Stress–Energy Response and Background Variation

For a sufficiently regular theory, the infinitesimal response of relative Cauchy evolution to a compact metric perturbation is generated by stress–energy. This statement is representation-dependent and weak: it concerns derivatives of represented observables as quadratic forms on a common dense domain. The finite algebraic automorphism exists under weaker assumptions than this derivative.

Required background. Natural Transformations, Fields, and Subtheory Embeddings fixes local covariance of composite fields. Time-Slice Axiom and Relative Cauchy Evolution defines the automorphism being differentiated.

Helpful background. Renormalized Stress Tensor: Axioms and Ambiguities gives the curved-spacetime renormalization conditions. Spacetime Currents, Stress Tensors, and Charge Algebras reviews Ward identities. Shape Deformations and Perturbation, Response Kernels and the Stress Tensor, Gauge-Invariant Response Kernels, and First Laws and Physical-Process Variations show related response constructions with different domains.

The infinitesimal response theorem

Use the relative Cauchy evolution convention of the preceding page. Let $h$ be a compactly supported smooth symmetric tensor and assume $g+sh$ is an admissible globally hyperbolic metric for $|s|$ small. For the free Klein–Gordon theory, choose a quasifree Hadamard state $\omega$, its GNS representation $\pi_\omega$, and a common invariant dense domain $\mathscr V_\omega$ on which the differentiated fields and stress tensor are defined. Under the differentiability hypotheses of Brunetti, Fredenhagen, and Verch,

$$\left.\frac{d}{ds}\pi_\omega\!\left( \operatorname{rce}_M[sh](A)\right)\right|_{s=0} =-\frac{i}{2}\,[T_M(h),\pi_\omega(A)]$$

as a quadratic-form identity on $\mathscr V_\omega$ for the stated regular subalgebra, where

$$T_M(h)=\int_M T_{\mu\nu}h^{\mu\nu}\,d\mathrm{vol}_M.$$

The sign and factor follow the page’s definition of $\operatorname{rce}$ and the convention for metric variation; changing either convention changes the displayed form. The general divergence statement and the free-field commutator formula are Brunetti, Fredenhagen, and Verch 2003, Theorems 4.2–4.3, pp. 23–29.

This gives the concrete calculation used in Relative Cauchy Evolution and Background Response: differentiate the free-scalar automorphism and recover the commutator with the smeared renormalized stress tensor. It does not say that every interacting LCQFT has a stress tensor implementing a norm derivative.

Construction mechanism

At the classical level, vary the Klein–Gordon operator and the advanced/retarded Green operators. Differentiating the Møller scattering map gives a symplectic derivation whose bilinear form is the metric variation of the action. After quantization, that variation becomes insertion of $T_{\mu\nu}$. The Hadamard condition makes the renormalized local product meaningful, while the common domain permits the commutator to be read as a quadratic form.

Diffeomorphism covariance supplies a separate check. If $h=\mathcal L_Xg$ for compactly supported $X$, the background change is induced by an admissible infinitesimal relabeling. Integration by parts gives

$$T_M(\mathcal L_Xg)=-2\int_M X^\nu\nabla^\mu T_{\mu\nu}\,d\mathrm{vol}_M.$$

Consequently invariance under such changes requires $\nabla^\mu T_{\mu\nu}=0$ in the relevant weak sense. This Ward-identity check is independent of the explicit Green-operator calculation.

Renormalization and failure boundaries

Permitted curvature counterterms in the renormalized stress tensor are local, covariant, symmetric, and conserved. They can change $T_{\mu\nu}$ without changing the covariance principle; c-number terms also drop out of the commutator, though they matter for expectation values and semiclassical gravity. An arbitrarily chosen nonconserved curvature tensor is not an admissible ambiguity. It would make $T(\mathcal L_Xg)$ nonzero for a pure background diffeomorphism and therefore cannot generate covariant metric response.

Other boundaries are equally important. A finite $\operatorname{rce}[h]$ need not be differentiable in a chosen topology. A derivative in one representation is not automatically an inner derivation of the abstract C*-algebra. Finally, a metric response theorem does not by itself determine electromagnetic, source, or boundary responses; those require a source category carrying the corresponding background.

What the first derivative does not determine

Knowing the derivation for every compact $h$ fixes the tangent response at the original metric. It does not by itself integrate to the finite automorphism for a large perturbation: domains may vary, differentiability may fail at intermediate metrics, and path ordering matters when the infinitesimal generators do not commute. A second variation additionally contains stress-tensor response and contact terms; it is not obtained merely by squaring $T(h)$.

The factor $1/2$ supplies a dimensional and tensorial check. A metric variation couples through $\frac12\int T_{\mu\nu}\,\delta g^{\mu\nu}d\mathrm{vol}$, so $T(h)$ has the dimensions of an action in units with $\hbar=1$, and its commutator is a dimensionless derivation. If a proposed formula couples an antisymmetric part of $h$ or depends on coordinates outside its support, it cannot be the metric derivative.

Finally, expectation values and automorphisms answer different questions. A c-number curvature ambiguity changes $\omega(T_{\mu\nu})$ but commutes with every represented observable, so rce cannot determine that part of the renormalized expectation value. Semiclassical backreaction must retain the ambiguity even though the algebraic response derivation does not see it.

Exercise

Assume the response formula and let $h=\mathcal L_Xg$ with compactly supported $X$. Show that conservation of $T_{\mu\nu}$ makes the infinitesimal response vanish.

Solution

Symmetry of $T$ gives $T_{\mu\nu}\mathcal L_Xg^{\mu\nu}=2T_{\mu\nu}\nabla^\mu X^\nu$ up to the sign fixed by varying the inverse metric. Integrating by parts produces a term proportional to $-2\int X^\nu\nabla^\mu T_{\mu\nu}$; the boundary term vanishes because $X$ is compactly supported. Conservation makes $T(h)$ vanish, so its commutator and the response vanish.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Physics.” Communications in Mathematical Physics 237 (2003): 31–68. DOI; Open PDF.
• Hollands, Stefan, and Robert M. Wald. “Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime.” Communications in Mathematical Physics 223 (2001): 289–326. DOI; Open PDF.