Wavefront-Set Products, Pullbacks, and Pushforwards

Products, restrictions, parameter families, fiber integrals, and kernel compositions force singular distributions to interact. Wavefront calculus gives robust sufficient conditions under which the usual smooth operation extends canonically, together with an upper bound on the singular covectors that survive. The central habit is simple but exacting: include zero covectors over supports, apply the cotangent map, and test whether a forbidden total zero can occur.

Required background. Microlocal calculus for quantum fields supplies wavefront sets and the site’s Fourier convention; local and microcausal functionals with Peierls brackets shows why these cone tests matter for QFT kernels.

Helpful background. Scaling degree and extension of distributions treats products at forbidden diagonals; domains, signatures, supports, and regularity fixes the analytic setting; singular support and wavefront sets develops the general theory.

The chapter overview provides three reusable guides: the dependency map shows where these operations enter the proof chain, the hypothesis–conclusion table separates the local cone tests from support assumptions, and the failure map shows what must be repaired when a test fails.

For a direct route through this page, begin with products, pullbacks, pushforwards, and kernel composition. Then compare the timelike application with the null failure, and finish with the reusable check and exercises.

Products as diagonal pullbacks

Let $M$ be a smooth manifold and let $u,v\in\mathcal D'(M)$. The ordinary product of smooth functions may be written as a restriction:

$$uv=\Delta^*(u\otimes v), \qquad \Delta:M\longrightarrow M\times M, \qquad \Delta(x)=(x,x).$$

For distributions, the tensor product always exists. To state its wavefront bound without repeatedly treating smooth factors as special cases, define the wavefront set with the support zero section adjoined,

$$\operatorname{WF}_0(u) = \operatorname{WF}(u) \cup \{(x,0):x\in\operatorname{supp}u\}.$$

Then

$$\operatorname{WF}(u\otimes v) \subset \Bigl(\operatorname{WF}_0(u)\times\operatorname{WF}_0(v)\Bigr) \setminus \{(x,0;y,0)\}.$$

The conormal bundle of the diagonal is

$$N^*\Delta = \{(x,x;k,-k):k\ne0\}.$$

Consequently, the diagonal pullback is licensed when no singular covector of one factor cancels a singular covector of the other:

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

When this condition holds, the output cone is obtained by adding covectors over the same base point:

$$\operatorname{WF}(uv) \subset \left\{ \begin{array}{c} (x,k+\ell): (x,k)\in\operatorname{WF}_0(u),\\[2pt] (x,\ell)\in\operatorname{WF}_0(v), \quad k+\ell\ne0 \end{array} \right\}.$$

The zero-section terms reproduce the original wavefront sets when the other factor is smooth; the nonzero sums describe genuinely new directions. This construction agrees with smooth multiplication and is sequentially continuous when the two wavefront sets stay inside fixed admissible cones. See Brunetti, Fredenhagen, and Köhler 1996, Theorem 2.6, pp. 5–6 (Open PDF) and Hörmander 1990, Theorem 8.2.10.

Two one-dimensional tests make the condition memorable. The square of $(t-i0)^{-1}$ is allowed because both factors have only positive covectors, whereas the product of $(t-i0)^{-1}$ and $(t+i0)^{-1}$ fails the test because their covectors can sum to zero. Similarly, $\delta_0^2$ fails because $\operatorname{WF}(\delta_0)$ contains both orientations. These are conclusions about this canonical product theorem: failure of its condition is not a proof that no separately prescribed nonlinear operation could ever be introduced.

Pullbacks, restrictions, and parameter families

Let $X$ and $Y$ be smooth manifolds, let $F:X\to Y$ be smooth, and let $u\in\mathcal D'(Y)$. The differential of $F$ sends tangent vectors forward, while its transpose sends a covector at $F(x)$ back to $x$:

$$(dF_x)^{\mathsf T}:T^*_{F(x)}Y\longrightarrow T_x^*X, \qquad \bigl((dF_x)^{\mathsf T}\eta\bigr)(v) = \eta(dF_xv).$$

The normal set of the map consists of the nonzero target covectors that this transpose kills:

$$N_F = \left\{ (F(x),\eta): \eta\ne0,\quad (dF_x)^{\mathsf T}\eta=0 \right\}.$$

If

$$N_F\cap\operatorname{WF}(u)=\varnothing,$$

then the pullback $F^*u$ is defined and

$$\operatorname{WF}(F^*u) \subset \left\{ \bigl(x,(dF_x)^{\mathsf T}\eta\bigr): (F(x),\eta)\in\operatorname{WF}(u) \right\}.$$

A submersion has injective transpose, so its normal set is empty and every distribution pulls back. For a diffeomorphism the displayed inclusion is an equality after applying the cotangent lift. For a general map it is only an upper bound: different source contributions can disappear after pullback. These statements are the content of Hörmander 1990, Theorem 8.2.4.

