Contact Terms, Operator Domains, and Perturbative Error Budgets

A modular-response coefficient is reliable only when four logically distinct obligations are met: the correlation kernel is completed as a distribution, every unbounded product has a common domain, regulator limits are taken in a declared order, and the omitted perturbative remainder is bounded. A small quadrature residual cannot compensate for a missing contact term or an undefined commutator.

Required background. Response kernels and stress-tensor insertions distinguish Euclidean, retarded, and modular orderings and show where explicit source variations enter.

Helpful background. Nonlocal modular generators supply resolvent, energy-window, and truncation controls for unbounded or nonlocal generators.

The chapter’s differentiable-family map shows where normalization, contact, and domain data enter before a derivative is interpreted. Its independent validity gates keep those obligations separate rather than compressing them into one undifferentiated error bar.

Contact terms complete the response distribution

At separated points, a current or stress-tensor Ward identity may look homogeneous. Differentiating a time-ordered product also differentiates step functions and transformed insertions, producing terms supported at coincidence. In schematic form,

$$\partial_\mu \left\langle T\left\{j^\mu(x)\prod_i\mathcal O_i(x_i)\right\}\right\rangle =\sum_i\delta^{(d)}(x-x_i) \left\langle T\left\{\mathcal O_1\cdots \delta_j\mathcal O_i\cdots\right\}\right\rangle +\text{anomaly or Schwinger terms},$$

with signs and factors of $i$ fixed by the Euclidean or Lorentzian convention. The original Ward–Takahashi derivation makes the coincidence contribution explicit Takahashi 1957, pp. 371–375. Stress-tensor identities additionally receive metric, measure, and operator variations Osborn and Petkou 1994, §6, eqs. (6.1)–(6.6).

For a moving entangling surface, the regulator itself has support: a split collar, brick wall, replica defect, lattice cut, or mode boundary may move with the surface. Bulk, boundary, contact, and counterterm pieces should therefore be combined before extrapolation:

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

Only the sum constrained by the Ward identity is a candidate continuum observable. Deformed-half-space modular calculations provide a concrete example in which the shape diffeomorphism, cutoff surface, and its boundary conditions must be tracked together Faulkner et al. 2016, §2.1 and appendix A, especially eqs. (99)–(101).

Unbounded products need a named common domain

For unbounded $K$ and $O$, the commutator applied to a vector requires

$$\xi\in\mathcal D(KO)\cap\mathcal D(OK),$$

meaning $O\xi\in\mathcal D(K)$ and $K\xi\in\mathcal D(O)$ in addition to the first actions being defined. Nested commutators require further intersections. Differentiating $e^{-isK}Oe^{isK}$ on a core also requires the core to be invariant under the relevant flow. Dense sets of analytic vectors are one standard route to such control Nelson 1959, pp. 572–579.

Always state the strength of the conclusion:

• a bounded-operator identity;
• equality on a specified invariant core;
• equality of closable quadratic forms;
• or a distributional identity only after smearing.

These are not interchangeable. In particular, agreement of a few matrix elements does not extend an unbounded identity to all Hilbert-space vectors.

A modular-circle second-response benchmark

Put a free scalar field in a finite box with spatial mode cutoff $\Lambda$ and isolate one oscillator mode. On its Fock space choose the dimensionless reference modular Hamiltonian

$$K_0=\kappa\left(N+\frac12\right), \qquad q=\frac{a+a^\dagger}{\sqrt{2\kappa}}, \qquad \sigma=Z_0^{-1}e^{-K_0},$$

with $\kappa>0$. Let $\tau\in[-1/2,1/2)$ be Euclidean modular time on the unit circle. A smooth real, reflection-symmetric periodic source $J(\tau)=J(-\tau)$ prepares a Hermitian Gaussian density operator by inserting $e^{\int_0^1Jq}$ in its cut Euclidean path integral. For $K[J]=-\log\rho_J$, the scalar normalization response is

$$W[J]:=\log\frac{Z[J]}{Z[0]} =\frac12\int_0^1d\tau\,d\tau'\, J(\tau)G_\kappa(\tau-\tau')J(\tau'),$$

where the periodic modular correlator is

$$G_\kappa(\tau) =\frac{\cosh[\kappa(1/2-\lvert\tau\rvert)]} {2\kappa\sinh(\kappa/2)}, \qquad -\frac12\leq\tau<\frac12.$$

Thus $G_\kappa$ is an explicit second functional derivative of the modular Hamiltonian’s normalization term. The benchmark tests that second-order modular response; it does not pretend that this scalar term is the full operator-valued $\delta^2K[J]$. Because the theory is Gaussian, third and higher derivatives of this scalar normalization response vanish at this fixed regulator.

