Coincident Products and Contact Terms

Coincident products are singular because quantum fields and their correlators are distributions, not functions whose arguments may simply be set equal. A collision therefore asks for a restriction to a diagonal or an extension across it; either operation can fail or require new local data. Contact terms are the delta functions and their derivatives supported on those collision sets. They are invisible at separated points but indispensable in differentiated time-ordered products, source identities, and Ward identities.

Required background. Local and Composite Operator Insertions supplies source-generated insertions, smearing, full-versus-connected normalization, and the regulated meaning of a formal composite.

Helpful background. Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards supplies the weak derivative and pullback language. Products, Scaling Degree, and Extensions of Singular Distributions supplies the precise extension theorems used here only as orientation.

Coincidence is not pointwise substitution

Let $t(x,y)$ denote a scalar distribution on two copies of spacetime, and let

$$\Delta:M\longrightarrow M\times M, \qquad \Delta(x)=(x,x)$$

be the total diagonal. The notation $t(x,x)$ would mean the pullback $\Delta^*t$ if that pullback exists. For a singular two-point distribution, it often does not. Replacing $y$ by $x$ in a formal kernel is not a substitute for the missing pullback.

A related but distinct problem starts with a product $t_0(x,y)$ defined only for $x\ne y$. One may seek an extension $t$ to all of $M\times M$ that agrees with $t_0$ off the diagonal. The extension can exist without being unique. Pullback asks for a distribution on the diagonal; extension supplies a distribution across a deleted diagonal. Both require hypotheses beyond ordinary algebra.

Smearing makes the distinction operational. If $f(x,y)$ has support away from $x=y$, then $t_0(f)$ is already fixed. A family of test functions whose support approaches the diagonal probes new short-distance information. A divergent or profile-dependent limit means that a coincidence prescription is still missing.

Scaling degree detects a local ambiguity

In four spacetime dimensions, the free Feynman two-point function has the leading short-distance behavior

$$D_F(r)\sim\frac{1}{4\pi^2(-r^2+i0)}, \qquad \operatorname{sd}_0D_F=2.$$

Its square is a well-defined boundary-value distribution for $r\ne0$ and has

$$\operatorname{sd}_0 D_F^2=4.$$

The number $4$ equals the codimension of the diagonal in $M\times M$. An extension preserving this scaling degree is therefore not unique: two such extensions can differ by

$$c\,\delta^{(4)}(r).$$

More generally, a distribution on $\mathbb R^N\setminus\{0\}$ with scaling degree below $N$ has a unique extension with the same scaling degree. At or above $N$, derivatives of $\delta$ up to the allowed singular order can appear. This criterion diagnoses the available local ambiguity; symmetry, field equations, and normalization conditions may reduce or fix it. The precise theorem and the propagator example are given in Brunetti and Fredenhagen 2000, §§ 5.1–5.2, PDF pp. 21–25.

This example does not say that every collision is represented by $D_F^2$ or that scaling degree is the only criterion. Tensor structure, several partial diagonals, derivative couplings, gauge identities, and wavefront conditions add information. Those theorem-level questions remain outside this Foundations treatment.

Contact terms live on collision sets

Locally, a scalar contact distribution—or, componentwise, an operator-valued contact distribution before taking matrix elements—supported on the diagonal has the schematic form

$$C(x,y) =\sum_{|\alpha|\leq N} c_\alpha(x)\, \partial_{(x-y)}^\alpha\delta^{(d)}(x-y).$$

Here the coefficients $c_\alpha$ may themselves be distributions—or operator-valued distributions—along the diagonal; they are ordinary local coefficient functions only in simpler settings.

Such a term pairs to zero with every test function supported away from $x=y$. Consequently, separated-point correlators cannot determine its coefficient. At coincidence it is not “zero almost everywhere”; distributions are defined by their action on test functions, and a delta-supported contribution can be the entire answer.

Contact terms arise in several logically different ways:

• differentiating an ordering prescription can expose an equal-time commutator;
• differentiating an explicit local source term produces delta functions between source arguments;
• extending a singular product can introduce local normalization freedom;
• changing a renormalized operator basis can redistribute local terms while preserving separated-point data.