If $i:S\hookrightarrow Y$ is an embedded submanifold, then $N_i=N^*S\setminus0$, where

$$N^*S = \{(x,\eta):x\in S,\ \eta(v)=0\ \text{for every }v\in T_xS\}.$$

Thus the standard restriction theorem licenses $u|_S=i^*u$ when

$$\operatorname{WF}(u)\cap N^*S=\varnothing.$$

The same theorem controls distribution-valued parameter families. Let $\Lambda$ be a parameter manifold, let $U\in\mathcal D'(X\times\Lambda)$, and let $j_\lambda(x)=(x,\lambda)$. The fiber $U_\lambda=j_\lambda^*U$ exists if the wavefront set contains no pure parameter covector over that fiber:

$$\operatorname{WF}(U) \cap \{(x,\lambda;0,\zeta):\zeta\ne0\} = \varnothing.$$

If this exclusion holds throughout an open parameter region, then for every $f\in C^\infty_c(X)$ the scalar function

$$\lambda\longmapsto \langle U_\lambda,f\rangle$$

is smooth there. One way to see this is to multiply $U$ by $f$ and push it forward along $X\times\Lambda\to\Lambda$: compact support in the $X$ direction supplies properness, while the exclusion of pure parameter covectors removes every possible nonzero output wavefront covector.

Pushforwards, critical values, and proper support

Pushforward has a different license from pullback. Let $X$ and $Y$ be smooth manifolds, fix compatible density conventions, take $F:X\to Y$ smooth, and let $u\in\mathcal D'(X)$. If $F$ restricted to $\operatorname{supp}u$ is proper—preimages of compact subsets of $Y$ are compact in $\operatorname{supp}u$—then one may define

$$\langle F_*u,\varphi\rangle = \langle u,\chi(\varphi\circ F)\rangle,$$

where $\chi\in C^\infty_c(X)$ equals one near the compact set $\operatorname{supp}u\cap F^{-1}(\operatorname{supp}\varphi)$. The value is independent of the choice of $\chi$. If $u$ itself is compactly supported, this is customarily abbreviated as $\langle u,\varphi\circ F\rangle$. Properness is what makes the definition work; it is a support condition, not a transversality condition.

The wavefront estimate is

$$\operatorname{WF}(F_*u) \subset \left\{ \begin{array}{c} (y,\eta):\eta\ne0,\ \text{there is }x\in\operatorname{supp}u,\\[2pt] F(x)=y,\quad \bigl(x,(dF_x)^{\mathsf T}\eta\bigr) \in\operatorname{WF}_0(u) \end{array} \right\}.$$

The use of $\operatorname{WF}_0(u)$ is essential. If $(dF_x)^{\mathsf T}\eta=0$ at a critical point, the source entry is the zero covector over $\operatorname{supp}u$; it can still produce a nonzero target covector. Properness does not remove this contribution. See Hörmander 1990, Theorem 8.2.12.

Here is the elementary model. Let $F:\mathbb R\to\mathbb R$ be $F(x)=x^2$, and push forward the smooth compactly supported density $a(x)\,dx$, with $a(0)\ne0$. Away from the critical value $y=0$, changing variables on the two branches gives

$$F_*\bigl(a(x)\,dx\bigr) = \frac{a(\sqrt y)+a(-\sqrt y)}{2\sqrt y}\,dy, \qquad y>0,$$

and the density vanishes for $y<0$. With $H$ denoting the Heaviside function, its coefficient behaves as $a(0)H(y)/\sqrt y$ near the origin and is singular there, even though the input is smooth. The pushforward bound predicts exactly this possibility: at $x=0$, $dF_0=0$, so every nonzero target covector pulls back to the zero covector included in $\operatorname{WF}_0(a\,dx)$.

