions, and the difference between local and integrated redundancy statements. It stops before the following developed theories:

Question beyond the chapter	Developed treatment
How are interacting composite insertions renormalized and mixed?	Renormalized Composite-Operator Insertions
How do operator and Wilson-coefficient bases evolve?	Dual Evolution of Operators and Wilson Coefficients
When is an OPE convergent in a conformal theory, and what data does it contain?	From the Local OPE to Conformal Data
How are local Wick powers and products defined under theorem-level hypotheses?	Local Covariant Wick Powers and Operator Products
How does point splitting work for renormalized observables in curved spacetime?	Wick Polynomials and Point Splitting

Free normal ordering is an example, not a universal interacting prescription. Likewise, the free OPE preview is a short-distance expansion in a declared free setting; it does not imply convergence of a general Lorentzian interacting OPE.

Check your preparation

The overview has no prerequisite gate. Use these observable checks to decide where to enter.

Can you do this?	If yes	If unsure, repair here
Explain why $\phi(x)$ is an operator-valued distribution and why smearing matters	start with local insertions	Quantum Fields as Operator-Valued Distributions and Test-Function Spaces, Distributions, Support, and Convergence
Differentiate a source functional and track whether the result is full or connected	use the source-coupling route directly	The Generating Functional
Compute free contractions and distinguish a contraction from a pointwise product	enter the Wick-product or free-OPE route	Wick’s Theorem and Free Gaussian Factorization
Interpret derivatives and delta functions weakly	check contact terms	Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards
State when an integration by parts or field equation may be used in an action	take the local-versus-integrated branch	The Action Principle and Field Equations

For a broader repair sequence, use Fourier analysis, distributions, and Green functions repair or Variational and classical-field reasoning repair, then return to the route table below.

Choose a route

The arrows in this table are suggested reading orders; each linked leaf states its own hard preparation.

Reader goal	Route	What you should be able to decide afterward
Insert a local expression and diagnose contact terms	Local and Composite Operator Insertions → Coincident Products and Contact Terms → Local versus Integrated Operator Redundancies when field equations or integrations by parts matter	whether a statement concerns separated insertions, diagonal contact support, or an integrated observable
Define a selected free composite	local insertions → coincident products → Free Wick Products and Point Splitting, with Wick’s theorem as required preparation	what was subtracted, which reference state or two-point function was used, and why the result is not yet an interacting prescription
Understand the OPE handoff	local insertions → coincident products → Free-Field OPE Preview, with Wick’s theorem and distribution extensions available	which statement is a free asymptotic preview and which later volume supplies interacting, conformal, or rigorous meaning

From separated insertions to short-distance structure

The conceptual map is a sequence of questions, not a claim that every step is automatic.

Stage	Mathematical object	Decisive question	Typical failure if skipped
separated insertion	a smeared local field or a source derivative at distinct points	is the operator domain and ordering specified?	treating $\phi(x)$ as a bounded point operator
collision	a distribution on configuration space approaching a diagonal	does the restriction or extension to coincident points exist?	multiplying singular distributions as ordinary functions
contact analysis	terms supported on partial or total diagonals, such as derivatives of delta functions	did a derivative, source variation, or renormalized product generate a local term?	applying a field equation inside an ordered correlator and dropping the contact
free subtraction	a Wick product or point-split limit relative to declared free data	which singular two-point contribution was subtracted?	calling reference-dependent normal ordering a universal renormalization
redundancy comparison	a local insertion, an integrated functional, or an on-shell matrix element	which boundary, source, and contact terms survive in that object?	declaring an equation-of-motion operator identically zero
short-distance expansion	an asymptotic expansion of a separated product in local operators	in which theory, ordering, regime, topology, and convergence sense is the symbol $\sim$ used?	importing conformal convergence or interacting Wilson coefficients into the free preview

Fields are distributions before products are considered. Fewster and Rejzner explain the smeared-field viewpoint and the role of local algebras in Fewster and Rejzner 2020, §§ 4.1–4.2, PDF pp. 13–16. The additional problem at a collision is not removed by merely renaming the formal monomial a “composite operator.”

One four-dimensional free-field checkpoint makes the extension problem quantitative. Near the origin,

$$D_F(x)\sim\frac{1}{4\pi^2(-x^2+i0)}, \qquad \operatorname{sd}_0 D_F=2.$$

The punctured product $D_F^2$ therefore has scaling degree $4$, equal to the spacetime dimension. Extending it through $x=0$ while preserving that scaling degree is nonunique by a term $c\,\delta^{(4)}(x)$ before symmetries and normalization conditions are imposed. This is the simplest numerical signal that a collision can require new local data; the extension theorem and its hypotheses are developed in Brunetti and Fredenhagen 2000, §§ 5.1–5.2, PDF pp. 21–25.

