Local Observables, Stress Tensors, and Anomalies

A renormalized local observable is not a bare expression with its infinity erased. It is an operator-valued distribution or state expectation defined by a local subtraction, a finite renormalization prescription, and the Ward identities appropriate to the background. This chapter develops that construction for Wick powers, stress tensors, currents, anomalies, boundaries, numerical representations, null projections, and separated-point stress correlations.

The recurring scalar operator is $P_\xi=\Box_g+m^2+\xi R$, with $\xi_{\mathrm{conf}}=-1/6$ in four dimensions. The causal convention remains $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$ and $[\Phi(f),\Phi(h)]=-iE(f,h)\mathbf1$. At the first use of metric variation we adopt the local sign

$$\delta S_{\mathrm m} = \frac12\int\mathrm d^4x\,\sqrt{-g}\, T_{\mu\nu}\,\delta g^{\mu\nu}, \qquad T_{\mu\nu} = \frac{2}{\sqrt{-g}} \frac{\delta S_{\mathrm m}}{\delta g^{\mu\nu}}.$$

Every effective-action variation and finite curvature shift below uses this sign. A source using the opposite definition must be translated before its coefficients are compared.

Helpful background. Hadamard Parametrix and Short-Distance Structure supplies the universal ultraviolet form; Interacting Stress Tensors and Ward Identities extends the free construction; Entropy Counterterms and Renormalization Ambiguities gives a neighboring application of geometric counterterms.

From singular two-point data to a local observable

For a Hadamard state, write the two-point function locally as

$$W_\omega(x,x')=H_\ell(x,x')+w_{\omega,\ell}(x,x'),$$

where $H_\ell$ contains the state-independent singularity and $w_{\omega,\ell}$ is smooth near coincidence. A point-split observable applies a bidifferential operator to $W_\omega-H_\ell$, takes the coincidence limit, and then adds the allowed local finite terms. The subtraction is universal; the smooth remainder retains the state dependence. Hollands and Wald characterize the locally covariant finite freedom of Wick powers under precise scaling and regularity hypotheses Hollands and Wald 2001, Theorem 5.1.

The structure map displays this order. Inspect its boundary note: finite local ambiguity and anomaly coefficients enter at different stages and cannot be exchanged.

The construction retains state dependence while classifying local finite freedom and checking conservation, anomalies, and boundaries; it is schematic and not to scale.

Chapter guide

1. Wick Polynomials and Hadamard Point Splitting constructs local composite fields.
2. Renormalized Stress Tensor: Axioms and Curvature Ambiguities classifies the conserved finite shifts.
3. Curved-Space Renormalization Schemes: Domains and Translation aligns point splitting, adiabatic, DeWitt–Schwinger, and dimensional prescriptions.
4. Conservation, Local Covariance, and the Backreaction Source imposes the Ward identities needed by semiclassical gravity.
5. Trace Anomalies and Convention Translation separates invariant anomaly data from scheme-shiftable total derivatives.
6. Spin, Gauge, and Gravitational-Anomaly Responses treats consistent and covariant currents on curved backgrounds.
7. Renormalized Currents and Charge Density gives a gauge-covariant point split.
8. Vacuum Polarization and Curved-Space Casimir Effects identifies scheme-resistant differences and forces.
9. Mode-Sum and Numerical Renormalization controls high-mode subtraction and residual tails.
10. Geometric Discretization and Continuum Checks couples field refinement to geometric refinement.
11. Boundaries, Surface Counterterms, and Boundary Stress separates bulk and surface renormalization.
12. Null-Projected and Smeared Stress Observables fixes sampling and affine normalization.
13. Stress Bi-Tensors and Noise-Kernel Input constructs the matter correlation delivered to stochastic gravity.

Domain and failure conditions

This is the canonical chapter comparison. Agreement means equality after the listed finite translation and within the common domain—not equality of unrenormalized summands or regulators.

Prescription	Natural domain and observable	Subtraction data and order	State dependence retained	Finite local freedom	Decisive Ward or boundary check	Translation and failure condition
Hadamard point splitting	General Hadamard states; Wick powers, currents, stress	Local parametrix; enough derivatives for the observable	Smooth remainder $W_\omega-H_\ell$	Covariant curvature polynomials of the correct dimension	Field equation, gauge covariance, conservation, trace convention	Align $\ell$ and local terms; fails if a full reference-state correlator is subtracted
DeWitt–Schwinger	Local massive-field or short-distance expansion; Green functions and stress	Heat-kernel coefficients through the differentiated coincidence order	Added through the exact state-dependent smooth solution	Same local gravitational counterterm basis	Conservation after the complete coefficient set is included	Match proper-time scale and finite coefficients; asymptotic expansion is not a globally exact state
Adiabatic subtraction	Homogeneous mode backgrounds; mode observables	Observable-specific WKB order, fourth order for four-dimensional scalar stress	Exact modes minus their high-momentum adiabatic terms	Local curvature shifts after scale alignment	Mode-by-mode identity must integrate to covariant conservation	Match canonical variable, scale, and finite terms; state adiabatic order is not subtraction order
Dimensional renormalization	Covariant perturbation theory and effective actions	Poles in $d=4-2\epsilon$ plus declared finite subtraction	Finite state or contour data in the chosen functional	Finite local counterterms and coupling coordinates	Background Ward identities and anomaly after regulator removal	Translate $\mu$ and gravitational couplings; minimal subtraction alone is not a physical condition
Analytic mode sum	Static or separable geometries; field square and stress	Analytic large-mode singular tail plus low-mode treatment	Boundary condition and state encoded in exact modes	Same local terms as the covariant prescription	Residual tail, conservation, state difference, independent split	Match degeneracy and density conventions; fails when a fitted tail is extrapolated without control
Numerical or geometric discretization	Nonseparable backgrounds or refined meshes; usually differences first	Local lattice/FEM counterterms and a joint geometry–field continuum path	Prepared discrete state, with finite-volume and boundary errors	Regulator-symmetry counterterms that approach the continuum basis	Refinement, covariance restoration, volume, boundary, and benchmark tests	Translate only after a stable joint limit; a single plateau is not continuum evidence

The validity map turns each last-column condition into a stopping rule. A finite answer is not enough: conservation, convention translation, surface terms, smearing, and regulator removal must also pass.

Insufficient subtraction or a failed Ward, boundary, smearing, or continuum check narrows the result to a regulator-dependent diagnostic; the map is schematic and not to scale.

What may be exported

For a state $\omega$ and prescription $\mathcal R$, report

$$\begin{gathered} \bigl(\mathcal O_{\mathcal R},\ \omega,\ H_\ell, \Delta\mathcal O_{\mathrm{local}},\\ \text{Ward identities},\ \text{smearing or boundary data}, \text{regulator error}\bigr). \end{gathered}$$

A state difference in one prescription can cancel the universal local ambiguity; an absolute expectation generally cannot. An anomaly coefficient fixed by field content is not a choice of state. A total derivative can move under a finite counterterm without changing type-A or type-B data. A nonlocal anomaly-induced action belongs to Chapter 8, and the causal gravitational response to Chapter 9. Wald’s axiomatic analysis gives the foundational stress-tensor consistency conditions and their remaining local freedom Wald 1977, pp. 1–19.

References

• Stefan Hollands 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, arXiv:gr-qc/0103074.
• Robert M. Wald, “The Back Reaction Effect in Particle Creation in Curved Spacetime,” Communications in Mathematical Physics 54 (1977), 1–19, DOI.