Composition of singular kernels

Let $K_{12}\in\mathcal D'(X\times Y)$ and $K_{23}\in\mathcal D'(Y\times Z)$. Formally,

$$K_{13}(x,z) = \int_Y K_{12}(x,y)K_{23}(y,z)\,dy.$$

The rigorous construction separates three operations on $X\times Y\times Z$:

$$K_{13} = (\pi_{13})_* \left[ (\pi_{12})^*K_{12}\, (\pi_{23})^*K_{23} \right].$$

The two pullbacks always exist because the coordinate projections are submersions. The product fails its standard wavefront test only if there are $x,y,z$ and $\eta\ne0$ such that

$$(x,0;y,\eta)\in\operatorname{WF}(K_{12}), \qquad (y,-\eta;z,0)\in\operatorname{WF}(K_{23}).$$

Why must both outer covectors vanish? After lifting, the two covectors are $(0,\eta,0)$ and $(0,-\eta,0)$; only then does their entire sum on $X\times Y\times Z$ vanish. By contrast, a matched pair $\eta$ and $-\eta$ with a nonzero outer covector is not an obstruction—it is precisely a contribution that can survive integration over $y$.

There is also a global support condition. Set

$$\mathcal S = \{(x,y,z):(x,y)\in\operatorname{supp}K_{12},\ (y,z)\in\operatorname{supp}K_{23}\}.$$

The projection $\pi_{13}:\mathcal S\to X\times Z$ must be proper. In the ordering used here, the simpler sufficient hypothesis in Khavkine and Moretti is that $\operatorname{supp}K_{23}\to Z$ is proper. Local cone compatibility cannot replace either support condition.

Once the product and pushforward are licensed, the support-refined raw wavefront bound is

$$\begin{aligned} \operatorname{WF}(K_{13}) \subset \bigl\{(x,\xi;z,\zeta):\;&(\xi,\zeta)\ne(0,0),\ \text{there are }y,\eta\text{ with}\\ &(x,\xi;y,\eta)\in\operatorname{WF}_0(K_{12}),\\ &(y,-\eta;z,\zeta)\in\operatorname{WF}_0(K_{23}) \bigr\}. \end{aligned}$$

This formula has three channels: two genuine singular covectors compose through opposite raw $Y$ covectors; a partial-zero covector of $K_{12}$ can yield $(x,\xi;z,0)$ when paired with support of $K_{23}$; and support of $K_{12}$ can pair with a partial-zero covector of $K_{23}$ to yield $(x,0;z,\zeta)$. Dropping either partial-zero channel can make an apparently elegant canonical-relation calculation false.

Operator kernels are often described with the sign in the second slot reversed:

$$\operatorname{WF}'(K) = \{(x,\xi;y,\eta):(x,\xi;y,-\eta)\in\operatorname{WF}(K)\}.$$

Define the two outer marginal cones

$$\begin{aligned} \operatorname{WF}_X(K_{12}) &= \{(x,\xi):(x,\xi;y,0)\in\operatorname{WF}(K_{12}) \text{ for some }y\},\\ \operatorname{WF}'_Z(K_{23}) &= \{(z,\zeta):(y,0;z,\zeta)\in\operatorname{WF}'(K_{23}) \text{ for some }y\}. \end{aligned}$$

A standard, slightly coarser relation bound is then

$$\begin{aligned} \operatorname{WF}'(K_{13}) \subset{}& \operatorname{WF}'(K_{12})\circ\operatorname{WF}'(K_{23})\\ &{}\cup \bigl(\operatorname{WF}_X(K_{12})\times Z\times\{0\}\bigr)\\ &{}\cup \bigl(X\times\{0\}\times\operatorname{WF}'_Z(K_{23})\bigr). \end{aligned}$$

The shared covector has the same sign in the two primed relations, so their ordinary relation composition encodes the opposite raw signs above. The criterion and the two indispensable marginal terms are stated in Khavkine and Moretti 2015, Theorem 8 and Eq. (73), p. 47 (Open PDF) and, in operator order, Brunetti, Fredenhagen, and Köhler 1996, Theorem 2.7, p. 6 (Open PDF).