The first two mechanisms are derived below. The last two are only identified here; interacting normalization and mixing continue in Renormalization and EFT.

Time ordering produces the free equation-of-motion contact

For a continuum free real scalar with $m>0$, avoiding an unrelated infrared qualification, define

$$D_F(x-y)=\langle0|\mathrm T\{\phi(x)\phi(y)\}|0\rangle, \qquad P_x=\Box_x+m^2.$$

The first time derivative of the ordered product has no contact because the equal-time field commutator vanishes:

$$\partial_{x^0}D_F(x-y) =\langle0|\mathrm T\{\dot\phi(x)\phi(y)\}|0\rangle.$$

The second derivative does meet the canonical commutator. With

$$[\phi(t,\mathbf x),\dot\phi(t,\mathbf y)] =i\delta^{(d-1)}(\mathbf x-\mathbf y),$$

one obtains

$$\partial_{x^0}^2D_F(x-y) =\langle0|\mathrm T\{\ddot\phi(x)\phi(y)\}|0\rangle -i\delta^{(d)}(x-y).$$

The spatial derivatives and mass term combine with the ordered first term through the free field equation, leaving the distributional identity

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

The minus sign follows from $[\dot\phi(x),\phi(y)]=-i\delta^{(d-1)}(\mathbf x-\mathbf y)$, while the factor of $i$ matches the site’s momentum-space numerator $i/(p^2-m^2+i0)$. Schwartz derives the same time-ordering contact and its interacting generalization in Schwartz 2014, § 14.7.1, pp. 273–274.

The distinction is essential:

$$\big\langle\mathrm T\{(P\phi)(x)\phi(y)\}\big\rangle=0 \quad\text{for the free operator equation,}$$

but

$$P_x\big\langle\mathrm T\{\phi(x)\phi(y)\}\big\rangle =-i\delta^{(d)}(x-y).$$

Applying $P_x$ outside the time-ordering operation differentiates its step functions. Inserting the equation-of-motion operator inside the already defined ordering is a different operation.

Schwartz makes this operator-versus-path-integral differentiation warning explicit in Schwartz 2014, § 14.7.2, p. 275.

The same contact appears from a source variation

For the normalized free functional with action $S_0=-\tfrac12\phi\!\cdot\!P\!\cdot\!\phi$ and source sign $+J\!\cdot\!\phi$, the formal continuum distributional functional equation is

$$\left[ iP_x\frac{\delta}{\delta J(x)}+J(x) \right]Z_0[J]=0.$$

Differentiate once with respect to $J(y)$ and set $J=0$. The derivative of the explicit source is local:

$$\frac{\delta J(x)}{\delta J(y)} =\delta^{(d)}(x-y).$$

Using $\delta^2Z_0/\delta J(x)\delta J(y)=-D_F(x-y)$ then gives

$$-iP_xD_F(x-y)+\delta^{(d)}(x-y)=0,$$

which is again $P_xD_F=-i\delta$. Zinn-Justin presents the regularization assumptions, Euclidean functional identity, and first contact hierarchy in Zinn-Justin 2021, §§ 7.5–7.5.1, pp. 133–135; the signs above are the translation to $e^{iS+iJ\cdot\phi}$.

The full free hierarchy follows by repeating the same operation. Treat $x,x_1,\ldots,x_n$ as independent variables and read the result as a smeared distributional identity:

$$\begin{aligned} P_xG^{(n+1)}_0(x,x_1,\ldots,x_n) =-i\sum_{r=1}^n \delta^{(d)}(x-x_r)\, G^{(n-1)}_0(x_1,\ldots,\widehat{x_r},\ldots,x_n), \end{aligned}$$

where the hat means omission and $G_0^{(0)}=1$. Each contact records the contraction in which the field at $x$ meets one external insertion. Away from all partial diagonals $x=x_r$, the right-hand side vanishes. This is an exact identity of smeared distributions with the selected Feynman boundary condition, not the assertion that an infinite hierarchy is solved or closed.

