Contact Terms, Equal-Time Commutators, and Schwinger Terms

Contact terms are the distributional memory of operator insertions. When a derivative acts on a time-ordered current product, it also differentiates the step functions that impose temporal order. The resulting delta functions multiply equal-time commutators, and those commutators encode the symmetry action found by the regulated path-integral derivation on the preceding page.

Products of currents at the same point require more care. Their regulated equal-time commutators can contain additional local distributions traditionally called Schwinger terms. Such a term may integrate away, may survive only for nonconstant smearing or at a boundary, or may be accompanied by seagull interactions. Its presence alone establishes neither a central extension nor an anomaly.

Required background. Localized Transformations and Ward–Takahashi Identities supplies the regulated contact identity and sign conventions. Coincident Products and Contact Terms supplies the prescription required for equal-point products.

Helpful background. Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards supplies distributional differentiation and the delta-function identities used below.

Differentiating a time-ordered product

Keep the convention

$$U(\epsilon)=e^{-i\epsilon^aQ_a}, \qquad \delta_a\mathcal O=-i[Q_a,\mathcal O],$$

so

$$[Q_a,\mathcal O]=i\delta_a\mathcal O.$$

The current is bosonic. Thus no graded sign appears when it is moved past an insertion in the elementary time-ordering calculation; fermionic insertions retain their displayed order.

Begin with one local operator. For $x^0\neq y^0$,

$$\begin{aligned} \mathrm T\{j_a^0(x)\mathcal O(y)\} ={}&\theta(x^0-y^0)j_a^0(x)\mathcal O(y) \\ &+\theta(y^0-x^0)\mathcal O(y)j_a^0(x). \end{aligned}$$

Differentiate distributionally. Since $\partial_{x^0}\theta(x^0-y^0)=\delta(x^0-y^0)$,

$$\begin{aligned} &\partial_{x^0} \mathrm T\{j_a^0(x)\mathcal O(y)\} \\ ={}&\mathrm T\{(\partial_{x^0}j_a^0)(x)\mathcal O(y)\} \\ &+\delta(x^0-y^0) [j_a^0(x),\mathcal O(y)]. \end{aligned}$$

Spatial derivatives do not differentiate the time-ordering step functions. Combining all components gives

$$\begin{aligned} &\partial_\mu \mathrm T\{j_a^\mu(x)\mathcal O(y)\} \\ ={}&\mathrm T\{(\partial_\mu j_a^\mu)(x)\mathcal O(y)\} \\ &+\delta(x^0-y^0) [j_a^0(x),\mathcal O(y)]. \end{aligned}$$

For several insertions $\mathcal X=\mathcal O_1(x_1)\cdots\mathcal O_n(x_n)$, there is one such commutator for every $x_k$. Weinberg derives this theta-function identity explicitly for the QED current with Dirac-field insertions in Weinberg 1995, Vol. I, § 10.4, pp. 447–448.

This formula is an identity of distributions. Writing $\partial_\mu\mathrm T\{j^\mu\mathcal X\} =\mathrm T\{(\partial_\mu j^\mu)\mathcal X\}$ without the second line is wrong precisely at coincidence.

From the charge action to local contacts

At equal time, the spatial integral of the current density is the charge. The required normalization is therefore

$$\begin{aligned} &[j_a^0(t,\mathbf x),\mathcal O(t,\mathbf y)] \\ ={}&i\delta^{(d-1)}(\mathbf x-\mathbf y) \\ &\quad\times \delta_a\mathcal O(t,\mathbf y) \\ &+\mathscr S_{a,\mathcal O}(\mathbf x,\mathbf y). \end{aligned}$$

Here $\mathscr S_{a,\mathcal O}$ denotes any additional local distribution allowed by the regulated composite product. Consistency with the declared charge action requires

$$\int\mathrm d^{d-1}x\, \mathscr S_{a,\mathcal O}(\mathbf x,\mathbf y)=0$$

under the same constant smearing and boundary conditions used to define $Q_a$. Derivatives of spatial delta functions are the simplest possibility. On a region with boundary, even that integral can leave a surface contribution.