Pulling a Hadamard two-point distribution to a timelike worldline

Let $(M,g)$ be a time-oriented globally hyperbolic spacetime and let $W_2\in\mathcal D'(M\times M)$ be the scalar two-point distribution of a Hadamard state. Let $\gamma:I\to M$ be a smooth future-directed timelike worldline parametrized by proper time, so $u(\tau)=\dot\gamma(\tau)$ is unit future timelike. Define

$$\iota:I^2\longrightarrow M^2, \qquad \iota(\tau,\tau') = \bigl(\gamma(\tau),\gamma(\tau')\bigr).$$

The transpose differential is computed slot by slot:

$$(d\iota_{(\tau,\tau')})^{\mathsf T}(k,k') = \bigl(k(u(\tau)),k'(u(\tau'))\bigr).$$

A nonzero causal covector cannot annihilate a timelike vector. In an orthonormal frame whose time axis is $u$, a null covector has nonzero time component, and that component is exactly its evaluation on $u$. The Hadamard wavefront set contains paired nonzero null covectors, so

$$N_\iota\cap\operatorname{WF}(W_2)=\varnothing.$$

Therefore the worldline two-point distribution

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

is well defined. If the Hadamard element is

$$\bigl(\gamma(\tau),k;\gamma(\tau'),-k'\bigr), \qquad (\gamma(\tau),k)\sim(\gamma(\tau'),k'),$$

with $k^\sharp$ future directed, then the pulled-back covector is

$$(\tau,\zeta;\tau',-\zeta'), \qquad \zeta=k(u(\tau))>0, \qquad \zeta'=k'(u(\tau'))>0.$$

Hence

$$\operatorname{WF}(W_\gamma) \subset \{(\tau,\zeta;\tau',-\zeta'):\zeta>0,\ \zeta'>0\}.$$

This calculation is worked out in Fewster 2000, §3, Eqs. (3.5)–(3.8), pp. 10–11 (Open PDF). At coincidence, parallel transport is trivial and the allowed form reduces to $(\tau,\zeta;\tau,-\zeta)$. Distinct sufficiently close points on the worldline are not null related, so the local singular support is diagonal. Globally, a spacetime with lensing or compact spatial directions may admit a returning null geodesic between distinct worldline points; then $\zeta$ and $\zeta'$ remain positive but need not be equal.

The Hadamard pairing diagram makes the transported covector and the second-slot minus sign visible. The result supplies the distributional input used in detector response along curved and accelerated worldlines. Positive type also passes to the pullback, so smooth compact switching gives well-defined quadratic pairings; see Fewster 2000, Theorem 2.2, pp. 8–9 (Open PDF). Sharp or noncompact switching, infrared limits, and long-time transition rates require additional analysis.

For an independent flat-space check in four dimensions, take the inertial worldline $\gamma(\tau)=(\tau,\mathbf0)$ and write $s=\tau-\tau'$. The mass-shell representation gives, as a distribution,

$$\begin{aligned} W_\gamma(s) &= \int_{\mathbb R^3} \frac{d^3\mathbf p}{(2\pi)^3\,2E_{\mathbf p}}\, e^{-iE_{\mathbf p}s}\\ &= \frac{1}{4\pi^2} \int_m^\infty \sqrt{\omega^2-m^2}\, e^{-i\omega s}\,d\omega . \end{aligned}$$

With the site’s forward Fourier transform, the proper-time frequency is supported on $\omega\ge m$ and its high-frequency singular direction has $\zeta>0$. Since $s=\tau-\tau'$, the two slots carry $(\zeta,-\zeta)$, exactly as the cotangent calculation predicts.

Why the null restriction fails the test

Now let $\gamma$ be a smooth null curve with future null tangent $\ell$. At any point $x=\gamma(\tau)$, set $k=\ell^\flat=g(\ell,\cdot)$. This is a nonzero null covector with

$$k(\ell)=g(\ell,\ell)=0.$$

The diagonal Hadamard cone contains

$$(x,k;x,-k).$$

For $\iota=\gamma\times\gamma$, its pullback is

$$(d\iota_{(\tau,\tau)})^{\mathsf T}(k,-k) = \bigl(k(\ell),-k(\ell)\bigr) =(0,0).$$

Thus this Hadamard wavefront element lies in $N_\iota$, and the hypothesis of the general pullback theorem fails. The theorem therefore supplies no canonical restriction of $W_2$ to the null curve. Transverse smearing, a boundary-value prescription, or a renormalized extension may define another object, but each adds information not contained in the naïve pullback.

