Modular Response Kernels and Stress-Tensor Insertions

Modular response can be represented by Euclidean, retarded, or modular-time kernels, but these kernels answer different questions. Euclidean kernels differentiate a prepared state or partition function, retarded kernels describe causal real-time response, and modular kernels differentiate logarithms or relative modular operators. Stress-tensor Ward identities and contact terms are needed to translate among them.

Required background. Shape-deformation perturbation theory supplies the moving-region stress and displacement insertions.

Helpful background. The entanglement first law identifies the fixed-reference linear response.

Three response problems

Consider an observable $O$ and an operator $B$ coupled to a source. Three common derivatives are:

a derivative of a Euclidean path-integral state with respect to a Euclidean source;
a real-time response to a perturbation of the Hamiltonian;
a derivative of $K=-\log\rho$ or of relative modular flow.

Their kernels are related by analytic continuation only when the state, contour, ordering, and contact prescriptions are matched.

For a Euclidean action $I_\lambda=I_0+\lambda\int d^dx\,J(x)B(x)$ ,

\left.\frac{d}{d\lambda}\langle O\rangle_\lambda\right|_0 =-\int d^dx\,J(x) \langle O\,B(x)\rangle_{\!c} +\langle\delta_\lambda O\rangle.

The last term includes explicit operator, metric, and measure variations. If $B=T_{\mu\nu}$ arises from a metric source, varying the stress tensor itself generates contact terms.

For a Lorentzian perturbation $H(t)=H_0+\lambda f(t)B(t)$ switched on from the past, the linear-response construction of Kubo 1957, pp. 570–576 gives

\delta\langle O(t)\rangle =\lambda\int_{-\infty}^{t}dt'\, G^R_{OB}(t,t')f(t'),

with

G^R_{OB}(t,t') =-i\theta(t-t')\langle[O(t),B(t')]\rangle.

This kernel has causal support. A Euclidean time-ordered correlator does not acquire that support until the continuation and boundary prescription are specified.

The structural map places Modular Response Kernels and Stress-Tensor Insertions 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.

Modular logarithm kernel

For a faithful regulated reference $\rho$ and tangent $X=\delta\rho$ ,

\delta K =-\int_0^\infty d\beta\; (\rho+\beta)^{-1}X(\rho+\beta)^{-1}.

The resolvent expression can be transformed into a modular-frequency or modular-time integral. In the eigenbasis of $\rho$ , the kernel is the divided difference

(\delta K)_{mn} =-\frac{\log p_m-\log p_n}{p_m-p_n}X_{mn}.

This formula fixes the noncommutative ordering. A modular-time representation must reproduce it mode by mode, including the coincident limit $-X_{nn}/p_n$ and the commuting zero mode.

For a response observable $O$ one then encounters integrals schematically of the form

\delta\langle O\rangle =\int_{-\infty}^{\infty}ds\; g(s)\, \langle O\,\sigma_s(X_{\rm prep})\rangle_c +\text{contact terms}.

The distribution $g(s)$ depends on whether one is differentiating a logarithm, relative entropy, or a cocycle. It is not interchangeable with a retarded step function.

Stress-tensor sources and Ward identities

A background-metric variation is normalized by

\delta I_E =\frac12\int d^dx\,\sqrt g\; T^{\mu\nu}\delta g_{\mu\nu}

for the stated Euclidean stress-tensor convention. Diffeomorphism invariance yields a Ward identity whose separated-point part is $\nabla_\mu\langle T^{\mu\nu}(x)\mathcal O_1\cdots\rangle=0$ , but derivatives acting on the time ordering and on transformed insertions produce delta-function contacts.

For shape response, the metric perturbation generated by a diffeomorphism can be moved by the Ward identity to the entangling defect or causal horizon. The result includes:

stress flux through future and past null boundaries;
displacement-operator insertions on the entangling surface;
equal-time commutators when the contour crosses an insertion;
variations of normals, measure, and local counterterms.

Discarding the bulk pure-gauge metric perturbation before retaining its boundary and contact terms incorrectly sets the shape response to zero.

Analytic continuation and ordering

To continue a Euclidean kernel to a retarded one, specify the operator ordering and frequencies. For a thermal or modular KMS state, boundary values on opposite sides of the strip encode different orderings. A spectral representation gives the clean procedure:

construct the Euclidean or modular spectral density with its contact polynomial separated;
continue the frequency to $\omega+i0$ for the retarded boundary value;
restore local contact terms fixed by Ward identities;
verify causal support and the KMS relation independently.

Contact polynomials are invisible to separated-point spectral densities but affect local response and sum rules. They must be fixed by the renormalization scheme and symmetry, not guessed from the continuation.

Support and error checks

A trustworthy response kernel reports:

the source support and switching prescription;
whether the comparison algebra is fixed or moving;
Euclidean, Wightman, retarded, or modular ordering;
the stress-tensor and Fourier-sign conventions;
contact and boundary terms;
the regulator and the domain of unbounded insertions;
numerical quadrature, finite-window, and continuation error if computed.

Causal support is an especially useful cross-check: a proposed retarded kernel that responds outside the future of the source has either the wrong ordering, a missed contact term, or a regulator artifact.

Common pitfalls

Calling every integrated two-point function a susceptibility. The contour and kernel determine which quadratic form is being computed.

Continuing only the nonlocal part. Local contact terms can survive continuation and are required by Ward identities.

Replacing modular order by real-time order. Modular parameter is not physical time in a generic region. Their kernels coincide only after an additional geometric or thermal identification.

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

Kubo, Ryogo. “Statistical-Mechanical Theory of Irreversible Processes. I.” Journal of the Physical Society of Japan 12 (1957): 570–586. DOI.