Substituting the universal first term into the time-ordering identity gives

$$\begin{aligned} &\partial_\mu \left\langle \mathrm T\{j_a^\mu(x)\mathcal X\} \right\rangle \\ ={}&-\left\langle \mathrm T\{\mathcal B_a(x)\mathcal X\} \right\rangle \\ &+i\sum_k\delta^{(d)}(x-x_k) \\ &\quad\times \left\langle \mathrm T\{\mathcal X_{a,k}\} \right\rangle \\ &+\mathcal S_a(x;\mathcal X). \end{aligned}$$

The first two terms exactly match the localized change-of-variables identity. Schwartz derives those insertion-variation contacts directly in Schwartz 2014, §§ 14.8.1–14.8.2, pp. 278–280. The last term collects additional regulated equal-time contacts. In a complete Ward identity they must be matched by the chosen current-product prescription, local counterterms, operator variations, and any seagull vertices. They must not be added a second time to a path-integral formula that already includes them.

Current algebra and Schwinger terms

Suppose currents transform in the adjoint representation, $\delta_a j_b^\nu=f_{ab}{}^c j_c^\nu$. A locally supported commutator compatible with this transformation law has the schematic form

$$\begin{aligned} &[j_a^0(t,\mathbf x),j_b^\nu(t,\mathbf y)] \\ ={}&i f_{ab}{}^c j_c^\nu(t,\mathbf y) \\ &\quad\times \delta^{(d-1)}(\mathbf x-\mathbf y) \\ &+\mathscr S_{ab}^\nu(\mathbf x,\mathbf y). \end{aligned}$$

The first term integrates to $[Q_a,j_b^\nu]=i f_{ab}{}^c j_c^\nu$. The Schwinger term $\mathscr S_{ab}^\nu$ is supported at coincident points and can contain derivatives of delta functions multiplied by local operators, or in special cases a $c$-number distribution. The cited sources do not classify its most general form.

This distinction matters:

• A local derivative-delta term can vanish for constant smearing while changing commutators of local or Fourier-mode charges.
• A boundary can convert an integration by parts into a surface term.
• Only a term that survives in the relevant integrated algebra and commutes with the represented symmetry generators is a candidate central extension. It becomes a scalar only after restricting to an appropriate irreducible sector.
• An anomaly is an obstruction to imposing the quantum symmetry identity; it is not defined merely by the presence of an equal-time contact.

Possible projective global actions belong to Quantum Implementations, Projective Actions, and Central Extensions. Boundary charge algebras belong to Charge Algebras, Central Terms, and Corners, while affine current algebras belong to Affine Current Algebras and WZW Models.

What the QED example actually establishes

For a neutral electric current one might naively set $[j^0(t,\mathbf x),j^\nu(t,\mathbf y)]=0$. That step uses a same-point composite operator and therefore requires regulation. Weinberg shows in a bounded QED and scalar-QED analysis that regulated current commutators can acquire Schwinger terms. Some arise from the regulator, while charged-scalar currents can also have regulator-independent terms; in the full multiphoton amplitude these are cancelled by additional interactions quadratic in the electromagnetic field. See Weinberg 1995, Vol. I, § 10.5, pp. 448–450.

The correct lesson is not that every Schwinger term is regulator-dependent. It is that the current commutator and the accompanying interaction vertices must be regulated and tested together. The cancellation in this example is evidence that a Schwinger term by itself is not a universal anomaly diagnostic.

Threaded complex-scalar calculation

For the exact complex scalar,

$$\begin{aligned} j^0 &=i\left(\phi^\dagger\pi^\dagger-\pi\phi\right), \\ \pi &=\partial_0\phi^\dagger, \\ \pi^\dagger &=\partial_0\phi. \end{aligned}$$

The canonical equal-time relations give

$$\begin{aligned} &[j^0(t,\mathbf x),\phi(t,\mathbf y)] \\ ={}&-\delta^{(d-1)}(\mathbf x-\mathbf y) \\ &\quad\times \phi(t,\mathbf y), \\ &[j^0(t,\mathbf x),\phi^\dagger(t,\mathbf y)] \\ ={}&+\delta^{(d-1)}(\mathbf x-\mathbf y) \\ &\quad\times \phi^\dagger(t,\mathbf y). \end{aligned}$$

Inserted into the derivative of the time-ordered product, these reproduce the negative contact at every $\phi$ and the positive contact at every $\phi^\dagger$ derived on the preceding page.

