Contact Terms, Operator Domains, and Perturbative Error Budgets

Modular-response calculations combine distributions, unbounded operators, moving boundaries, analytic continuation, and perturbative truncation. A compact final coefficient is trustworthy only when each source of ambiguity has been isolated. Contact terms can change local response, domain failure can make formal commutators undefined, and noncommuting limits can invalidate a small numerical residual.

Required background. Response kernels and stress-tensor insertions supply the Euclidean, retarded, and modular orderings.

Helpful background. Nonlocal modular generators supply resolvent and truncation controls.

Contact terms are part of the distribution

Ward identities hold as distributional equations; Osborn and Petkou 1994, §§2–3 gives a detailed stress-tensor example. For a conserved current and a local insertion,

\partial_\mu \langle T\{j^\mu(x)\mathcal O(y)\}\rangle =\delta^{(d)}(x-y) \langle\delta_j\mathcal O(y)\rangle +\cdots,

where the omitted terms include other insertions and possible anomalies or Schwinger terms. At separated points the right-hand side vanishes, but an integrated response can be determined precisely by the contact.

For stress-tensor response, contacts arise from:

varying the operator as well as the action;
derivatives acting on time ordering;
equal-time commutators;
variation of the metric, measure, normals, or entangling defect;
local source and boundary counterterms.

A separated-point correlator plus an arbitrary subtraction constant is not a complete distribution. Symmetry, renormalization conditions, and a declared scheme fix the local terms.

The structural map places Contact Terms, Operator Domains, and Perturbative Error Budgets along the perturbative sequence from normalized state or shape variations to information metrics and modular transport.

A normalized family yields the fixed-reference first law and a positive quadratic response, which branches into state susceptibility, shape kernels, metric choices, and modular holonomy. — At fixed comparison algebra, normalization removes the linear term in relative entropy. The second response supports several inequivalent constructions: state and shape tangents, stress-tensor kernels, monotone metrics, and zero-mode-projected transport. The diagram is schematic and not to scale.

Common operator domains

Let $K$ be an unbounded modular generator and $O$ an unbounded smeared field. The commutator $[K,O]\xi$ requires at least

\xi\in\mathcal D(KO)\cap\mathcal D(OK).

Nested commutators require a smaller common domain, and differentiating modular flow requires invariance of that domain under $e^{-isK}$ . A practical choice is a dense set of analytic vectors or an energy-bounded core stable under all smeared operators in the calculation.

Equality of matrix elements on a few states does not prove equality of unbounded operators. State whether the result holds:

as a bounded-operator identity;
as equality on a named core;
as a closable quadratic form;
or only after smearing inside correlation functions.

The weakest sufficient formulation is often the most honest one.

Boundary and regulator bookkeeping

In a region problem, the regulator has support: a lattice cut, split collar, brick wall, replica cone, or mode boundary. State whether it moves with the entangling surface. The regulated response can be decomposed schematically as

R_\epsilon =R_{\rm bulk,\epsilon} +R_{\partial,\epsilon} +R_{\rm contact,\epsilon} +R_{\rm ct,\epsilon}.

Only the sum satisfying the Ward identity should be extrapolated. Divergences may cancel among the terms. Taking $\epsilon\to0$ separately can erase the cancellation or leave a scheme-dependent finite remainder.

When a quantity is claimed universal, identify the invariant part: a logarithmic coefficient, a derivative that removes local polynomials, a relative difference at matched regulator, or a renormalized value fixed by an explicit condition.

Noncommuting limits

Modular response often involves several limits:

\epsilon\to0,\qquad L\to\infty,\qquad T_{\rm mod}\to\infty,\qquad n\to1,\qquad \lambda\to0.

They need not commute. Examples include a spectral gap closing before the perturbative derivative, a replica continuation developing a contact term at $n=1$ , or a finite modular-time window smoothing a continuum spectrum.

A convergence study varies at least the two most consequential controls independently. If

\lim_{\epsilon\to0}\lim_{\lambda\to0}R_{\epsilon,\lambda} \ne \lim_{\lambda\to0}\lim_{\epsilon\to0}R_{\epsilon,\lambda},

the claimed derivative must state the chosen order and its physical meaning.

Perturbative error budget

For an observable reported through order $N$ , organize the uncertainty as

R_{\rm reported} =R^{(0)}+\cdots+\lambda^N R^{(N)} +\delta_{\rm trunc} +\delta_{\rm reg} +\delta_{\rm domain} +\delta_{\rm contact} +\delta_{\rm num}.

These terms have different status:

$\delta_{\rm trunc}$ is bounded by the next order, a resolvent estimate, or variation of fit order;
$\delta_{\rm reg}$ comes from cutoff extrapolation and competing regulator checks;
$\delta_{\rm domain}$ records excluded high-energy or nonsmeared directions;
$\delta_{\rm contact}$ is scheme uncertainty after allowed local counterterms;
$\delta_{\rm num}$ includes quadrature, conditioning, finite-window, and analytic-continuation error.

Adding all five in quadrature is justified only when they are independent statistical uncertainties. Theoretical bounds and scheme ranges are usually better reported separately.

Deliberate failure tests

A strong calculation includes at least one case designed to fail:

omit the connected subtraction and confirm that normalization produces a false linear term;
reverse the modular KMS shift and show failure in a two-level model;
drop a Ward contact and show violation of charge conservation;
approach a zero eigenvalue and expose the divergent logarithmic kernel;
fit beyond the perturbative window and show loss of order-by-order stability.

These tests identify which assumption protects each conclusion.

Final review checklist

Before interpreting a coefficient, answer:

What algebra, state family, region, and reference are fixed?
Which tangent, support, smearing, and common operator domain are used?
Which Euclidean, Lorentzian, or modular ordering defines the kernel?
Which local, boundary, and contact terms complete the distribution?
Which metric or divergence fixes the normalization?
Which limits are taken, and in what order?
What controls the perturbative remainder?
Which part is regulator independent and which part is scheme dependent?

If any answer is missing, narrow the claim to the regulated formula actually established.

Common pitfalls

Treating contact terms as removable nuisances. They can enforce the symmetry and determine integrated response. Remove only terms allowed by a stated renormalization condition.

Using formal commutators as operator identities. Name a common invariant core or downgrade the statement to a quadratic form or correlator.

Combining unlike uncertainties into one error bar. Keep truncation, regulator, scheme, and numerical errors separately interpretable.

Before interpreting this response coefficient, use the validity map to check normalization, tangent domains, contact and regulator terms, metric choice, and analyticity independently.

A reported susceptibility must pass separate checks for normalized families, common tangent domains, contact and regulator terms, and an explicit metric and analytic strip before receiving a physical interpretation. — Normalization, domain, contact-term, metric, and analyticity checks are independent. A failed gate narrows or withdraws the physical claim; it is not a change of notation. The decision map is schematic and not to scale.

References

Osborn, Hugh, and Andreas Petkou. “Implications of Conformal Invariance in Field Theories for General Dimensions.” Annals of Physics 231 (1994): 311–362. DOI; arXiv.