Chapter guide

1. Local and Composite Operator Insertions. Learn how an auxiliary source labels local expressions and how separated-point free-scalar examples such as $\phi^2$ and the stress tensor enter correlators. It requires the generating functional and operator-valued distributions; test-function distributions are useful support. Continue to the contact-term page before taking coincidence limits, and to Renormalization and EFT before claiming an interacting definition.

2. Coincident Products and Contact Terms. Diagnose why a product defined away from the diagonal may fail at coincidence and how weak differentiation or source variation creates delta-supported terms. Local insertions are required; delta derivatives and singular-distribution extensions are useful preparation. The page stops before wavefront-set criteria, Epstein–Glaser induction, and renormalized mixing matrices.

3. Free Wick Products and Point Splitting. Define the free-scalar Wick square by subtracting a declared two-point singularity and track its reference-state dependence. Coincident products and Wick’s theorem are required. The result is a controlled free construction, not a solution of the interacting or curved-spacetime problem.

4. Local versus Integrated Operator Redundancies. Compare a local field-equation insertion with an integrated field redefinition, keeping contact terms, sources, boundaries, and on-shell matrix elements distinct. Local insertions and the action principle are required; the contact-term page is helpful. Developed EFT basis reduction continues with Integration by Parts and Equation-of-Motion Redundancy.

5. Free-Field OPE Preview. Expand a separated free-scalar product into Wick composites and learn the vocabulary of coefficient singularity, operator basis, and short-distance remainder. Coincident products and Wick’s theorem are required; distribution extensions and wavefront sets are useful depth. Interacting coefficient evolution, conformal OPE data, and theorem-level products continue in the exact routes named on that page.

Conventions that change the claim

The chapter inherits the site’s metric, Fourier, source, and $+i0$ conventions. Four additions must always be stated locally:

• Ordering and state. Wightman, time-ordered, Euclidean, and contour-ordered products are different distributions. A subtraction tied to one state or ordering is not silently transferred to another.
• Regulator and subtraction data. A formal $\mathcal O_{A,\Lambda}$, a free Wick product, and a renormalized interacting operator are three different objects. The regulator, reference two-point function, scheme, and scale appear at the first claim they affect.
• Equality type. An identity away from coincident points, an equality of extended distributions, an insertion identity modulo contacts, and an asymptotic expansion are not interchangeable.
• Boundary and support. A total derivative integrates away only under conditions that remove the boundary contribution. A contact term vanishes only against test functions or observables that actually annihilate its support.

For an interacting basis, renormalization is generally matrix-valued and may include lower-dimensional operators allowed by the symmetries. Products of already-renormalized insertions can still require additional diagonal counterterms. Zinn-Justin makes both qualifications explicit in Zinn-Justin 2021, § 11.1, pp. 241–244. Those results set the boundary of this Foundations chapter; their calculation belongs downstream.

The scalar thread

One free scalar connects the five leaves without making the free case universal.

1. Couple a source to the regulated expression $\phi^2(x)$ and use source differentiation to define separated insertions.

2. Bring insertion points together and test the diagonal. The identity $(\Box+m^2)D_F=-i\delta$ exposes the first contact term.

3. Relative to the same free vacuum, define the Wick square schematically by

   $$:\!\phi^2\!:_0(x) =\lim_{y\to x} \left[ \phi(x)\phi(y) -\langle0|\phi(x)\phi(y)|0\rangle\mathbf1 \right],$$

   where the limit is understood only after specifying the distributional prescription and smearing. The operator form of Wick’s theorem and normal ordering is proved in Schwartz 2014, §§ 7.A.1–7.A.2, pp. 100–103.

4. Compare the local insertion $(\Box+m^2)\phi(x)$ with its integral against a test function or with an on-shell matrix element. Contact and boundary terms decide which simplification is valid.

5. At separated points in the free vacuum,

   $$\mathrm T\{\phi(x)\phi(0)\} =D_F(x)\mathbf1 +:\!\phi(x)\phi(0)\!:,$$

   and the remaining normal-ordered bilocal factor has the declared formal or asymptotic local expansion in free Wick composites. This is the chapter’s preview, not the general interacting Wilson OPE. Zinn-Justin’s perturbative four-dimensional short-distance expansion and its operator-mixing qualifications appear in Zinn-Justin 2021, §§ 11.3–11.6, pp. 248–257.

What the five pages establish together

The same notation can conceal different mathematical claims:

Statement	Valid reading	Invalid shortcut
“insert $\mathcal O(x)$”	differentiate a declared source functional or smear a defined operator-valued distribution	assume every formal polynomial is already a continuum operator
“take $y\to x$”	specify a restriction, extension, subtraction, or regulated limit	substitute coincident coordinates into singular distributions
“normal order it”	subtract contractions relative to declared free reference data	infer scheme-independent interacting renormalization
“use the equation of motion”	retain contacts locally and state the integrated/on-shell conditions used	set the local insertion to zero in every correlator
“drop a total derivative”	verify support, boundary, and source conditions	erase a local divergence operator
“apply the OPE”	state theory, ordering, regime, operator basis, remainder, and convergence or asymptotic sense	treat one free Wick expansion as a universal convergent theorem

That is the chapter-scale result: short-distance QFT is controlled by the type and domain of each object as much as by its formal algebra. The five leaves progressively make those domains visible and then stop at the point where renormalization, conformal symmetry, or theorem-level microlocal structure becomes essential.