Local source terms generate local insertion terms

Suppose a permissible change of normalization adds a local quadratic source term to the connected functional,

$$W_{mathrm{loc}}[K] =\frac{c}{2}\int\mathrm d^d x\,K(x)^2.$$

Then

$$\frac{\delta^2W_{\mathrm{loc}}} {\delta K(x)\delta K(y)} =c\,\delta^{(d)}(x-y),$$

and, because $G_c^{(2)}=-iW^{(2)}$ in the site’s Lorentzian convention, the connected two-insertion function shifts by $-ic\,\delta^{(d)}(x-y)$. Source terms with derivatives generate derivatives of delta functions. Thus two definitions can agree at every separated pair and differ only by contact terms.

This freedom is constrained rather than arbitrary. Locality, covariance, internal symmetries, dimensional analysis, field equations, and chosen normalization conditions restrict which source polynomials are allowed. Determining those constraints in an interacting renormalized theory belongs to the later composite-operator treatment.

Ordering, smearing, and regulators change the statement

Choice	Consequence at coincidence
time ordering	derivatives can act on step functions and produce equal-time commutators
Wightman ordering	the free equation acts homogeneously on each field argument; the time-ordering contact above is absent
Euclidean ordering	the elliptic Green function has its own delta normalization, with no copied Lorentzian factor of $i$
finite regulator	the identity may contain a regulated kernel $\delta_\Lambda$ and regulator-dependent local terms
hard boundary or noninvariant cutoff	boundary flux or symmetry-breaking terms can supplement the local contact
coincident composite	even one insertion such as $\phi^2(x)$ contains an internal collision requiring a definition

All identities should therefore be paired with test functions before a continuum limit is taken. A regulator that preserves the relevant symmetry can make its Ward identity exact at finite $\Lambda$; a generic regulator may require compensating terms. No continuum delta function should be inferred merely by erasing the regulator label.

Common mistakes at a diagonal

“The delta term vanishes because I am interested in generic points.” That is true only after restricting every test function or observable away from the diagonal. Integrated identities and further source derivatives generally probe the contact support.

“The classical equation of motion sets the ordered correlator to zero.” The equation annihilates the field insertion, not the step functions in the ordering operation. Differentiating the ordered correlator restores the commutator contact.

“An extension ambiguity is arbitrary nonlocal physics.” Two acceptable extensions agree away from the collision set; their difference is local. Normalization and symmetry conditions then constrain its coefficient.

“A contact term is always a removable artifact.” Some coefficients are prescription dependent, while others are required by exact Ward identities or canonical commutators. The support alone does not decide which case applies.

Check your understanding

Check 1: test the sign

Differentiate the time ordering twice and use $[\dot\phi(x),\phi(y)]=-i\delta^{(d-1)}(\mathbf x-\mathbf y)$. The result is $(\Box+m^2)D_F=-i\delta^{(d)}$, consistent with the momentum-space factor $i/(p^2-m^2+i0)$.

Check 2: separate pullback from extension

The symbol $t(x,x)$ asks for a pullback to the diagonal. Defining a product first on $x\ne y$ and then filling in the diagonal asks for an extension. Neither is justified by pointwise substitution, and success of one does not automatically imply the other.

Check 3: find what separated data cannot see

If $t'$ and $t$ differ by $c\delta^{(d)}(x-y)$, then $t'(f)=t(f)$ for every $f$ supported away from $x=y$. A normalization condition probing the diagonal is needed to distinguish them.

Check 4: differentiate a local source term

Two functional derivatives of $\tfrac c2\int K^2$ produce $c\delta(x-y)$. If this term belongs to $W$, multiply by $-i$ to obtain its contribution to the connected two-insertion correlator.

Continue from the collision problem

• Free Wick Products and Point Splitting gives a selected free-field subtraction after the diagonal problem is visible.
• Local versus Integrated Operator Redundancies tracks these contacts through equations of motion, integrations by parts, and field redefinitions.
• Contact Terms and Renormalized Operator Products develops interacting normalization at collisions, while Operator Mixing and Renormalization Matrices treats the resulting operator bases.
• Singular Support and Wavefront Sets gives the precise pullback criterion; Epstein–Glaser Induction and Products, Scaling Degree, and Extensions of Singular Distributions develop the extension machinery.

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.

• 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.