There is also a useful warning already in this simple model. Since

$$j^i =i\left[ (\partial_i\phi^\dagger)\phi -\phi^\dagger\partial_i\phi \right],$$

a formal canonical calculation gives the following contribution in a current-product prescription that preserves this normalization:

$$\begin{aligned} &[j^0(t,\mathbf x),j^i(t,\mathbf y)] \\ ={}&2i[\phi^\dagger\phi]_R(t,\mathbf y) \\ &\quad\times \partial_{y^i} \delta^{(d-1)}(\mathbf x-\mathbf y). \end{aligned}$$

The bracket $[\phi^\dagger\phi]_R$ emphasizes that the coefficient is a defined composite insertion. Finite local redefinitions of the coincident current product may supplement this canonical contribution. The displayed derivative contact integrates to zero under constant spatial smearing with no boundary, but it survives for nonconstant test functions. It is operator-valued, so it is not a central charge. It is a concrete Schwinger-type contact whose interpretation depends on the complete regulated identity.

With the controlled breaking

$$\mathcal B =iN\left( h\phi^N-h^*(\phi^\dagger)^N \right),$$

the field-insertion contacts remain, while $-\langle\mathrm T\{[\mathcal B]_R(x)\mathcal X\}\rangle$ is added as a bulk breaking insertion. A delta-supported contact and a nonzero separated-point divergence are different phenomena. The residual $\mathbb Z_N$ has a finite Ward identity but no infinitesimal current algebra.

Common pitfalls

Dropping the derivative of time ordering. Current conservation removes the time-ordered $\partial_\mu j^\mu$ insertion for an exact symmetry, not the derivatives of the theta functions.

Inferring a local commutator from the charge alone. The integrated charge fixes the spatial integral of the equal-time commutator. Derivative-delta terms can remain locally invisible to constant smearing.

Calling every Schwinger term a central extension. Operator-valued or derivative contacts need not commute with the symmetry generators and may integrate to zero. Centrality is an additional algebraic statement.

Calling every Schwinger term an anomaly. The anomaly test is failure of the complete regulated Ward identity after allowed counterterms. Regulated Jacobians and Measure Variation develops that obstruction.

Ignoring seagull vertices. A current-current commutator cannot be tested in isolation when the same regulator or Lagrangian produces quadratic source or gauge-field interactions.

Check your understanding

Use the complex-scalar current and canonical equal-time commutators to derive first $[j^0(\mathbf x),\phi(\mathbf y)]$, and then $[j^0(\mathbf x),j^i(\mathbf y)]$. Explain why the second result does not modify the global $U(1)$ charge action on a boundaryless equal-time slice.

Check

Only the term $-i\pi\phi$ in $j^0$ fails to commute with $\phi$, giving

$$[j^0(\mathbf x),\phi(\mathbf y)] =-\delta^{(d-1)}(\mathbf x-\mathbf y)\phi(\mathbf y).$$

Differentiate this relation with respect to $y^i$, use its conjugate, and apply the Leibniz rule to $j^i=i[(\partial_i\phi^\dagger)\phi-\phi^\dagger\partial_i\phi]$. The ordinary delta terms cancel, leaving the displayed canonical contribution in a prescription that preserves this normalization:

$$\begin{aligned} &[j^0(\mathbf x),j^i(\mathbf y)] \\ ={}&2i[\phi^\dagger\phi]_R(\mathbf y) \\ &\quad\times \partial_{y^i}\delta^{(d-1)}(\mathbf x-\mathbf y). \end{aligned}$$

For constant smearing, $\int\mathrm d^{d-1}x\,\partial_{y^i}\delta(\mathbf x-\mathbf y)=0$. Thus this derivative contact does not alter $[Q,j^i]$ when there is no boundary term. Nonconstant smearing instead probes the derivative of the test function. Finite local contact redefinitions of the renormalized current product remain possible and must be fixed by the full Ward identity.

What to carry forward

Insertion contacts, Schwinger terms, central extensions, and anomalies are four different layers. The first follows universally from the symmetry action and time ordering. The second belongs to regulated coincident current products. The third concerns the resulting integrated algebra. The fourth concerns whether the full quantum Ward identity can be imposed at all.

Current Sources and Generating Functionals packages these distributions through functional derivatives, where local counterterms and seagull terms can be tracked systematically. Detailed renormalization of contact identities belongs to Symmetry-Restoring Counterterms.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI