Wavefront-Set Products, Pullbacks, and Pushforwards

Products, restrictions, fiber integrals, and kernel compositions are defined only when their singular covectors meet transversely. Wavefront calculus turns that statement into explicit cone tests and bounds the singular directions of the result; failure of a sufficient test is a genuine warning, not permission to manipulate the formal expression.

Required background. Microlocal calculus for quantum fields supplies wavefront sets; local and microcausal functionals with Peierls brackets supplies the QFT kernel setting.

Helpful background. Scaling degree and extension of distributions treats forbidden diagonals; domains, signatures, supports, and regularity fixes domains; singular support and wavefront sets gives the general analysis.

Three transversality theorems

For $u,v\in\mathcal D'(M)$, the pointwise product is canonically defined if

$$(x,k)\in\operatorname{WF}(u)\quad\Longrightarrow\quad (x,-k)\notin\operatorname{WF}(v).$$

Then $\operatorname{WF}(uv)$ is contained in the union of the two original cones and their fiberwise sums. This is a sufficient canonical extension of smooth multiplication, sequentially continuous in fixed cone spaces; Brunetti, Fredenhagen, and Köhler 1996, Theorem 2.6, pp. 5–6.

For a smooth map $F:X\to Y$, define its normal set

$$N_F=\{(F(x),\eta): {}^t dF_x\eta=0\}.$$

The pullback $F^*u$ exists if $N_F\cap\operatorname{WF}(u)=\varnothing$, and its wavefront set is contained in ${}^tdF(\operatorname{WF}(u))$. Restriction to a submanifold is this theorem applied to the inclusion; the forbidden covectors are its conormal bundle.

For a smooth $F:X\to Y$, pushforward of a compactly supported distribution, or of one for which $F$ is proper on the relevant support, is defined by $F_*u(f)=u(f\circ F)$. Its wavefront bound retains those target covectors whose pullbacks occur in $\operatorname{WF}(u)$, with possible zero source covectors handled by the proper-support hypothesis. Kernel composition combines a product, a pullback to a triple product, and a pushforward; both covector matching and proper support are indispensable. The composition criterion and sign convention appear in Brunetti, Fredenhagen, and Köhler 1996, Theorem 2.7, pp. 6–7.

Pulling a Hadamard kernel to a timelike worldline

Let $\gamma:I\to M$ be a smooth future-directed timelike curve and $\iota=\gamma\times\gamma:I^2\to M^2$. Its normal set contains covectors $(k,k')$ satisfying $k(\dot\gamma(\tau))=k'(\dot\gamma(\tau'))=0$. A nonzero causal covector cannot annihilate a timelike vector. The Hadamard wavefront set consists of paired nonzero null covectors, so $N_\iota\cap\operatorname{WF}(W_2)=\varnothing$. Therefore

$$W_\gamma(\tau,\tau')=(\gamma\times\gamma)^*W_2$$

is a well-defined distribution. Its singular covectors are bounded by

$$(\tau,k(\dot\gamma(\tau));\tau',-k'(\dot\gamma(\tau'))),$$

where $(\gamma(\tau),k)\sim(\gamma(\tau'),k')$ and $k$ has the Hadamard future orientation. Along a timelike worldline, distinct points sufficiently close are timelike related, so the local singular support lies at coincidence; the two proper-time covectors are opposite and retain the positive-frequency orientation.

This verifies the distributional input used in detector response along curved and accelerated worldlines. Switching functions and response integrals require their own support and infrared conditions; the pullback theorem alone does not prove positivity or a transition rate.

An independent flat-space check takes $\gamma(\tau)=(\tau,\mathbf0)$. Substitution in the mass-shell representation gives an ordinary positive-frequency distribution in $\tau-\tau'$, whose first Fourier covector has one orientation only. This agrees with the cotangent pullback above.

The same bookkeeping explains the minus sign in kernel composition. If $K_1(x,y)$ and $K_2(y,z)$ are composed by integrating their product over $y$, then a singular covector $\eta$ contributed by $K_1$ in the shared variable must meet $-\eta$ from $K_2$. The product must first exist on $X\times Y\times Z$, and the projection to $X\times Z$ must be proper on the resulting support. Only then may the $y$ covectors cancel under pushforward. Thus “matching canonical relations” is not sufficient by itself: unmatched zero sums obstruct the product, while nonproper support can make the fiber integral diverge even when every local cone test passes. Checking cone transversality and checking support are logically independent parts of the theorem.

Adversarial null restriction

Let $\gamma$ instead be null with tangent $\ell$. The null covector $\ell^\flat$ annihilates $\ell$ because $g(\ell,\ell)=0$. A Hadamard singular covector parallel to the same null geodesic can therefore lie in the conormal set of the curve. The transversality condition fails, so the naïve restriction of $W_2$ to the null curve is not canonically defined. Smearing in transverse directions, using a boundary value, or specifying a renormalized extension may define a different object, but each is additional data.

The converse is limited: satisfying the wavefront criterion licenses the operation, not a unique renormalization across a locus where it fails. Conversely, failure of the sufficient criterion does not prove that no specially structured extension exists; it proves that the general pullback theorem supplies none.

Exercises

1. Squaring a delta distribution. Apply the product criterion to $\delta_0^2$ on $\mathbb R$.

Solution

$\operatorname{WF}(\delta_0)=\{(0,k):k\ne0\}$ contains both $k$ and $-k$. The zero sum occurs, so the canonical product theorem does not define $\delta_0^2$.

2. Restriction to a Cauchy surface. Which covectors obstruct restricting $u$ to a spacelike hypersurface $\Sigma$?

Solution

The conormal covectors to $\Sigma$ are multiples of its timelike normal covector. Restriction is allowed when $\operatorname{WF}(u)$ contains no such covector over $\Sigma$.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Michael Köhler. “The Microlocal Spectrum Condition and Wick Polynomials of Free Fields on Curved Spacetimes.” Communications in Mathematical Physics 180 (1996): 633–652. DOI. Open PDF.
• Hörmander, Lars. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. 2nd ed. Springer, 1990. DOI.