Renormalized Stress–Energy Tensor

The renormalized stress–energy tensor is the local observable that connects quantum fields to geometry, but its definition is more delicate than the Wick square. Differential point splitting removes the state-independent Hadamard singularity; a further local term restores exact conservation because a parametrix solves the field equation only up to a smooth remainder. Conservation, covariance, state dependence, and finite gravitational counterterms must be kept visible as separate ingredients.

Required background. Stress–Energy and Conservation provides the Hilbert definition and the classical conservation identity. Curvature supplies the curvature tensors in the improved stress tensor and its finite ambiguities.

Helpful background. Flat-Spacetime Quantum Field Theory defines vacuum normalization. Partial Differential Equations and Green Functions supplies bisolutions and parametrices. Causality and Global Structure develops the local geometric setting. Quantum Fields in Curved Spacetime provides the Hadamard and local-covariance framework.

From the classical tensor to a bidifferential operator

For a real scalar field we fix

$$S[\phi,g]=-\frac12\int_M d^4x\,\sqrt{-g} \left(\nabla_a\phi\nabla^a\phi+(m^2+\xi R)\phi^2\right), \qquad P=\Box-m^2-\xi R,$$

and define $T_{ab}=-(2/\sqrt{-g})\,\delta S/\delta g^{ab}$. Varying the action gives the useful unexpanded form

$$\begin{aligned} T_{ab}={}&\nabla_a\phi\nabla_b\phi -\frac12g_{ab}\bigl((\nabla\phi)^2+m^2\phi^2\bigr)\\ &+\xi\bigl(G_{ab}\phi^2+g_{ab}\Box\phi^2 -\nabla_a\nabla_b\phi^2\bigr). \end{aligned}$$

The derivatives in the second line act on $\phi^2$, including both factors. Expanding that line reproduces the improved tensor and the off-shell identity $\nabla^aT_{ab}=(P\phi)\nabla_b\phi$ in Décanini and Folacci 2008, §II.A, Eqs. (9) and (14), PDF. This checks the curvature-coupling sign and proves conservation on solutions.

Products such as $\nabla_a\Phi(x)\nabla_b\Phi(x)$ are singular, so one first replaces the classical quadratic expression by a bidifferential operator $D_{ab}(x,x')$. Use the symmetrized two-point function $W_\omega^{\mathrm s}=(W_\omega+W_\omega^T)/2$ and the same symmetrization of $H_\ell$. Put $w=W_\omega^{\mathrm s}-H_\ell^{\mathrm s}$, $F=[w]$, and $K_{ab}=[\nabla_a\nabla_{b'}w]$, with the primed index parallel-transported before coincidence. Then the classical point-split prescription can be written explicitly as

$$\begin{aligned} t^{(0)}_{ab} &=K_{ab}-\frac12g_{ab}(K^c{}_c+m^2F)\\ &\quad+\xi\bigl(G_{ab}F+g_{ab}\Box F-\nabla_a\nabla_bF\bigr). \end{aligned}$$

Here a derivative of $F(x)=w(x,x)$ differentiates both arguments of $w$; it is not $[\nabla_a w]$. This distinction is essential to the improvement terms. The expression is the coincidence form of $D_{ab}w$ below.

Given a Hadamard state $\omega$ and a declared four-dimensional parametrix $H_\ell$, the provisional expectation value is

$$t^{(0)}_{ab}(x) =\lim_{x'\to x}D_{ab}(x,x') \left[W_\omega^{\mathrm s}(x,x')-H_\ell^{\mathrm s}(x,x')\right].$$

Choose enough Hadamard terms that the subtraction has the derivatives required at coincidence. The result is real by symmetrization and the subtraction prescription is locally covariant because $D_{ab}$ and $H_\ell$ are constructed from the local geometry. Its state dependence resides entirely in the regular part of $W_\omega$.

Restoring conservation

The Hadamard parametrix is not an exact bisolution: $P_xH_\ell$ and $P_{x'}H_\ell$ are smooth but generally nonzero. Consequently, the classical identity used to prove conservation leaves a local remainder. Covariance and the transport equations imply

$$\nabla^a t^{(0)}_{ab}=\nabla_b Q,$$

for a state-independent local scalar $Q$ determined by the parametrix convention. Defining

$$\left\langle T_{ab}\right\rangle_{\omega,\ell} =t^{(0)}_{ab}-g_{ab}Q+C_{ab}$$

restores $\nabla^a\langle T_{ab}\rangle=0$ provided the finite ambiguity $C_{ab}$ is itself conserved. This equation defines the sign of $Q$ on this page; changing the convention for the parametrix remainder changes the named $Q$, not the conservation requirement.

