Local and Composite Operator Insertions

A local expression is inserted by coupling it to its own spacetime-dependent source and differentiating the regulated generating functional. This operation fixes the insertion’s label, ordering, normalization, and position, but it does not by itself construct a regulator-independent interacting composite operator. The clean starting point is therefore a family of regulated bosonic expressions $\mathcal O_{A,\Lambda}$, smeared sources, and insertion points kept away from one another and from any external fields.

Required background. The Generating Functional supplies normalized source differentiation and the selected in–out/Feynman boundary prescription. Quantum Fields as Operator-Valued Distributions supplies the smeared meaning of a field and its correlators.

Helpful background. Test-Function Spaces, Distributions, Support, and Convergence supplies precise support and convergence language for the sources used below.

A source labels a regulated local insertion

Let $\Lambda$ denote a finite regulator, and let $\mathcal O_{A,\Lambda}(x)$ be a declared bosonic local expression built from regulated fields and finitely many derivatives. Couple a smooth compactly supported source $K^A$ to each expression:

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

Here a local expression depends on fields and finitely many derivatives at one spacetime argument; it is composite when it is nonlinear in the basic fields, as in $\phi^2$; and an insertion is the distributional entry produced in a correlator by differentiating its source. None of these labels by itself makes the expression a physical observable. Dimensional homogeneity gives the immediate check

$$[K^A]=d-[\mathcal O_A].$$

The integration cycle $\mathcal C_F$, state selection, pole prescription, and normalization are the same in–out/Feynman data used on the generating-functional page. Define

$$Z_\Lambda[J,K] =\frac{\mathcal Z_\Lambda[J,K]}{\mathcal Z_\Lambda[0,0]}, \qquad W_\Lambda[J,K]=-i\log Z_\Lambda[J,K].$$

At the regulated level, differentiation is now ordinary calculus in the source variables:

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

Thus the source derivative inserts the expression at $x$. The source term also records which linear combination is meant when several expressions have the same quantum numbers. Zinn-Justin develops this local-source construction and shows why interacting insertions can mix with other expressions of the same or lower dimension in Zinn-Justin 2021, §§ 11.1–11.1.3, pp. 240–244. His displayed functional uses Euclidean weight; the factors of $i$ here follow the site’s Lorentzian convention.

The word “composite” is descriptive, not a proof of existence. A monomial such as $\phi^2(x)$ or $\partial_\mu\phi(x)\partial_\nu\phi(x)$ is well-defined at a finite regulator. Removing $\Lambda$ may require subtractions, mixing, and a renormalization prescription. This page keeps $\Lambda$ explicit whenever that distinction matters.

Functional derivatives are distributional notation

A point label is shorthand for a distribution. For a test function $f^A$,

$$\mathcal O_\Lambda(f) =\sum_A\int\mathrm d^d x\, f^A(x)\mathcal O_{A,\Lambda}(x),$$

and the invariant statement is the directional source derivative

$$\left. \frac{1}{i}\frac{\mathrm d}{\mathrm d\epsilon} \log Z_\Lambda[J,K+\epsilon f] \right|_{\epsilon=0} =\big\langle\mathcal O_\Lambda(f)\big\rangle_{J,K}.$$

Writing $\delta/\delta K^A(x)$ exposes the distributional kernel of this relation; it does not turn $\mathcal O_A(x)$ into a bounded operator at a mathematical point. Fewster and Rejzner formulate fields through smeared operators and explicitly warn against assuming an underlying point operator in Fewster and Rejzner 2020, §§ 4.1–4.2, PDF pp. 13–16.

For several insertions, first choose test functions with mutually separated supports. The resulting distribution is defined on the configuration space away from its partial diagonals. Shrinking supports until two insertion points collide is a new operation; it is not secretly performed by taking two independent source derivatives.

Full and connected insertion correlators

Repeated derivatives of $Z_\Lambda$ generate full time-ordered insertions:

$$G^{(n)}_{A_1\ldots A_n;J,K}(x_1,\ldots,x_n) =\frac{(-i)^n}{Z_\Lambda[J,K]} \frac{\delta^n Z_\Lambda[J,K]} {\delta K^{A_1}(x_1)\cdots\delta K^{A_n}(x_n)}.$$

The logarithm removes products of source-dependent lower moments. With $W_\Lambda=-i\log Z_\Lambda$,

$$G^{(n)}_{c,A_1\ldots A_n;J,K} =(-i)^n \frac{\delta^n\log Z_\Lambda} {\delta K^{A_1}\cdots\delta K^{A_n}} =(-i)^{n-1} \frac{\delta^n W_\Lambda} {\delta K^{A_1}\cdots\delta K^{A_n}}.$$

For $n=1$ the full and connected insertions agree. For $n=2$, $W_\Lambda^{(2)}=iG^{(2)}_{c}$ in this Lorentzian convention. A logarithm removes disconnected partitions; it does not subtract the short-distance singularities inside a local composite.

