Local Observables, Stress Tensors, and Anomalies

Local quantum observables in curved spacetime require a subtraction prescription and finite renormalization conditions. Stress tensors and currents must also satisfy the appropriate conservation laws, with anomalies and boundary terms treated consistently.

Enter this chapter

Begin with Wick Polynomials and Hadamard Point Splitting to construct composite fields from Hadamard-subtracted products.

Helpful background. Hadamard Parametrix and Short-Distance Structure and Interacting Stress Tensors and Ward Identities develop the definitions used by the chapter’s opening pages.

Topics

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.

How the ideas fit together

Hadamard subtraction removes the universal short-distance singularity while leaving the smooth, state-dependent part of a two-point function. The resulting Wick products retain a restricted freedom to add local covariant terms Hollands and Wald 2001, Theorem 5.1.

For the stress tensor, conservation further constrains these terms and is essential when it becomes a gravitational source. Trace anomalies and scheme-dependent total derivatives must be distinguished. Numerical agreement between subtraction methods means agreement after finite terms and conventions are matched, not equality of their bare mode sums.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

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

Where to go next

Continue to Matter Effective Actions in Curved Spacetime to define the matter operator, its domain, and the loop expansion.

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.