The standard Minkowski vacuum shows that this is a real divergence, not merely a gap in the theorem. In inertial coordinates let $z=x-x'=(\Delta t,\Delta\mathbf x)$, where $\Delta t=t-t'$ and $\Delta\mathbf x=\mathbf x-\mathbf x'$. For the four-dimensional massless two-point function, use the regularization

$$W_\epsilon(z) = \frac{1}{4\pi^2\!\left( \lvert\Delta\mathbf x\rvert^2 -(\Delta t-i\epsilon)^2 \right)}, \qquad \epsilon>0.$$

On the null line $\gamma(\lambda)=(\lambda,\lambda,0,0)$, with $s=\lambda-\lambda'$, the restricted smooth functions are

$$W_\epsilon\bigl(\gamma(\lambda),\gamma(\lambda')\bigr) = \frac{1}{4\pi^2\epsilon(\epsilon+2is)}.$$

Choose a test function supported where $s$ is positive and bounded away from zero. Its pairing with this expression grows as $1/\epsilon$ with a nonzero coefficient, so the family has no limit in $\mathcal D'(\mathbb R^2)$ as $\epsilon\downarrow0$. Defining null-line data therefore requires an additional transverse smearing, subtraction, or other prescription.

The contrast is geometric: the annihilator of a timelike tangent contains only spacelike covectors, missing the Hadamard null cone, whereas a null tangent’s annihilator contains its own metric-dual null covector. The chapter’s failure map places this conormal collision in the larger chain of microlocal checks.

A reusable cone-and-support check

For any proposed operation, use the same four steps.

1. Express the operation geometrically. Products are diagonal pullbacks; restrictions are inclusion pullbacks; fiber integrals are pushforwards; kernel composition is pull–multiply–push.
2. Lift every input cone, including support zero sections. Write the relevant transpose differentials and retain partial-zero kernel covectors.
3. Test the actual obstruction. For a pullback, exclude covectors annihilated by the transpose differential. For a product, exclude a total zero sum. Do not confuse a shared-variable cancellation that contributes to a pushforward with a zero of the full lifted covector.
4. Check global support and propagate the surviving cone. Properness licenses a pushforward. It does not erase critical-point or partial-zero contributions from the output bound.

Passing these steps licenses the standard canonical operation, with sequential continuity in fixed admissible cone spaces where the theorem asserts it. Failing one step identifies the missing hypothesis or the locus where additional extension or renormalization data are needed.

Common pitfalls

Treating a sufficient criterion as a converse. A Hörmander condition guarantees the standard canonical operation on specified cone classes, with the theorem’s stated continuity properties. Its failure says that theorem does not define the expression; it does not classify every special cancellation or every separately chosen extension.

Using properness as a microlocal cure. Properness prevents support from escaping along an integrated fiber. Critical points can still turn a smooth input into a singular pushforward, as $x\mapsto x^2$ demonstrates.

Deleting zero covectors too early. A wavefront set itself excludes the zero section, but tensor-product, pushforward, and composition bounds need $\operatorname{WF}_0$. Kernel covectors with one zero slot are genuine and generate the marginal terms.

Reading an inclusion as equality. Pullback and pushforward theorems usually give upper bounds. Proving that every allowed direction is actually present requires a lower-bound argument, a local normal form, or an explicit model calculation.

Exercises

1. Same- and opposite-oriented boundary values. Using

$$\operatorname{WF}\!\left((t-i0)^{-1}\right) =\{(0,\tau):\tau>0\}, \qquad \operatorname{WF}\!\left((t+i0)^{-1}\right) =\{(0,\tau):\tau<0\},$$

decide which of the products $\bigl[(t-i0)^{-1}\bigr]^2$ and $(t-i0)^{-1}(t+i0)^{-1}$ is licensed by the product theorem.

Solution

For two copies of $(t-i0)^{-1}$, both covectors are positive, so their sum cannot vanish. The product exists and agrees with the boundary value $(t-i0)^{-2}$. In the mixed product, choose $\tau>0$ from the first factor and $-\tau<0$ from the second; their sum is zero. The theorem does not license the mixed product.

2. Cauchy data versus characteristic data. Let $Pu=0$ for a normally hyperbolic operator, and use microlocal elliptic regularity to assume $\operatorname{WF}(u)\subset\operatorname{Char}(P)$. If $\Sigma$ is a smooth spacelike Cauchy surface, explain why $u|_\Sigma$ and its normal derivative are microlocally allowed. What changes if $\Sigma$ is null?

Solution

Microlocal elliptic regularity first places $\operatorname{WF}(u)$ in the null characteristic set; propagation of singularities then transports those directions along its Hamilton flow. The nonzero conormal covectors of a spacelike hypersurface are timelike, so $N^*\Sigma\cap\operatorname{WF}(u)=\varnothing$ and the restriction exists. Applying a differential operator does not enlarge the wavefront set, so the normal derivative may also be restricted. For a null hypersurface, the conormal is itself null and can meet the characteristic wavefront set; the general restriction theorem may then fail.

