Fields and Local Algebras on Curved Backgrounds

A curved-spacetime field theory has a definite physical domain only after its background, causal and boundary data, field bundle, differential operator, classical phase space, observable algebra, and quantum state have been distinguished. Geometry fixes propagation, but it does not choose a state. A gauge-fixed equation can expose a useful propagator, but it does not by itself identify the physical observables. A locally valid equation also need not give a globally predictive theory when global hyperbolicity, suitable boundary conditions, or control of zero modes is missing.

This chapter constructs that background-and-field layer. It begins by separating fixed-background QFT from semiclassical and quantum-gravitational approximations, then develops scalar, spinor, gauge, and differential-form systems through causal propagation, covariant phase space, algebraic quantization, local covariance, topology, and timelike boundaries. State selection and ultraviolet admissibility begin in the next chapter.

Helpful background. Hyperbolic Equations and Causal Propagators supplies the support properties used throughout; Levi–Civita Connections, Geodesics, and Riemann Curvature supplies the curvature convention; and Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supplies the bundle language used for spinors and gauge fields.

The background-and-field contract

For a linear field, the reusable input is not merely a metric. It is the collection

$$\mathfrak D = (M,g,\mathfrak o,\mathfrak t; E,P;\mathcal B;\mathcal S,\sigma;\mathcal A),$$

where $(M,g)$ is a Lorentzian spacetime, $\mathfrak o$ and $\mathfrak t$ are orientation and time orientation, $E\to M$ is the field bundle, $P$ is the field operator, $\mathcal B$ denotes any boundary or asymptotic condition, $\mathcal S$ is the space of solutions, $\sigma$ is its symplectic or Hermitian pairing, and $\mathcal A$ is the algebra of observables. Gauge systems require a complex of fields and gauge transformations in place of an unreduced solution space.

For the real scalar reference system,

$$P=\Box_g+m^2+\xi R,$$

and global hyperbolicity gives unique advanced and retarded Green operators on compactly supported sources. Their difference determines the classical bracket and the quantum commutator. A positive two-point function is additional state data; it cannot be reconstructed from $P$ and causal support alone. This separation is the central organizing fact of the chapter and of algebraic QFT on curved spacetime Hollands and Wald 2015, §2.1, pp. 9–19.

The construction map below displays that order. Inspect especially the separation between the local algebra and state-dependent expectation values, and the two checkpoints that feed into the construction.

The chapter’s controlled construction keeps causal, gauge, topological, boundary, algebraic, and state data in their logical order; the map is schematic and not to scale.

Choose a route

The arrows below are suggested reading routes, not a claim that every page hard-requires the page before it.

Goal	Route	Result
First curved-QFT construction	Regimes → global hyperbolicity → scalar field → Green operators → symplectic form → algebraic quantization	Specify and quantize a free scalar without choosing a preferred vacuum
Fermions	Global hyperbolicity → spinors and spin connection → conserved inner products → algebraic quantization	Identify the geometric data required by a Dirac field
Gauge fields and global sectors	Global hyperbolicity → gauge fields → differential forms → topology	Separate local radiative modes from gauge, harmonic, and flux sectors
Local covariance and response	Local algebras → local covariance → relative Cauchy evolution	Compare the same local theory on different backgrounds and compute its linear background response
Timelike boundaries	Global hyperbolicity → Green operators → symplectic flux → timelike boundaries	Decide whether a boundary condition yields predictive, flux-conserving, stable evolution
Conformal transport	Scalar curvature coupling → local covariance → conformal transformations	Transport a conformal field while identifying mass, state, and anomaly obstructions

Chapter guide

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.

Domain and failure conditions

The table is a compact comparison of the data that must accompany each field system. “Predictive” means predictive for the stated initial-boundary problem; it does not imply that a preferred state exists.

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

The claim-path map turns the last column into a decision rule. A successful local calculation licenses only the domain whose assumptions have passed; any listed witness forces a narrower statement or a different theory.

The omitted hypothesis sets the boundary of a curved-spacetime QFT claim; the validity and failure map is schematic and not to scale.

One calculation viewed at several levels

A nonminimally coupled scalar on spatially flat FLRW is the recurring test case. The geometry page decides whether constant-time slices are Cauchy surfaces. The scalar page derives the field equation. The Green-operator page fixes causal response. The symplectic page checks conservation of the Wronskian. Algebraic quantization converts that pairing into commutation relations. Only after those steps may a state select a two-point function or a Fock realization.

The same sequence exposes failures. Adding a timelike boundary changes the Green operators and introduces flux terms. Compactifying space introduces zero modes. Changing conformal frame transports the classical equation only in the conformally invariant case. None of these changes can be repaired by renaming a vacuum.

Review the chapter

Background specification. Given a vector field on a spacetime with boundary, list the background, bundle, operator, constraint, boundary, phase-space, and algebra data. A successful answer distinguishes geometric input from state data.

Causal versus statistical information. Explain why replacing a quasifree state changes its Wightman function but not the retarded Green operator of a fixed free equation. The invariant check is unchanged causal support and commutator.

Surface independence. Starting from a conserved symplectic current, integrate over the region between two Cauchy surfaces. The two pairings agree precisely when flux through every remaining boundary vanishes.

Topology test. Compare Maxwell theory on locally flat spacetimes with spatial slices $\mathbb R^3$ and $T^3$. Local curvature agrees, but harmonic one-forms and global Wilson loops need not.

Failure diagnosis. If a formal mode basis exists on an AdS patch but contains negative eigenvalues of the chosen spatial extension, completeness does not establish stability. The boundary-condition page gives the repair criterion.

Where this chapter stops

Volume I retains the geometry, global analysis, and self-adjointness theorems. States, Hadamard Structure, and Microlocal Control adds ultraviolet-admissible states without manufacturing a preferred vacuum. Local Observables, Stress Tensors, and Anomalies constructs renormalized composites. The theorem-first functorial, microlocal, and boundary analysis belongs to Mathematical QFT.

References

• Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society (2007), Open PDF, especially Chapters 3–4.
• Marco Benini, Claudio Dappiaggi, and Thomas-Paul Hack, “Quantum Field Theory on Curved Backgrounds: A Primer,” International Journal of Modern Physics A 28 (2013), 1330023, arXiv:1306.0527.
• Romeo Brunetti, Klaus Fredenhagen, and Rainer Verch, “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Field Theory,” Communications in Mathematical Physics 237 (2003), 31–68, arXiv:math-ph/0112041.
• Stefan Hollands and Robert M. Wald, “Quantum Fields in Curved Spacetime,” Physics Reports 574 (2015), 1–35, DOI, Open PDF.