On the finite-particle core, $q$ and $K_0$ have a common invariant domain. Distributionally on the circle,

$$D_\tau G_\kappa(\tau)=\delta_{\rm per}(\tau), \qquad D_\tau=-\partial_\tau^2+\kappa^2.$$

Away from $\tau=0$, the equation is homogeneous; the derivative jump $G_\kappa'(0^+)-G_\kappa'(0^-)=-1$ supplies the periodic delta function.

Point split with the periodic heat kernel

$$\delta_\epsilon(u) =\sum_{n\in\mathbb Z} e^{-2\pi^2\epsilon^2n^2}e^{2\pi inu}, \qquad G_{\kappa,\epsilon}=G_\kappa*\delta_\epsilon,$$

so $D_\tau G_{\kappa,\epsilon}=\delta_\epsilon$ exactly. Test two independent normalized smooth profiles,

$$f_1(\tau)=\sqrt2\cos(2\pi\tau), \qquad f_2(\tau)=\sqrt2\cos(4\pi\tau).$$

Define the smeared Ward matrix

$$C_{ab}(\epsilon) :=\int_0^1d\tau\,d\tau'\, f_a(\tau)[D_\tau G_{\kappa,\epsilon}](\tau-\tau')f_b(\tau').$$

Fourier orthogonality gives

$$C_{12}=C_{21}=0, \qquad C_{11}=e^{-2\pi^2\epsilon^2}, \qquad C_{22}=e^{-8\pi^2\epsilon^2}.$$

The zero-splitting target is $C_{ab}(0)=\delta_{ab}$. Define

$$R_W(\epsilon) :=\max_{a,b}\lvert C_{ab}(\epsilon)-\delta_{ab}\rvert.$$

$\epsilon$	$C_{11}$	$C_{22}$	$R_W$
$0.1000$	$0.820869$	$0.454041$	$0.545959$
$0.0500$	$0.951850$	$0.820869$	$0.179131$
$0.0250$	$0.987739$	$0.951850$	$0.048150$
$0.0125$	$0.996921$	$0.987739$	$0.012261$

The leading errors are profile dependent but controlled:

$$1-C_{11}=2\pi^2\epsilon^2+O(\epsilon^4), \qquad 1-C_{22}=8\pi^2\epsilon^2+O(\epsilon^4).$$

The two profiles therefore verify the predicted $O(\epsilon^2)$ convergence with the expected factor-of-four difference, rather than merely agreeing at one resolution. The control data are $\kappa$, $\epsilon$, the two fixed profiles, the spatial cutoff and mode embedding, and the common finite-particle core.

Adversarial contact and domain deletions

First delete the contact term and keep only the separated-point equation $D_\tau G_\kappa=0$ for $\tau\ne\tau'$. The resulting matrix is $\widetilde C_{ab}=0$, hence

$$\widetilde R_W =\max_{a,b}\lvert\widetilde C_{ab}-\delta_{ab}\rvert=1$$

at every resolution. Regulator refinement cannot repair the missing distributional term. This is a direct Ward-identity detector for the deliberate failure.

Now delete the common-domain restriction. In the oscillator number basis, take the Hilbert-space vector

$$\xi=\mathcal N\sum_{n=0}^\infty\frac{1}{n+1}\lvert n\rangle.$$

It is normalizable, but it is not in $\mathcal D(K_0)$ because $\sum_n n^2/(n+1)^2$ diverges. It is also not in the coordinate domain: the positive large-$n$ coefficients make $\lVert q\xi_N\rVert^2$ grow logarithmically for number-cutoff truncations $\xi_N$. Therefore $[K_0,q]\xi$ is undefined. A matrix-element calculation on $\xi_N$ that appears to converge before these graph norms are checked does not establish an operator identity.

An operational error budget

For a response through second order in a source amplitude $\lambda$, report

$$R(\lambda) =R^{(0)}+\lambda R^{(1)} +\frac{\lambda^2}{2}R^{(2)} +\mathcal R_3(\lambda)$$

together with separate controls:

Source	Control quantity	Acceptance statement
Contact/point splitting	$R_W(\epsilon)$ for at least two profiles	Tends to zero with the derived profile-dependent power
Operator domain	Graph norms on a named invariant core	Remain finite under the reported cutoff extrapolation
Spatial regulator	Matched spatial smearing under $\Lambda$ refinement	Same smeared observable and counterterm condition at every $\Lambda$
Numerical work	Quadrature and conditioning residuals	Smaller than the leading theoretical uncertainty
Perturbative remainder	$\lVert\mathcal R_3(\lambda)\rVert$ or a derivative bound	Scales as $O(\lvert\lambda\rvert^3)$ in the declared window

If $\sup_{\lvert u\rvert\le\lvert\lambda\rvert}\lVert R^{(3)}(u)\rVert\le M_3$, Taylor’s theorem gives $\lVert\mathcal R_3(\lambda)\rVert\le M_3\lvert\lambda\rvert^3/6$. Without such a bound, halving $\lambda$ should reduce a third-order-dominated residual by about eight; failure of that test narrows the perturbative window. In the Gaussian benchmark above, $M_3=0$ exactly.

