Boundaries, Nonglobally Hyperbolic Spacetimes, and State Obstructions

When global hyperbolicity fails, the local wave equation no longer determines unique global evolution. A timelike boundary must be supplied with boundary data that make the spatial operator self-adjoint and, for a stable ground-state construction, positive. Different admissible extensions can have different spectra, causal propagators, algebras, and states. Other failures of global hyperbolicity, such as compactly generated Cauchy horizons, may obstruct even the local Hadamard and locality properties needed to define the stress tensor.

Required background. Local covariance with boundaries and background structures explains how boundary data enter objects and morphisms. Green-hyperbolic operators and causal propagators supplies the boundaryless theorem that now requires replacement.

Helpful background. Existence and gluing of Hadamard states states the globally hyperbolic result that cannot simply be reused. No-natural-state results separates existence from preferred selection. Timelike boundaries and AdS boundary conditions gives the main application. Moving mirrors and the dynamical Casimir effect and boundary surface counterterms show dynamical and renormalization consequences. Cauchy horizons, chronology, and global hyperbolicity treats a different obstruction.

Static dynamics requires an operator domain

On a static spacetime write the Klein–Gordon equation schematically as

$$\partial_t^2\phi+A\phi=0,$$

where $A$ is a symmetric spatial differential operator on an initial dense domain. If the spatial slice is complete in the relevant metric and there is no timelike boundary, $A$ is often essentially self-adjoint. With a timelike boundary it may have a family of self-adjoint extensions $A_\zeta$, encoded for a scalar by Dirichlet, Neumann, Robin, or more general boundary data. A positive self-adjoint extension gives functional calculus

$$\phi(t)=\cos(tA_\zeta^{1/2})\phi_0 +A_\zeta^{-1/2}\sin(tA_\zeta^{1/2})\pi_0.$$

This evolution is unique only after $\zeta$ is fixed. Positivity excludes exponentially growing modes and permits a ground-state covariance; self-adjointness alone is not enough for stability. Ishibashi and Wald classify the admissible positive extensions for fields on static AdS and relate them to boundary conditions at conformal infinity in Ishibashi and Wald 2004, §§ 2–4, pp. 2984–3005. Their result is a static extension theorem, not a general theorem for moving or nonlinear boundaries.

The causal propagator $E_\zeta$ is built from the same evolution. Reflected null rays contribute to its support and singularities. Consequently the CCR form

$$\sigma_\zeta([f],[h])=\int fE_\zeta h\,\mathrm d\mathrm{vol}$$

and the algebra it defines depend on the boundary condition. A morphism between boundary spacetimes must preserve the boundary structure and operator domain; an embedding that changes $\zeta$ does not intertwine propagators.

The AdS-strip ground state

For a scalar on a static AdS strip, separate time and space after the usual conformal rescaling. In the mass range where both asymptotic radial behaviors are normalizable, the radial Sturm–Liouville operator admits a family of extensions. A Robin parameter may be expressed schematically as

$$\cos\zeta\,\phi+L\sin\zeta\,n^a\nabla_a\phi=0$$

at the timelike conformal boundary, with the precise weighted trace determined by the rescaled field. Only the parameter range giving a positive extension is used. If $A_\zeta u_{j,\zeta}=\omega_{j,\zeta}^2u_{j,\zeta}$, the static ground-state two-point function is

$$\omega_{2,\zeta}(t,x;t',x') =\sum_j\frac{e^{-i\omega_{j,\zeta}(t-t'-i0)}}{2\omega_{j,\zeta}} u_{j,\zeta}(x)\overline{u_{j,\zeta}(x')},$$

with an integral replacing the sum for continuous spectrum. Both frequencies and modes depend on $\zeta$. Dappiaggi and Ferreira construct the Poincaré-AdS ground two-point functions for admissible boundary conditions, identify possible bound states, and verify ordinary Hadamard form in every globally hyperbolic subregion in Dappiaggi and Ferreira 2016, §§ III–V, pp. 125016-4–125016-13.

Globally, reflected singularities require a boundary-adapted microlocal condition. For Dirichlet fields on asymptotically AdS spacetimes satisfying the Breitenlohner–Freedman bound, Wrochna proves existence and uniqueness modulo bulk-smooth terms for a holographic Hadamard condition based on the b-wavefront set Wrochna 2017, Theorems 1.1–1.2 and § 5, pp. 2295–2323. This does not authorize arbitrary Robin pseudodifferential data; later theorems specify wider classes.

The construction is worked out on timelike boundaries and AdS boundary conditions. An independent check compares two extensions: their eigenfrequencies must satisfy their respective Robin equations, and the equal-time commutator derived from $E_\zeta$ must reproduce the identity on that extension’s domain.

Cauchy horizons are a different obstruction

A boundary condition can restore evolution for a timelike boundary, but not every nonglobally hyperbolic spacetime is repaired that way. For a compactly generated Cauchy horizon, Kay, Radzikowski, and Wald prove that at certain base points no extension of the initial algebra can satisfy the expected local agreement, and the two-point distribution of any initially Hadamard state must fail the Hadamard condition there Kay, Radzikowski, and Wald 1997, Theorems 1–2, pp. 538–550. The conclusion is localized and theorem-specific; it is not a proof that every horizon or every nonglobally hyperbolic spacetime admits no QFT.

Failure boundary: omit the boundary parameter

Choose two positive Robin extensions $A_{\zeta_1}$ and $A_{\zeta_2}$ with different parameters in the admissible mass window. Their radial eigenvalue equations differ, so generically $\omega_{j,\zeta_1}\neq\omega_{j,\zeta_2}$. The ground-state two-point functions and causal propagators therefore differ even though the bulk metric and differential expression $P$ are identical. Local field equations and the formal CCR do not select between them.

The strongest surviving claim without boundary data is only the local differential equation in the interior. There is no unique global solution map, algebra, ground state, or stress tensor. Conversely, selecting one Robin condition constructs one theory; it does not prove covariance under maps that change the boundary condition or positivity for every Robin parameter.

Exercises

Why does a negative eigenvalue of a self-adjoint spatial extension obstruct the static ground-state construction?

Solution

If $A_\zeta u=-\kappa^2u$, the mode equation is $\ddot q-\kappa^2q=0$ and has exponential solutions rather than oscillations of positive frequency. The operator $A_\zeta^{1/2}$ is not a positive one-particle Hamiltonian, and the covariance $(2A_\zeta^{1/2})^{-1}$ is not a positive ground-state covariance. Self-adjointness guarantees unitary functional calculus only for the first-order formulation with an appropriate energy; it does not make the Klein–Gordon energy positive.

References

• Dappiaggi, Claudio, and Hugo R. C. Ferreira. “Hadamard States for a Scalar Field in Anti-de Sitter Spacetime with Arbitrary Boundary Conditions.” Physical Review D 94 (2016): 125016. DOI; Open PDF.
• Ishibashi, Akihiro, and Robert M. Wald. “Dynamics in Non-Globally-Hyperbolic Static Spacetimes III: Anti-de Sitter Spacetime.” Classical and Quantum Gravity 21 (2004): 2981–3014. DOI; Open PDF.
• Kay, Bernard S., Marek J. Radzikowski, and Robert M. Wald. “Quantum Field Theory on Spacetimes with a Compactly Generated Cauchy Horizon.” Communications in Mathematical Physics 183 (1997): 533–556. DOI; Open PDF.
• Wrochna, Michał. “The Holographic Hadamard Condition on Asymptotically Anti-de Sitter Spacetimes.” Letters in Mathematical Physics 107 (2017): 2291–2331. DOI; Open PDF.