Modular Response Kernels and Stress-Tensor Insertions

Euclidean, retarded, and modular-logarithm kernels differentiate different objects. They can be related only after the state preparation, ordering, analytic boundary value, source normalization, and contact prescription are matched. Stress-tensor response is especially unforgiving because the operator itself varies with the metric.

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

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

The chapter overview introduces the differentiable-family convention and the independent validity gates. Here, contour choice and Ward contacts are separate gates.

Three kernels, three questions

For a Euclidean action

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

normalized path-integral differentiation gives

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

The second term includes explicit variations of the operator, measure, metric, normal vectors, and counterterms.

For the Lorentzian convention

$$H(t)=H_0+\lambda f(t)B(t),$$

switched on from the past, linear response is

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

with

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

Kubo 1957, pp. 570–576 derives this causal response. Reversing the sign of the Hamiltonian source reverses the response sign.

For a faithful regulated density operator $\rho$ and normalized tangent $X=\delta\rho$, the logarithm instead has the resolvent kernel

$$\delta K =-\int_0^\infty dt\, (\rho+t)^{-1}X(\rho+t)^{-1}.$$

If $\rho=\sum_np_n\lvert n\rangle\langle n\rvert$, then

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

with diagonal value $-X_{nn}/p_n$. This divided difference fixes the noncommutative ordering.

What each response kernel differentiates.

Route	Input	Ordering or kernel	Support statement	Local terms
Euclidean	Prepared state or partition function	Euclidean time ordering and connected subtraction	No Lorentzian causal claim before continuation	Metric, measure, and operator variations
Retarded	Real-time Hamiltonian source	Minus i times the future-supported commutator	Vanishing before the source and outside causal propagation	Equal-time and seagull contacts
Modular logarithm	Density-operator or normal-state tangent	Resolvent divided difference, or an equivalent modular-time distribution	Modular support; not physical time in a generic region	Normalization zero mode and source-preparation contacts

A stress-tensor source in a CFT ball

Let $B_R$ be the radius-$R$ ball at $t=0$ in a CFT vacuum. The local vacuum modular Hamiltonian is

$$K_{B_R} =2\pi\int_{\lvert\mathbf x\rvert<R} d^{d-1}x\, \frac{R^2-\lvert\mathbf x\rvert^2}{2R} T_{00}(0,\mathbf x)+c.$$

Casini, Huerta, and Myers 2011, §2.1, eqs. (20)–(22) derive this generator by conformally mapping the Rindler wedge to the ball. In continuum QFT it defines modular flow on the local algebra; the following density matrix is only a type-I regulator.

Choose a real smooth tensor $h_{\mu\nu}$ compactly supported a distance $\delta>0$ from the entangling surface, and define the Hermitian smeared insertion

$$B_h=\frac12\int_{B_R}d^{d-1}x\, h_{\mu\nu}(\mathbf x)T^{\mu\nu}(0,\mathbf x).$$

At finite cutoff this is an operator. For the explicit calculation below, also impose a finite-dimensional spectral regulator; then $K_{B_R}+\lambda B_h$ is self-adjoint and its exponential is trace class. An infinite-dimensional type-I variant instead requires a declared $\lvert\lambda\rvert$ window in which $B_h$ is relatively form-bounded with respect to $K_{B_R}$, the form sum is self-adjoint and bounded below, and $Z_\lambda=\operatorname{Tr}e^{-(K_{B_R}+\lambda B_h)}$ is finite. In the continuum, $B_h$ is understood as a quadratic form on a common energy-bounded core; if sharp-time smearing is not available in the theory, replace it by a smooth time test function before removing the cutoff.

At fixed regulator, consider the modular Gibbs family

$$\rho_\lambda =\frac{e^{-(K_{B_R}+\lambda B_h)}} {\operatorname{Tr}e^{-(K_{B_R}+\lambda B_h)}}, \qquad \rho_0=\sigma.$$

Define $B_h^c=B_h-\langle B_h\rangle_\sigma\mathbf1$. Euclidean/Duhamel differentiation gives

$$X=\dot\rho_0 =-\int_0^1du\, \sigma^{1-u}B_h^c\sigma^u.$$

In a modular-energy eigenbasis $K_{B_R}\lvert n\rangle=k_n\lvert n\rangle$, with $p_n=e^{-k_n}/Z$,

$$X_{mn} =-\frac{p_m-p_n}{\log p_m-\log p_n} (B_h^c)_{mn}.$$

Applying the modular-logarithm divided difference cancels this logarithmic mean mode by mode:

$$(\delta K_E)_{mn}=(B_h^c)_{mn}.$$

Thus

$$\boxed{\delta K_E=B_h-\langle B_h\rangle_\sigma\mathbf1}$$

in every finite faithful regulator. This is an explicit modular response kernel for the smeared stress-tensor perturbation, including its normalization zero mode.

