The Renormalized Stress Tensor: Conservation and Ambiguities

The expectation value of the stress tensor is not obtained by simply evaluating the two-point function at coincidence. Hadamard subtraction removes its universal singularity, a local correction enforces the stress-tensor Ward identity, and finite renormalization remains. In four dimensions, locality, covariance, conservation, scaling, and the usual continuity conditions reduce that freedom to a finite-dimensional family of conserved local curvature tensors. The state-dependent part is then unambiguous once the prescription is fixed.

Required background. Time-ordered products and the renormalized stress tensor supplies point splitting and Ward identities. Existence, deformation, and gluing of Hadamard states supplies the state domain on which coincidence limits exist.

Helpful background. Stress–energy response and background variation relates the tensor to relative Cauchy evolution. Local covariant Wick powers and operator products classifies finite local terms. Renormalized stress-tensor axioms and ambiguities develops the physical prescription. Conservation, local covariance, and the backreaction source explains the Bianchi constraint. Vacuum polarization and curved-space Casimir effects and mode-sum numerical renormalization show how the same local subtraction enters concrete calculations.

Point splitting and the conservation condition

For a real Klein–Gordon field $P\phi=(\Box_g+m^2+\xi R)\phi=0$ in a Hadamard state $\omega$, choose a local Hadamard parametrix $H_\ell(x,x')$ with length scale $\ell$. If $D_{ab}(x,x')$ is the bidifferential operator obtained from the classical stress tensor, the renormalized expectation has the form

$$\langle T_{ab}(x)\rangle_{\omega,\ell} =\lim_{x'\to x}D_{ab}(x,x') \bigl(\omega_2(x,x')-H_\ell(x,x')\bigr)+C_{ab}(x).$$

The difference in parentheses is smooth, so the limit exists. However, the parametrix is only a local bisolution modulo a smooth remainder. Applying the classical bidifferential expression alone can therefore leave a local divergence. The tensor $C_{ab}$ is chosen locally and covariantly so that

$$\nabla^a\langle T_{ab}\rangle_{\omega,\ell}=0.$$

This is a Ward identity, not a state-by-state adjustment: the same prescription works for every Hadamard state. Wald’s axiomatic analysis identifies conservation and local geometric dependence as decisive conditions Wald 1977, Theorems 1–2, pp. 3–11. In the locally covariant Wick-polynomial framework, the conservation correction and finite freedom arise from the same naturality and scaling constraints Hollands and Wald 2001, Theorems 5.1–5.2, pp. 312–321.

If $\omega$ and $\omega'$ are Hadamard, their two-point difference is smooth. Consequently

$$\langle T_{ab}\rangle_\omega- \langle T_{ab}\rangle_{\omega'} =\lim_{x'\to x}D_{ab}(x,x') \bigl(\omega_2-\omega'_2\bigr)(x,x')$$

contains neither the parametrix nor the purely geometric correction. This difference is the cleanest check that state dependence and renormalization freedom have not been mixed.

The four-dimensional ambiguity theorem

Assume a four-dimensional boundaryless spacetime, a scalar field, local covariance, conservation, the specified almost-homogeneous scaling behavior, and smooth or analytic dependence on the metric and parameters as in the classification theorem. Two admissible prescriptions then differ by

$$\Delta T_{ab} =\alpha_1m^4g_{ab}+\alpha_2m^2G_{ab} +\alpha_3I_{ab}+\alpha_4J_{ab},$$

where the $\alpha_i$ are dimensionless constants and

$$I_{ab}=\frac{2}{\sqrt{|g|}}\frac{\delta}{\delta g^{ab}} \int R^2\sqrt{|g|}\,\mathrm d^4x, \qquad J_{ab}=\frac{2}{\sqrt{|g|}}\frac{\delta}{\delta g^{ab}} \int R_{cd}R^{cd}\sqrt{|g|}\,\mathrm d^4x.$$

All four tensors are local, symmetric, and conserved; in four dimensions the variation of the Euler density removes a third independent curvature-squared tensor. The formula assumes the site’s curvature convention and a definite sign for metric variation; defining $I_{ab}$ and $J_{ab}$ variationally prevents a hidden sign translation. With boundaries, surface terms create additional ambiguities and this list is incomplete.

The directional conclusion is finite renormalization freedom, not uniqueness. Conversely, adding an arbitrary conserved state-dependent tensor is not allowed: it would violate locality in the background fields or the prescribed dependence on the state. Nor does conservation alone imply that a proposed subtraction is locally covariant.

FLRW application and an independent check

For the massless conformally coupled scalar on a spatially flat FLRW spacetime, the construction on renormalized stress-tensor axioms and ambiguities subtracts $H_\ell$, adds the conservation term, and separates the finite state-dependent part from the allowed curvature shifts. Homogeneity and isotropy force

$$\langle T^a{}_b\rangle=\operatorname{diag}(\rho,-p,-p,-p),$$

and the Ward identity becomes $\dot\rho+3H(\rho+p)=0$. This scalar equation is an independent check on every term. Each ambiguity tensor satisfies it separately by diffeomorphism invariance of its defining action. A numerical mode sum that violates the equation after regulator removal has not implemented a complete covariant prescription.

Failure boundary: changing the subtraction scale

Under $\ell\mapsto e^s\ell$, the Hadamard parametrix changes by a smooth local term. The renormalized tensor therefore shifts by a definite member of the curvature family above. If one changes $\ell$ in the matter calculation but holds the coefficients of the cosmological, Einstein–Hilbert, and curvature-squared terms fixed, the split between “matter” and “gravity” acquires spurious scale dependence. The total semiclassical equation is scale consistent only when the corresponding gravitational couplings run oppositely.

This failure need not appear as a nonzero divergence—the legitimate scale shift is conserved. Apparent nonconservation instead signals that the Ward-identity correction or a required boundary term was omitted. Thus “conserved” and “renormalization-scale consistent” are two independent checks.

Exercises

Show that a finite $R^2$ counterterm does not spoil conservation of the stress tensor it induces.

Solution

The functional $S[g]=\int R^2\sqrt{|g|}\,\mathrm d^4x$ is invariant under compactly supported diffeomorphisms. For $\delta g^{ab}=\nabla^aX^b+\nabla^bX^a$, its variation is proportional to $\int I_{ab}\nabla^aX^b$. Integrating by parts and using arbitrary compactly supported $X^b$ gives $\nabla^aI_{ab}=0$. No field equation is needed.

References

• 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.
• Moretti, Valter. “Comments on the Stress-Energy Tensor Operator in Curved Spacetime.” Communications in Mathematical Physics 232 (2003): 189–221. DOI; Open PDF.
• Wald, Robert M. “The Back Reaction Effect in Particle Creation in Curved Spacetime.” Communications in Mathematical Physics 54 (1977): 1–19. DOI.