Local Covariance with Boundaries and Background Structures

Spin structures, gauge bundles, external sources, and timelike-boundary conditions are part of a field equation’s definition. A locally covariant theory containing them must enlarge its source objects and require every morphism to preserve that data. Keeping the ordinary Loc arrows while silently changing a boundary condition or bundle turns the supposed functorial map into an ill-typed comparison.

Required background. Domains, Signatures, Supports, and Regularity supplies explicit domain control. Globally Hyperbolic Spacetimes and the Loc Categories, Locally Covariant QFT as a Functor, and Time-Slice Axiom and Relative Cauchy Evolution give the construction being extended.

Helpful background. Timelike Boundaries, AdS, and Boundary Conditions develops the PDE issues. Boundaries, Flux, Boundary Ward Identities, and Edge Modes and Topology, Zero Modes, and Global Sectors explain global and boundary degrees of freedom.

Enriched objects and preserving morphisms

Replace a bare spacetime by an enriched object

$$\boldsymbol M_B=(M,g,o,t;B),$$

where $B$ may include a spin structure, principal bundle with connection, external scalar or vector source, or a timelike boundary with a specified self-adjoint boundary condition. A morphism consists of a Loc-type embedding together with a lift or pullback map preserving every component of $B$. Examples include a spin-preserving embedding, a bundle morphism intertwining connections, or a boundary-preserving embedding that keeps a Robin parameter fixed.

The theory is then a functor from this enriched category to Phys. Relative Cauchy evolution may vary only backgrounds declared variable while holding the others fixed, and only along perturbations that remain objects. External-source models provide a rigorous example: the source is part of the object, affine fields replace an untyped linear shift, and source variation has its own relative Cauchy evolution Fewster and Schenkel 2015, §§ 3 and 6–7, pp. 7–12 and 24–34.

A restricted timelike-boundary construction

Consider a chosen class of anti-de Sitter Poincaré patches or slabs for which the Klein–Gordon initial-boundary-value problem is known to possess causal propagators. Attach a fixed admissible Robin condition, written schematically as

$$n^a\nabla_a\phi+\kappa\phi=0$$

at the timelike conformal boundary, with the appropriate weighted trace in the actual functional setting. Permit only embeddings that preserve the boundary and $\kappa$. If the corresponding propagators $E_{M,\kappa}$ intertwine under those embeddings and the induced symplectic maps are injective, the CCR prescription defines a candidate functor on that restricted category. Cauchy and relative-evolution claims additionally require the boundary-value problem to have the needed time-slice maps.

For a massive scalar on the Poincaré domain, Dappiaggi and Ferreira construct causal propagators and on-shell algebras for admissible boundary conditions and establish causality, the time-slice axiom, and F-locality in their setting Dappiaggi and Ferreira 2017, §§ 3.2 and 4.1–4.3. This supports the concrete test in Timelike Boundaries, AdS, and Boundary Conditions: fix one Robin prescription and test covariance under boundary-preserving embeddings. It is not a theorem that arbitrary AdS slabs and arbitrary boundary-changing embeddings form a standard LCQFT.

Gauge and topological sectors

Gauge theories add another choice: retain universal topological charges or reduce by a radical. For Maxwell theory, embeddings can change de Rham cohomology, so the universal assignment can send a nonzero flux class to zero and cease to be injective. The reduced theory restores injective covariance and dynamical locality by removing those classes Fewster and Lang 2016, §§ 5–6, pp. 13–22. Boundaries may instead require edge or flux data in an enlarged target; ordinary bulk observable algebras need not contain them faithfully.

Failure boundary and independent checks

The independent check is to draw the proposed morphism square for the differential operator and propagator:

$$P_{N,B_N}\psi_*=\psi_*P_{M,B_M}, \qquad E_{N,B_N}\psi_*=\psi_*E_{M,B_M}.$$

Changing $\kappa$ while pretending that the same $\psi_*$ intertwines the Green operators fails the second equality: a solution satisfying one boundary condition need not satisfy the other. The CCR map then fails to preserve its commutator. Likewise, forgetting the lift of a spin or gauge bundle leaves no defined map on sections.

A second check uses symplectic flux through the boundary. For a self-adjoint boundary condition, the flux term in Green’s identity must vanish on the operator domain. If an embedding or background variation changes that domain, repeat the boundary form calculation; a nonzero residue shows that the proposed symplectic map and hence the algebra homomorphism are unavailable. This diagnoses the failure without assuming any global AdS reconstruction.

The strongest surviving conclusion may be local. F-locality says that restriction to a suitable globally hyperbolic interior region reproduces an ordinary local algebra. It does not supply covariance between arbitrary boundary spacetimes or prove that boundary degrees of freedom are captured by the bulk algebra.

Exercise

Why is a map from a Dirichlet scalar theory to a Robin scalar theory not supplied merely by the identity map on the interior spacetime?

Solution

The identity on interior points does not preserve the operator domain. Dirichlet and Robin solutions obey different boundary traces and generally have different Green operators. Therefore extension of test functions need not intertwine causal propagators, so it need not preserve the symplectic form or CCR. A separate, proved comparison map would be required.

References

• Dappiaggi, Claudio, and Hugo R. C. Ferreira. “On the Algebraic Quantization of a Massive Scalar Field in Anti-de Sitter Spacetime.” Reviews in Mathematical Physics 30 (2018): 1850004. Open article; Open PDF.
• Fewster, Christopher J., and Benjamin Lang. “Dynamical Locality of the Free Maxwell Field.” Annales Henri Poincaré 17 (2016): 401–436. DOI; Open PDF.
• Fewster, Christopher J., and Alexander Schenkel. “Locally Covariant Quantum Field Theory with External Sources.” Annales Henri Poincaré 16 (2015): 2303–2365. DOI; Open PDF.