The same Euclidean preparation can be written as a real modular-time integral, but the source must describe the same family. The modular-Gibbs deformation is a uniform Euclidean-angle source, not an insertion at one fixed angle. Define

$$B_{h,E}(\tau) =e^{\tau K_{B_R}/(2\pi)}B_he^{-\tau K_{B_R}/(2\pi)}.$$

The interaction-picture identity is

$$e^{-(K_{B_R}+\lambda B_h)} =e^{-K_{B_R}}\, \mathcal T_\tau\exp\!\left[ -\frac{\lambda}{2\pi} \int_0^{2\pi}d\tau\,B_{h,E}(\tau) \right].$$

Thus, in a Euclidean path-integral convention with weight $e^{-I_E+\int jB_h}$, the matching source density is $j_\lambda(\tau)=-\lambda/(2\pi)$. Now define real modular rapidity by

$$O(s)=e^{isK_{B_R}/(2\pi)}Oe^{-isK_{B_R}/(2\pi)}.$$

For one source element at Euclidean modular angle $\tau$, the first-order kernel is

$$f_{(1)}(s+i\tau) =\frac{1}{4\sinh^2[(s+i\tau)/2]}, \qquad 0<\tau<2\pi.$$

Balakrishnan and Parrikar 2020, §2.2, eqs. (25)–(28) deform the Euclidean contour to this manifestly real modular flow for Rindler path-integral states. Applied to the uniform source above, their first-order formula is

$$\delta K_L =-\frac{1}{2\pi}\int_0^{2\pi}d\tau \int_{-\infty}^{\infty}ds\, f_{(1)}(s+i\tau)B_h(s) -\langle B_h\rangle_\sigma\mathbf1.$$

The $\tau$ integral must be treated as a distribution, not as ordinary endpoint quadrature. Let $\mathcal C_\eta$ follow $0<\tau<2\pi$ and indent the endpoint poles with the same $i0$ boundary prescription used in the shifted modular contour. After smearing in $s$,

$$-\frac{1}{2\pi}\lim_{\eta\downarrow0} \int_{\mathcal C_\eta}d\tau\, f_{(1)}(s+i\tau) =\delta(s).$$

This identity returns the nonconstant term $B_h$ with the displayed source sign. The last term in $\delta K_L$ is the derivative of the normalization constant. Hence $\delta K_L=B_h^c$, in agreement with the Duhamel and resolvent calculation. A fixed-$\tau$ insertion instead defines a different state family and generally gives a nonlocal modular-time integral. The conformal map used above transfers the Rindler construction to the ball only when the source, stress tensor, conformal factors, and contacts are transformed together.

For a spectral implementation with retained indices $m,n\leq N$, choose and report a positive scale $B_*$ with the same units as $B_h$, held fixed as $N$ changes, and use

$$\epsilon_{\rm route} =\max_{m,n\leq N} \frac{\left\lvert (\delta K_E-\delta K_L)_{mn} \right\rvert} {B_*+\lvert(B_h^c)_{mn}\rvert}.$$

When the uniform-source contact identity is imposed analytically, exact divided-difference arithmetic gives $\epsilon_{\rm route}=0$ at every finite regulator. Floating-point work should report this residual and repeat at larger $N$. The continuum statement is weaker: it requires both regulated forms to converge on the same smeared finite-energy core. No universal convergence rate follows from conformal symmetry alone. The control data are $\lvert\lambda\rvert$, the smearing scale, $\delta/R$, the UV cutoff, the spectral cutoff, and the route residual.

Physical-time continuation is an extra step

Real modular rapidity is not ordinary Lorentzian time for a generic region. For the vacuum ball it is geometric inside the causal diamond, but converting a Euclidean stress correlator to a physical retarded response still requires a choice of boundary value. In frequency space, separate local contact polynomials, continue the nonlocal spectral function to the retarded side, and then restore the Ward-fixed contacts. A proposed retarded kernel must independently satisfy future support.

The modular-log benchmark above does not claim that $f_{(1)}$ is a retarded step function. It compares Euclidean state preparation with a Lorentzian-operator representation of the same modular derivative.

Adversarial test: improvement and contacts

When a scalar operator $L$ of dimension $d-2$ is allowed, a conserved stress tensor may be shifted by

$$T'_{\mu\nu} =T_{\mu\nu} +c\left(\partial_\mu\partial_\nu -\eta_{\mu\nu}\Box\right)L.$$

This is the standard improvement ambiguity; Callan, Coleman, and Jackiw 1970, pp. 42–50 constructs the improved tensor.

To test that ambiguity without introducing a sharp-time distribution, replace the equal-time insertion used in the ball benchmark by the smooth spacetime smearing

$$B_h^{\rm spacetime} =\frac12\int d^dx\,h_{\mu\nu}(x)T^{\mu\nu}(x),$$

where $h_{\mu\nu}$ is smooth and compactly supported. Two integrations by parts then give

$$\left(B_h^{\rm spacetime}\right)' -B_h^{\rm spacetime} =\frac{c}{2}\int d^dx\,L(x) \left[ \partial_\mu\partial_\nu h^{\mu\nu} -\Box h^\mu{}_{\mu} \right].$$

Therefore a generic stress-response kernel changes. It is invariant under this shift only after fixing the stress-tensor scheme and source action, or for a smearing that annihilates the displayed differential combination, such as a compact transverse-traceless test tensor. Calling the unspecialized kernel “improvement independent” fails this direct test.

