Propagation of the Hadamard Property

Hadamard singularities propagate with the field equation. For a normally hyperbolic Klein–Gordon operator on a smooth, time-oriented, globally hyperbolic spacetime, a bisolution that has the Hadamard wavefront set in a neighborhood of one Cauchy surface has it throughout the spacetime. This result converts a local construction near Cauchy data into a global ultraviolet statement; it does not generate positivity or a state from arbitrary data.

Required background. Hadamard Admissibility and the Two-Point Wavefront Criterion states the directional condition. Green Functions and Causal Propagators supplies well-posed hyperbolic evolution.

Helpful background. Hyperbolic Equations and Causal Propagators develops the PDE setting. Singular Support and Wavefront Sets supplies the cotangent geometry.

The propagation statement

Let $(M,g)$ be smooth, time oriented, and globally hyperbolic, let $\Sigma$ be a smooth spacelike Cauchy surface, and let

$$P_\xi=\Box+m^2+\xi R$$

be normally hyperbolic. Suppose $\omega_2\in\mathcal D'(M\times M)$ is a two-point bisolution,

$$P_{\xi,x}\omega_2=P_{\xi,x'}\omega_2=0,$$

with the canonical antisymmetric part. If $\omega_2$ is Hadamard on $N\times N$ for some open neighborhood $N$ of $\Sigma$, then it is Hadamard on all of $M\times M$. In the older local-parametrix language this is the Fulling–Sweeny–Wald global singularity theorem Fulling, Sweeny, and Wald 1978, pp. 257–264; in microlocal language it follows from propagation of singularities together with the commutator.

The neighborhood must straddle a complete Cauchy surface. Hadamard form in an arbitrary small diamond does not determine singularities carried by null geodesics that never meet that diamond.

Why null directions propagate

The principal symbol of $P_\xi$ is

$$p(x,k)=g^{ab}(x)k_a k_b.$$

For a distributional solution, the propagation-of-singularities theorem says that its wavefront set inside the characteristic set $p=0$ is a union of maximally extended null bicharacteristics Duistermaat and Hörmander 1972, pp. 183–269. Projected to spacetime, these are null geodesics; in cotangent space, $k$ is parallel transported. Applying the theorem in each argument of $\omega_2$ transports the paired covectors in the Hadamard relation.

Two ingredients fix more than the characteristic set alone. The local Hadamard condition chooses the future branch in the first argument, and the antisymmetric part

$$\omega_2-\omega_2^{\mathsf T}=-iE$$

prevents an unaccounted opposite branch from appearing. Global hyperbolicity ensures that every inextendible causal curve, hence every relevant null generator, meets $\Sigma$ and that Cauchy evolution is unique.

First application: a Cauchy-band certificate

Assume a quasifree state has been constructed on a band $N$ around $\Sigma$. A reproducible propagation check has four parts:

1. verify $N$ contains all of $\Sigma$, not merely a compact subset;
2. verify the kernel is a bisolution and has antisymmetric part $-iE$;
3. verify the Hadamard orientation on $N\times N$;
4. evolve both arguments with the unique Cauchy propagator.

No new ultraviolet branch can develop away from $N$: any proposed extra covector would run backward along its bicharacteristic to an extra covector in the Cauchy band. The conclusion is global Hadamard admissibility. Positivity must already hold for the Cauchy covariance and is preserved by the induced algebraic evolution; propagation of singularities alone does not prove it.

What fails beyond the hypotheses

No global hyperbolicity. A Cauchy horizon or timelike boundary can support characteristics not controlled by the original data. At compactly generated Cauchy horizons, Hadamard extensions can fail at base points Kay, Radzikowski, and Wald 1997, Theorem 2.

No normally hyperbolic operator. An unfixed gauge operator has a degenerate principal symbol and constraint directions. One must pass to a gauge-fixed Green-hyperbolic complex and then restore the physical quotient; the scalar theorem cannot simply be quoted.

Boundary without boundary conditions. Reflected rays and boundary wavefront sets depend on the boundary condition. A declaration that the interior equation is hyperbolic does not specify their propagation.

The adversarial conclusion is therefore exact: breaking global hyperbolicity, hyperbolicity of the operator, or well-posed boundary data removes the global theorem, even if a local parametrix remains meaningful.

Domain and failure conditions

Propagation is the fourth stage of the construction map. Once a genuine state kernel has Hadamard form on a complete Cauchy neighborhood, normally hyperbolic evolution transports the allowed singular relation; it does not revisit the positivity or selection steps.

The propagation arrow is licensed by global hyperbolicity, normally hyperbolic bisolution equations, and complete Cauchy control. Schematic; not to scale.

Loss of global hyperbolicity, an unfixed gauge degeneracy, or unspecified boundary conditions removes the characteristics needed for this theorem. In the failure map such omissions set the boundary of the global claim even if local Hadamard form remains valid.

A local Hadamard kernel can survive where the global propagation theorem fails; the conclusion must then be restricted to the controlled domain. Schematic; not to scale.

For neighboring state claims and their failure conditions, see Domain and failure conditions.

Handoffs

Constructing Hadamard States by Deformation and Gluing uses this theorem to transfer reference states. Boundary and gauge cases begin with their own Green-hyperbolic analysis. The proof-level Hamilton-flow statement continues in Propagation of Singularities for Hyperbolic Fields.

References

• Duistermaat, Johannes J., and Lars Hörmander. “Fourier Integral Operators II.” Acta Mathematica 128 (1972): 183–269. DOI.
• Fulling, Stephen A., Mark Sweeny, and Robert M. Wald. “Singularity Structure of the Two-Point Function in Quantum Field Theory in Curved Spacetime.” Communications in Mathematical Physics 63 (1978): 257–264. DOI.
• 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.