The safe order for the benchmark is: take the response derivative at fixed $\Lambda,\kappa,\epsilon$ and fixed profiles; then send $\epsilon\to0$ on the modular circle; finally, for a full smeared-field extension, study $\Lambda\to\infty$ with matched spatial mode embeddings, smearings, and counterterms. The displayed single-mode identity is independent of $\Lambda$ once that mode is retained. A different order is a different claim and must be tested independently.

Common pitfalls

Treating a separated-point identity as the full distribution. Integrated response can be determined entirely by coincidence support. Verify a smeared Ward target.

Writing a formal commutator on every vector. Name a common invariant core, or weaken the result to a quadratic-form or smeared-correlator statement.

Combining unlike errors in quadrature. Regulator ranges, scheme choices, analytic bounds, and sampling errors have different meanings and should remain separately visible.

Exercises

1. Recover the oscillator contact term

Show distributionally on the unit circle that $(-\partial_\tau^2+\kappa^2)G_\kappa(\tau)=\delta_{\rm per}(\tau)$ by checking the homogeneous equation away from zero and integrating across a small interval around zero.

Solution

Away from zero, $G_\kappa''=\kappa^2G_\kappa$, so the expression vanishes. Integrate from $-\eta$ to $\eta$:

$$\int_{-\eta}^{\eta}(-G_\kappa''+\kappa^2G_\kappa)d\tau =-[G_\kappa'(\eta)-G_\kappa'(-\eta)] +\kappa^2\int_{-\eta}^{\eta}G_\kappa d\tau.$$

As $\eta\to0$, the integral term vanishes. Since $G_\kappa'(0^+)=-1/2$ and $G_\kappa'(0^-)=+1/2$, the jump term is $-[-1/2-1/2]=1$. Periodicity matches the values and first derivatives at $\tau=\pm1/2$, so there is no additional boundary contact. The distribution is therefore $\delta_{\rm per}(\tau)$.

2. Compute the two-profile Ward matrix

Evaluate $C_{11}$ and $C_{22}$ for the periodic heat kernel and the two cosine profiles. Extract their leading errors for $\epsilon\ll1$.

Solution

Convolution by $\delta_\epsilon$ multiplies the Fourier mode $e^{2\pi in\tau}$ by $e^{-2\pi^2\epsilon^2n^2}$. The profile $f_1$ contains only $n=\pm1$, while $f_2$ contains only $n=\pm2$. Orthonormality therefore gives

$$C_{11}=e^{-2\pi^2\epsilon^2}, \qquad C_{22}=e^{-8\pi^2\epsilon^2}, \qquad C_{12}=C_{21}=0.$$

Expanding $e^{-x}=1-x+O(x^2)$ gives

$$1-C_{11}=2\pi^2\epsilon^2+O(\epsilon^4), \qquad 1-C_{22}=8\pi^2\epsilon^2+O(\epsilon^4).$$

The higher-frequency profile has four times the leading point-splitting error.

3. Separate three failures

For the benchmark, state what happens to the Ward residual when (a) the contact is omitted, (b) the vector $\xi$ is used without a domain check, and (c) a nonzero cubic residual fails to decrease under $\lambda\mapsto\lambda/2$.

Solution

(a) The separated kernel gives $\widetilde C=0$, so $\widetilde R_W=1$ for every $\epsilon$; this is a contact failure. (b) The truncated vectors have diverging $K_0$ graph norm and logarithmically growing $q$ norm, so the limiting commutator is undefined; no Ward residual can promote it to an operator identity. (c) A cubic-dominated remainder should fall by $2^3=8$. If it does not, higher orders, regulator effects, or numerical errors dominate, and the claimed second-order window must be reduced. These diagnoses should not be combined into one scalar uncertainty.

References

• Faulkner, Thomas, Robert G. Leigh, Onkar Parrikar, and Huajia Wang. “Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition.” Journal of High Energy Physics 2016, no. 9 (2016): 038. DOI; Open PDF.
• Nelson, Edward. “Analytic Vectors.” Annals of Mathematics 70 (1959): 572–615. DOI.
• Osborn, Hugh, and Andreas Petkou. “Implications of Conformal Invariance in Field Theories for General Dimensions.” Annals of Physics 231 (1994): 311–362. DOI; Open PDF.
• Takahashi, Yasushi. “On the Generalized Ward Identity.” Il Nuovo Cimento 6 (1957): 371–375. DOI.