Now impose the Euclidean convention

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

For a compactly supported infinitesimal diffeomorphism, $\delta g_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}$, invariance of a correlator $\langle\mathcal X\rangle$ requires

$$-\frac12\int d^dx\sqrt g\, \delta g_{\mu\nu}(x) \langle T^{\mu\nu}(x)\mathcal X\rangle_c +\langle\delta_\xi\mathcal X\rangle=0.$$

Osborn and Petkou 1994, §§2 and 6 derive the stress-tensor Ward identities and their contact constraints. If one keeps only separated-point conservation and drops delta-function contacts, the first term is left equal to $-\langle\delta_\xi\mathcal X\rangle$ rather than zero. Likewise, the Euclidean two-stress kernel must include $\delta T^{\mu\nu}/\delta g_{\rho\sigma}$ contacts for symmetry under interchange of the two metric variations. Dropping them can break both the diffeomorphism Ward identity and kernel symmetry.

The strongest surviving claim is scheme-aware: Euclidean, modular, and retarded descriptions agree on their common nonlocal spectral data after matching source signs and ordering, while improvement, seagull, boundary, and anomaly contacts must be supplied separately. Separated-point agreement alone does not establish equality of full response kernels.

Common pitfalls

Using a retarded label for a modular-time distribution. Causal time and modular parameter coincide only with an additional geometric or thermal identification.

Continuing only the nonlocal correlator. Local polynomials and equal-time contacts affect Ward identities and sum rules.

Changing the stress tensor without changing the source scheme. An improvement is a change of background coupling unless the smearing makes it vanish.

Exercises

1. Derive the Duhamel tangent for $\rho_\lambda\propto e^{-(K+\lambda B)}$ and verify $\delta K=B-\langle B\rangle$ in every matrix element.

Solution

The derivative of an operator exponential is

$$\left.\frac{d}{d\lambda}e^{-(K+\lambda B)}\right|_0 =-\int_0^1du\,e^{-(1-u)K}Be^{-uK}.$$

Differentiating the normalization replaces $B$ by $B^c=B-\langle B\rangle$. Dividing by $Z$ gives

$$X=-\int_0^1du\,\sigma^{1-u}B^c\sigma^u.$$

In the eigenbasis of $\sigma$,

$$X_{mn} =-\frac{p_m-p_n}{\log p_m-\log p_n}B^c_{mn}.$$

Multiplication by the logarithm divided difference yields $(\delta K)_{mn}=B^c_{mn}$, including the diagonal limit.

2. Show that the improvement contribution vanishes for a compact transverse-traceless source.

Solution

If $\partial_\mu h^{\mu\nu}=0$ and $h^\mu{}_{\mu}=0$, then

$$\partial_\mu\partial_\nu h^{\mu\nu} -\Box h^\mu{}_{\mu}=0.$$

Compact support removes the boundary term in the two integrations by parts. Hence $\left(B_h^{\rm spacetime}\right)'-B_h^{\rm spacetime}=0$. For a generic source either condition can fail, so the cancellation is not universal.

3. Why is the contact term in a two-stress response required for kernel symmetry?

Solution

Let $W[g]=-\log Z[g]$. The full response is a second functional derivative,

$$\frac{\delta^2W}{\delta g_{\mu\nu}(x)\delta g_{\rho\sigma}(y)}.$$

Commuting the two derivatives makes this expression symmetric under exchanging $(x,\mu\nu)$ and $(y,\rho\sigma)$. Differentiating the first stress expectation produces both a connected product $\langle T^{\mu\nu}(x)T^{\rho\sigma}(y)\rangle_c$ and the local term $\langle\delta T^{\mu\nu}(x)/\delta g_{\rho\sigma}(y)\rangle$. The separated product alone need not transform symmetrically because the definition of $T$ and the measure also vary. The local term restores the symmetry required by the commuting functional derivatives and the Ward identity.

References

• Balakrishnan, Srivatsan, and Onkar Parrikar. “Modular Hamiltonians for Euclidean Path Integral States.” arXiv preprint arXiv:2002.00018 (2020). arXiv.
• Callan, Curtis G., Sidney Coleman, and Roman Jackiw. “A New Improved Energy-Momentum Tensor.” Annals of Physics 59, no. 1 (1970): 42–73. DOI.
• Casini, Horacio, Marina Huerta, and Robert C. Myers. “Towards a Derivation of Holographic Entanglement Entropy.” Journal of High Energy Physics 2011, no. 5 (2011): 036. DOI; arXiv.
• Kubo, Ryogo. “Statistical-Mechanical Theory of Irreversible Processes. I: General Theory and Simple Applications to Magnetic and Conduction Problems.” Journal of the Physical Society of Japan 12, no. 6 (1957): 570–586. DOI.
• Osborn, Hugh, and Andreas Petkou. “Implications of Conformal Invariance in Field Theories for General Dimensions.” Annals of Physics 231, no. 2 (1994): 311–362. DOI.