Mixed $J$ and $K$ derivatives insert basic fields and declared local expressions in the same ordered correlator. Their ordering remains the in–out time ordering encoded by $\mathcal C_F$; these derivatives do not automatically produce retarded response functions or in–in expectation values.

A regulated φ² insertion

Take a centered free real scalar and add

$$\frac12\int\mathrm d^d x\,K_2(x)\phi_\Lambda^2(x)$$

to the source action. Since $[\phi^2]=d-2$, the source has $[K_2]=2$. For the algebraic constant-source check—or for a source constant on a regulated finite region—the quadratic action reads

$$\frac12(\partial\phi_\Lambda)^2 -\frac12(m^2-K_2)\phi_\Lambda^2,$$

so the source convention passes the round-trip check $m^2\mapsto m^2-K_2$. At $J=K_2=0$, a mixed derivative gives

$$\frac{(-i)^3}{Z_\Lambda} \frac{\delta^3 Z_\Lambda} {\delta K_2(x)\,\delta J(y)\,\delta J(z)} =\frac12\big\langle0\big|\mathrm T\{\phi_\Lambda^2(x)\phi_\Lambda(y)\phi_\Lambda(z)\}\big|0\big\rangle.$$

For distinct $x,y,z$, free Gaussian factorization yields

$$\begin{aligned} \frac12\big\langle\mathrm T\{\phi_\Lambda^2(x)\phi_\Lambda(y)\phi_\Lambda(z)\}\big\rangle ={}&\frac12D_{F,\Lambda}(x,x)D_{F,\Lambda}(y,z)\\ &+D_{F,\Lambda}(x,y)D_{F,\Lambda}(x,z), \end{aligned}$$

whereas the connected insertion correlator is

$$\frac12\big\langle\mathrm T\{\phi_\Lambda^2(x)\phi_\Lambda(y)\phi_\Lambda(z)\}\big\rangle_c =D_{F,\Lambda}(x,y)D_{F,\Lambda}(x,z).$$

The three-pairing four-point formula underlying this calculation is displayed in Schwartz 2014, § 14.3.2, p. 263.

This comparison separates two issues. Taking the logarithm removes the disconnected factor $\tfrac12D_{F,\Lambda}(x,x)D_{F,\Lambda}(y,z)$. It does not define the continuum operator $\phi^2(x)$: the coincident contraction $D_{F,\Lambda}(x,x)$ still records regulator dependence, and products at $x=y$ or $x=z$ raise a further extension problem. Free Wick subtraction and point splitting are developed only after that collision problem has been isolated.

A stress-tensor insertion at separated points

A symmetric tensor source can label a more structured composite. For the stable free scalar with $m^2\geq0$, take the minimally coupled continuum tensor with no improvement term and evaluate the following expression in the regulated calculation:

$$T^{(0)}_{\mu\nu,\Lambda} =\partial_\mu\phi_\Lambda\,\partial_\nu\phi_\Lambda -\eta_{\mu\nu} \left( \frac12\partial_\rho\phi_\Lambda\partial^\rho\phi_\Lambda -\frac12m^2\phi_\Lambda^2 \right),$$

and couple $K^{\mu\nu}T^{(0)}_{\mu\nu,\Lambda}$. We use the symmetric-source convention

$$\frac{\delta K^{\rho\sigma}(u)} {\delta K^{\mu\nu}(x)} =\delta^\rho{}_{(\mu}\delta^\sigma{}_{\nu)} \delta^{(d)}(u-x),$$

so differentiating $\int K^{\rho\sigma}T^{(0)}_{\rho\sigma}$ inserts $T^{(0)}_{\mu\nu}$ without an extra factor of two. This is a bookkeeping source for a specified tensor, not yet a claim that all improvement, metric-variation, or renormalization choices are equivalent.

Because $[T^{(0)}_{\mu\nu}]=d$, its source is dimensionless. The sign convention also passes the energy-density check:

$$T^{(0)}_{00,\Lambda} =\frac12\left[ (\partial_0\phi_\Lambda)^2 +|\boldsymbol\nabla\phi_\Lambda|^2 +m^2\phi_\Lambda^2 \right].$$

The underlying canonical tensor and translation charges are derived in Schwartz 2014, § 3.3.1, pp. 34–36; the regulated correlator below follows by applying free Gaussian contractions to that declared tensor.