The construction implements the stress-tensor axioms: local covariance, conservation, the correct commutator or response to metric variation, and agreement with the usual normal-ordered result in Minkowski spacetime after a normalization choice. Wald’s original analysis is in Wald 1977, pp. 1–19; the trace and uniqueness qualification must be read with Wald 1978, p. 1477, which corrects the overly strong original uniqueness statement.

Trace and state dependence

For $m=0$ and conformal coupling $\xi=1/6$, the classical on-shell trace vanishes. The renormalized tensor cannot in general preserve conservation, locality, covariance, and a vanishing trace simultaneously. Its state-independent geometric trace has the four-dimensional structure

$$\left\langle T^a{}_a\right\rangle =\frac{1}{(4\pi)^2} \left(c\,C_{abcd}C^{abcd}-a\,E_4+b\,\Box R\right),$$

with coefficients depending on field content and normalization. The $\Box R$ coefficient can be shifted by a finite $R^2$ counterterm, whereas the type-A and type-B anomaly coefficients are not removed that way. The next chapter derives these statements rather than treating the displayed structure as a substitute for the heat-kernel calculation.

For two Hadamard states evaluated with the same prescription,

$$\Delta\langle T_{ab}\rangle =\lim_{x'\to x}D_{ab}(W_\omega-W_{\omega'}).$$

All local subtraction and ambiguity terms cancel. This finite state difference is often more robust than either absolute value. It also provides an independent implementation check: compute the difference without subtracting two large separately renormalized mode sums whenever possible.

Finite freedom and checks

The conserved $C_{ab}$ is not an arbitrary tensor. In four dimensions it is a linear combination of metric variations of local gravitational terms such as $\int\sqrt{-g}$, $\int\sqrt{-g}R$, and curvature-squared actions, with mass factors supplying the required dimensions. Those terms correspond to finite redefinitions of the cosmological, Newton, and higher-curvature couplings. They are classified on Renormalization Ambiguities and Gravitational Counterterms.

In the massless flat-space thermal state, subtracting the vacuum gives a particularly transparent normalization check:

$$\rho=\int\frac{d^3\boldsymbol k}{(2\pi)^3} \frac{k}{e^{\beta k}-1} =\frac{\pi^2T^4}{30}, \qquad p=\frac{\rho}{3}.$$

The integral follows from $\int_0^\infty x^3/(e^x-1)\,dx=\pi^4/15$. Isotropy sets the momentum flux to zero and gives $\langle k_i k_j\rangle=\delta_{ij}k^2/3$. Thus the orthonormal tensor is $\operatorname{diag}(\rho,p,p,p)$, with trace $-\rho+3p=0$ and vanishing divergence. The improvement term vanishes because the thermal Wick square is spatially and temporally constant. This checks normalization, dimensions, signature, and state subtraction together.

A credible result should pass four checks. First, its divergence must vanish to the claimed analytic or numerical accuracy. Second, its flat-spacetime vacuum value must match the declared normalization. Third, its trace must reproduce the field equation and any anomaly without deleting the scheme-dependent part. Fourth, differences between states must agree with direct evaluation of their smooth two-point-function difference.

This page defines a quantum observable on a prescribed background. Inserting it into a metric equation and solving for backreaction is a separate semiclassical-gravity problem; conservation is necessary for a consistent metric equation but does not solve it.

Exercise

Suppose a provisional point-split tensor obeys $\nabla^at^{(0)}_{ab}=\nabla_bQ$. Show that adding $-g_{ab}Q$ restores conservation, and explain why adding an arbitrary state-dependent conserved tensor would violate the intended prescription.

Solution

Metric compatibility gives $\nabla^a(g_{ab}Q)=\nabla_bQ$, so the two divergences cancel exactly. One may still add a local, covariant, state-independent conserved curvature tensor $C_{ab}$ because that is the classified finite renormalization freedom. An arbitrary state-dependent conserved tensor would change the observable separately for each state, would not arise from the common Hadamard subtraction, and would destroy the meaningful scheme-independent formula for differences between states.

References

• Décanini, Yves, and Antoine Folacci. “Hadamard Renormalization of the Stress-Energy Tensor for a Quantized Scalar Field in a General Spacetime of Arbitrary Dimension.” Physical Review D 78 (2008): 044025. 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.
• Wald, Robert M. “The Back Reaction Effect in Particle Creation in Curved Spacetime.” Communications in Mathematical Physics 54 (1977): 1–19. DOI.
• Wald, Robert M. “Trace Anomaly of a Conformally Invariant Quantum Field in Curved Spacetime.” Physical Review D 17 (1978): 1477–1484. DOI.