Misconception clinic

“Local” means pointwise. Locality concerns spacetime support and commutation structure; quantum fields are still operator-valued distributions. A coordinate label does not make arbitrary products at that point well-defined.

Normal ordering is renormalization. It is a useful free reference-dependent subtraction. Interacting composites generally mix and depend on a renormalization prescription.

The field equation vanishes inside every correlator. Ordered correlators acquire contact terms when the equation-of-motion operator meets another insertion. The free propagator checkpoint displays the missing delta function.

An integrated redundancy is a local identity. Integration by parts, field redefinitions, and on-shell simplifications require conditions on boundaries, sources, observables, and contact terms. They do not erase the local insertion.

Every OPE is convergent. The meaning ranges from a free formal or asymptotic short-distance expansion to a convergent conformal expansion or a theorem under additional hypotheses. The chapter labels only the first of these.

Review the chapter

Review mode	Task	Successful response	Repair
retrieval	Define a local composite insertion without treating it as a pointwise bounded operator.	Names the source or smearing, ordering, regulator/definition, and separated or coincident domain.	Local and Composite Operator Insertions
explanation	Explain why two individually defined insertions can need a new counterterm when they collide.	Identifies the diagonal as a new singular locus and local distributions as the extension ambiguity.	Coincident Products and Contact Terms and Products, Scaling Degree, and Extensions of Singular Distributions
derivation check	Recover the sign in $(\Box+m^2)D_F=-i\delta$.	Uses the site’s propagator numerator $i$, Fourier convention, or the time-ordering derivative; both routes agree.	Coincident Products and Contact Terms and Scalar Propagators, Ordered Correlators, and Sources
representation change	Translate one $K$ derivative of $\mathcal Z$ into an insertion.	Identifies $\mathcal Z$ as unnormalized, divides by it for the full source-dependent insertion, and uses derivatives of $W=-i\log Z$ for connected insertions.	The Generating Functional
comparison	Contrast $:\!\phi^2\!:_0$ with a renormalized interacting $[\phi^2]_R$.	States reference dependence for the first and regulator, mixing, scheme, and scale data for the second.	Free Wick Products and Point Splitting and Renormalized Composite-Operator Insertions
transfer	Decide whether $\int\mathrm d^d x\,\partial_\mu J^\mu$ may be dropped.	Asks about test-function support or boundary flux before simplifying; does not declare $\partial_\mu J^\mu(x)=0$.	Local versus Integrated Operator Redundancies
failure diagnosis	Find the error in applying $(\Box+m^2)\phi=0$ inside $\langle\mathrm T\phi(x)\phi(y)\rangle$.	Restores the delta-supported contact at $x=y$.	Coincident Products and Contact Terms
synthesis	Place a proposed short-distance expansion in the correct later treatment.	Distinguishes free preview, interacting Wilson/coefficient evolution, conformal convergence/data, and theorem-level product frameworks.	Free-Field OPE Preview and the three handoffs below

Where to continue

• Return to the volume map: Foundations shows how the operator-valued-distribution doorway connects this chapter to the rest of the free-field Rosetta stone.
• Continue with interacting operators: Renormalization and Effective Field Theory develops renormalized insertions, mixing, basis reduction, anomalous dimensions, and Wilson coefficients.
• Specialize to conformal data: From the Local OPE to Conformal Data adds primary normalization, blocks, convergence domains, and crossing.
• Strengthen the mathematical hypotheses: Products, Scaling Degree, and Extensions of Singular Distributions develops the extension problem, while Local Covariant Wick Powers and Operator Products treats theorem-level QFT constructions.
• Choose an ordering and state: Lorentzian, Euclidean, and In–In Formulations separates vacuum amplitudes, Euclidean data, causal response, and initial-state expectation values before products are compared.

References

• Brunetti, Romeo, and Klaus Fredenhagen. “Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds.” Communications in Mathematical Physics 208 (2000): 623–661. DOI. Open PDF, arXiv:math-ph/9903028.

• Fewster, Christopher J., and Kasia Rejzner. “Algebraic Quantum Field Theory—an Introduction.” In Progress and Visions in Quantum Theory in View of Gravity, 1–61. Cham: Birkhäuser, 2020. DOI. Open PDF, arXiv:1904.04051v2.

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. First ed. Cambridge: Cambridge University Press, 2014. DOI.

• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. Fifth ed. Oxford: Oxford University Press, 2021. DOI.