At separated $x,y,z$, its connected insertion between two scalar fields is the following; every displayed derivative acts on the $x$ argument:

$$\begin{aligned} &\big\langle\mathrm T\{T^{(0)}_{\mu\nu,\Lambda}(x) \phi_\Lambda(y)\phi_\Lambda(z)\}\big\rangle_c\\ ={}& \partial_\mu D_{F,\Lambda}(x,y)\, \partial_\nu D_{F,\Lambda}(x,z) +\partial_\nu D_{F,\Lambda}(x,y)\, \partial_\mu D_{F,\Lambda}(x,z)\\ &-\eta_{\mu\nu} \left[ \partial_\rho D_{F,\Lambda}(x,y)\, \partial^\rho D_{F,\Lambda}(x,z) -m^2D_{F,\Lambda}(x,y)D_{F,\Lambda}(x,z) \right]. \end{aligned}$$

Every term connects the insertion at $x$ to both external points. If a divergence is applied and $x$ approaches $y$ or $z$, derivatives of time ordering generate contact distributions. Away from those diagonals, the continuum free equation gives conservation after a controlled $\Lambda\to\infty$ limit. At finite $\Lambda$ this holds exactly only for a regulator–tensor pair satisfying the corresponding translation Ward identity; otherwise regulator terms remain. The contact terms, Ward-identity form, and ordering qualifications belong to the next page.

The choice of stress tensor matters locally. An improvement term such as $(\eta_{\mu\nu}\Box-\partial_\mu\partial_\nu)\phi^2$ changes the insertion even when an integrated charge or selected matrix element is unchanged under suitable boundary conditions. That distinction is developed later in this chapter rather than hidden inside the source notation.

What the source construction does and does not establish

The construction fixes	It does not establish
which regulated expression is coupled	existence of its regulator-independent interacting limit
source normalization and Lorentzian phases	a universal subtraction prescription
ordering, state, contour, and boundary data	retarded or in–in semantics
full versus connected insertion correlators	ultraviolet finiteness of either one
separated-support distributions	an extension to coincident insertion points
the chosen tensor or operator basis	scheme-independent mixing coefficients or anomalous dimensions

For fermionic expressions, the sources are Grassmann-valued and derivative order carries signs. For gauge theories, a source may couple to a gauge-invariant observable or to a gauge-fixed auxiliary field; the latter does not become a physical observable merely because it has a source. Neither extension changes the basic lesson: the source defines what is being differentiated, while the physical and renormalized status of the insertion requires additional structure.

Check your understanding

Check 1: recover the Lorentzian phase

Differentiate $e^{i\int K^A\mathcal O_A}$ once. The result is $i\mathcal O_A$ times the integrand, so $(i\mathcal Z)^{-1}\delta\mathcal Z/\delta K^A=\langle\mathcal O_A\rangle$. Repeating the operation gives the factor $(-i)^n$ in the definition of a full $n$-insertion correlator.

Check 2: separate connectedness from renormalization

In the free $\phi^2$ example, passing from $Z$ to $\log Z$ removes $\tfrac12D_{F,\Lambda}(x,x)D_{F,\Lambda}(y,z)$ because it factorizes into two disconnected blocks. It does not make the formal local product regulator independent. Connectedness classifies correlation structure; renormalization defines ultraviolet-sensitive local products.

Check 3: test the stress-tensor formula

Contract each of the two fields in $T^{(0)}_{\mu\nu}(x)$ with one external scalar. The two assignments give the first line of derivatives; contracting the Lagrangian term gives the trace contribution. No $D_F(x,x)$ term survives in the connected correlator.

Check 4: diagnose a proposed coincidence limit

If two source derivatives are evaluated at independent points, no coincidence has occurred. Setting those points equal afterward asks whether the resulting distribution restricts or extends to the diagonal. That is a new mathematical question, not an algebraic consequence of differentiation.

Continue from separated insertions

• Coincident Products and Contact Terms asks what changes when insertion points meet and derives the first delta-supported terms.
• Local versus Integrated Operator Redundancies compares local insertions with integrations by parts, field redefinitions, and selected on-shell matrix elements.
• Renormalized Composite-Operator Insertions develops interacting definitions, operator mixing, scheme and scale dependence, and anomalous dimensions.

References

• 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.