Fields and Local Algebras on Curved Backgrounds

Quantum fields on curved spacetime require a causal propagation problem before a quantum state can be chosen. Geometry, field equations, and boundary conditions determine the observable algebra; they do not generally select a vacuum.

Enter this chapter

Begin with Fixed-Background, Semiclassical, Gravitational-EFT, and Quantum-Gravity Regimes to distinguish prescribed geometry from semiclassical response and quantized gravity.

Helpful background. Hyperbolic Equations and Causal Propagators and Levi–Civita Connections, Geodesics, and Riemann Curvature provide causal Green operators and the geometric meaning of curvature.

Topics

1. Fixed-Background, Semiclassical, Gravitational-EFT, and Quantum-Gravity Regimes — Classifies what is quantized and which expansion controls each approximation.
2. Curved Spacetimes, Cauchy Surfaces, and Global Hyperbolicity — Identifies the causal hypothesis behind predictive evolution.
3. Covariant Scalar Fields and Curvature Coupling — Derives the scalar operator, conformal coupling, boundary variation, and FLRW mode equation.
4. Spinors, Tetrads, and Spin Connections — Constructs the curved Dirac operator and conserved current.
5. Gauge Fields, Gauge Fixing, and Ghosts on Curved Backgrounds — Separates a hyperbolic gauge-fixed representative from gauge-invariant content.
6. Differential-Form Fields and Reducible Gauge Systems — Adds reducibility, ghosts-for-ghosts, harmonic forms, and flux sectors.
7. Green Operators, Causal Propagators, and State-Dependent Two-Point Functions — Fixes the curved-space distribution dictionary by equations and support.
8. Covariant Symplectic Structure and Conserved Inner Products — Derives solution-space pairings and their boundary-flux condition.
9. Covariant Algebraic Quantization and Fock Realizations — Constructs CCR and CAR algebras before any representation is selected.
10. Local Field Algebras, Causality, and the Time-Slice Property — Assigns observables to regions without assuming tensor-product factorization.
11. Local Covariance, Isometries, and Background Embeddings — Explains how local observables are transported between admissible backgrounds.
12. Relative Cauchy Evolution and Background Response — Compares theories before and after a compact background perturbation.
13. Topology, Zero Modes, and Global Sectors — Shows why locally identical geometries can have different observable algebras.
14. Timelike Boundaries, Self-Adjoint Extensions, and AdS Boundary Conditions — Classifies the extra data required when causal curves can reach a boundary.
15. Conformal Transformations and Frame Changes — Distinguishes classical conformal transport from quantum equivalence.

How the ideas fit together

On a globally hyperbolic spacetime, suitable field equations propagate Cauchy data and define advanced and retarded Green operators. Their difference encodes the commutator structure used to construct the quantum algebra. A state then assigns expectation values to that algebra Hollands and Wald 2015, § 2.1, PDF.

This separation makes a failure informative. A timelike boundary demands additional boundary data; a gauge field requires reduction to physical observables; topology can introduce global sectors invisible to local equations. None of these missing inputs can be repaired merely by naming a preferred vacuum.

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

Domain and failure conditions

Field system	Operator or constraint	Additional data	Physical algebra	Characteristic failure
Real scalar	Normally hyperbolic $\Box_g+m^2+\xi R$	Cauchy data; boundary condition if present	CCR algebra modulo the equation	Loss of global hyperbolicity or uncontrolled boundary flux
Dirac field	First-order Dirac operator with normally hyperbolic square	Spin structure, tetrad orientation, spin connection	CAR algebra from the conserved Hermitian form	No spin structure, inconsistent adjoint, or orientation mismatch
Maxwell potential	Gauge-degenerate wave system	Gauge fixing, constraints, residual gauge quotient, boundary data	Gauge-invariant field-strength or reduced potential algebra	Zero modes or gauge-dependent quantities mistaken for observables
Differential $p$-form	de Rham complex with reducible gauge symmetry	Cohomology, harmonic modes, flux sector, reducibility stages	Local field strengths plus global flux observables	Local mode expansion omits harmonic or topological sectors
Global or topological sector	Kernel and cohomology of the local field complex	Spatial topology, large gauge group, flux lattice, zero-mode prescription	Sector-dependent algebra, possibly with a center	Locally identical geometries assigned the same global observables
Timelike-boundary problem	Chosen self-adjoint realization of the spatial operator	Extension and boundary condition, normal orientation	Algebra tied to that realization	Nonpositive spectrum, nonzero symplectic flux, or nonunique evolution
Locally covariant theory	Assignment over admissible background embeddings	Background category and preserved structures	Compatible family of local algebras	A noncausal embedding or changed boundary structure invalidates transport

Where to go next

Continue to States, Hadamard Structure, and Microlocal Control to identify the time-flow assumptions behind positive frequency and a vacuum.

References

• Stefan Hollands and Robert M. Wald, “Quantum Fields in Curved Spacetime,” Physics Reports 574 (2015), 1–35, DOI, Open PDF.