Propagation of Singularities for Hyperbolic Fields

For an operator of real principal type, singularities of a distributional solution cannot begin or end arbitrarily in the source-free characteristic region: they travel along the Hamiltonian flow of the principal symbol. For Klein–Gordon fields those bicharacteristics project to null geodesics, including through caustics where a coordinate geometric-optics amplitude may diverge.

Required background. Microlocal calculus for quantum fields supplies wavefront sets; Green-hyperbolic operators and causal propagators supplies global solutions.

Helpful background. Wavefront-set products, pullbacks, and pushforwards supplies cone operations; domains, signatures, supports, and regularity fixes the distributional domain; propagation of the Hadamard property gives the physical use.

Real principal type and bicharacteristics

Let $P$ be a scalar pseudodifferential operator with real homogeneous principal symbol $p$ and suppose $dp$ does not vanish on $p^{-1}(0)$. Its Hamilton vector field on $T^*M$ is

$$H_p=\frac{\partial p}{\partial k_a}\frac{\partial}{\partial x^a} -\frac{\partial p}{\partial x^a}\frac{\partial}{\partial k_a}.$$

If $Pu=f$, elliptic regularity first gives

$$\operatorname{WF}(u)\subset \operatorname{WF}(f)\cup\operatorname{Char}(P), \qquad \operatorname{Char}(P)=\{p=0\}.$$

The propagation theorem sharpens this: in $\operatorname{Char}(P)\setminus\operatorname{WF}(f)$, the wavefront set of $u$ is a union of maximally extended integral curves of $H_p$. Thus if one point on a source-free bicharacteristic is singular, every point on that connected bicharacteristic segment is singular; if one is microlocally smooth, all are smooth. The primary result is Duistermaat and Hörmander 1972, Theorem 6.1.1.

For $P=\Box_g+m^2+\xi R$, the principal symbol is $p(x,k)=g^{ab}k_ak_b$. Its characteristic set is the nonzero null cone. Hamilton’s equations project, up to reparametrization, to null geodesics and parallel transport the cotangent $k$. The mass and curvature coupling affect lower-order amplitudes and tails but not the characteristic directions.

A compact source and its fundamental solution

Let $f\in\mathcal E'(M)$ have compact support and set $u=G_+f$ on a globally hyperbolic spacetime. Away from $\operatorname{supp}f$, $Pu=0$. The kernel of $G_+$ has singular directions on future-directed null geodesic pairs; composing it with $f$ launches only the covectors compatible with $\operatorname{WF}(f)$ and the kernel relation. The propagation theorem then carries them along the complete source-free null bicharacteristics in $J^+(\operatorname{supp}f)$.

For a point source, this reproduces the light-cone singular front. Inside the cone, massive and curved-spacetime Green functions may have a smooth or less singular tail, but the leading wavefront remains null. The advanced solution gives the past branch. This is the mechanism used in propagation of the Hadamard property: establish the oriented Hadamard cone near a Cauchy surface, use the bisolution equation in each variable, and propagate it globally.

An independent flat-space check applies the Fourier transform. For $P=\Box+m^2$, singular directions of a fundamental solution must lie where the principal denominator fails to be elliptic at high frequency, $k^2=0$. Translational invariance keeps $k$ constant, so the Hamilton flow is a straight null ray. This matches the position-space light cone.

For a bisolution such as the commutator or a Hadamard two-point function, apply the theorem in one variable while holding the other microlocally fixed, and then repeat in the second variable. The first step propagates $(x,k)$ along the null Hamilton orbit from a neighborhood of a Cauchy surface; the second transports $(y,k')$. The field equation alone permits both time orientations. The commutator and the initial Hadamard condition select the paired orientation, while propagation proves that no source-free segment can acquire an isolated extra covector. This division of labor is essential: the propagation theorem transports an established condition but neither supplies positivity nor constructs the initial state.

Adversarial failure: caustics do not erase wavefronts

Geometric optics writes a local solution as $a(x)e^{i\lambda S(x)}$. At a conjugate point, the projected null congruence focuses and the coordinate amplitude or Van Vleck determinant representation may diverge. It is tempting to declare the approximation invalid and the exact solution smooth beyond the caustic. That conclusion is false: the Hamiltonian flow in cotangent space remains smooth, even when its projection to base space folds. The Lagrangian relation may acquire several branches and a Maslov phase, but the wavefront continues through the conjugate point.

The theorem does not determine amplitude, distributional order, or cancellation between several branches; it only constrains the singular support in phase space. A particular superposition can cancel a singularity, and a source can create new ones where $\operatorname{WF}(f)$ enters. Boundary reflections likewise require a boundary propagation theorem, not the boundary-free statement.

Exercises

1. Lower-order terms. Explain why adding $m^2+\xi R$ does not change the Klein–Gordon characteristic set.

Solution

The characteristic set is defined by the highest-order principal symbol. Multiplication by $m^2+\xi R$ is order zero, so the order-two symbol remains $g^{ab}k_ak_b$.

2. Smooth source region. If $Pu=f$ and $f$ is smooth near a characteristic bicharacteristic segment, can $u$ be singular at exactly one interior point of that segment?

Solution

No. In the source-free characteristic region, the propagation theorem makes $\operatorname{WF}(u)$ a union of whole bicharacteristic segments. An isolated interior singular point would violate invariance under $H_p$.

References

• Duistermaat, J. J., and Lars Hörmander. “Fourier Integral Operators. II.” Acta Mathematica 128 (1972): 183–269. DOI.
• Radzikowski, Marek J. “Micro-Local Approach to the Hadamard Condition in Quantum Field Theory on Curved Space-Time.” Communications in Mathematical Physics 179 (1996): 529–553. DOI.