3. A singularity created by a critical point. Let $F(x)=x^2$ and let $a\in C^\infty_c(\mathbb R)$ with $a(0)=1$. Derive the density of $F_*(a(x)\,dx)$ for $y>0$ and determine its leading behavior at $y=0$.

Solution

The preimages of $y>0$ are $\sqrt y$ and $-\sqrt y$, and the Jacobian magnitude on either branch is $2\sqrt y$. Therefore

$$F_*(a(x)\,dx) = \frac{a(\sqrt y)+a(-\sqrt y)}{2\sqrt y}\,dy$$

for $y>0$, with zero density for $y<0$. Smoothness of $a$ and $a(0)=1$ give a numerator $2+O(y)$, so the coefficient is $H(y)y^{-1/2}+O(H(y)y^{1/2})$. The critical point $dF_0=0$ accounts for the new singularity.

4. Empty wavefront set but no pushforward. Let $\pi:\mathbb R^2\to\mathbb R$ be $\pi(x,y)=x$ and let $u=1$. Why does $\operatorname{WF}(u)=\varnothing$ not make $\pi_*u$ a distribution by the usual definition?

Solution

For a compactly supported test function $\varphi(x)$, the pullback $\varphi\circ\pi=\varphi(x)$ has support $\operatorname{supp}\varphi\times\mathbb R$, which is not compact. The distribution $u=1$ is only required to act on compactly supported tests, and the formal fiber integral $\int_{\mathbb R}dy$ diverges. Thus the local wavefront condition is vacuous, but $\pi$ is not proper on $\operatorname{supp}u=\mathbb R^2$.

5. The identity kernel. On a manifold $Y$, the identity operator has kernel $I_Y(y,y')=\delta(y-y')$. Compute $\operatorname{WF}'(I_Y)$ and its partial-zero marginal cones. Explain what ordinary relation composition recovers and why the full kernel bound still needs a marginal term.

Solution

The raw wavefront set is

$$\operatorname{WF}(I_Y) = \{(y,\eta;y,-\eta):\eta\ne0\}.$$

Reversing the second sign gives

$$\operatorname{WF}'(I_Y) = \{(y,\eta;y,\eta):\eta\ne0\},$$

the diagonal relation on $T^*Y\setminus0$. Neither slot can be zero, so both partial-zero marginal cones of the identity kernel are empty. For a nonzero shared covector, relation composition with this diagonal returns the other relation.

This statement covers covectors for which the shared slot is nonzero. If the other kernel has a nonzero outer covector but a zero shared-slot covector, ordinary relation composition cannot see it because $\operatorname{WF}'(I_Y)$ excludes the zero section. The support-zero channel in the support-refined formula—or the corresponding marginal term in the coarser formula—recovers that part of $I_Y\circ K=K$ or $K\circ I_Y=K$.

6. Proper-time frequency and the null contrast. Starting from

$$W_\gamma(s) = \int\frac{d^3\mathbf p}{(2\pi)^3\,2E_{\mathbf p}} e^{-iE_{\mathbf p}s}, \qquad E_{\mathbf p}=\sqrt{\lvert\mathbf p\rvert^2+m^2},$$

derive the one-dimensional spectral integral in the text. Then explain why replacing a timelike tangent by a null tangent invalidates the conormal argument.

Solution

In spherical momentum coordinates,

$$d^3\mathbf p =4\pi p^2\,dp, \qquad \omega=\sqrt{p^2+m^2}, \qquad dp=\frac{\omega}{p}\,d\omega.$$

Hence

$$\frac{4\pi p^2\,dp}{(2\pi)^3\,2\omega} = \frac{\sqrt{\omega^2-m^2}}{4\pi^2}\,d\omega,$$

with $\omega\ge m$. The phase $e^{-i\omega s}$ has positive forward-Fourier frequency and pulls back through $s=\tau-\tau'$ to $(\zeta,-\zeta)$. A timelike tangent cannot be annihilated by a nonzero causal covector. For a null tangent $\ell$, however, $\ell^\flat(\ell)=0$, so a Hadamard null covector can lie in the normal set and the pullback theorem no longer applies.

Applications and next steps

• Green-hyperbolic operators and causal propagators applies support control to retarded, advanced, and causal kernels.
• Propagation of singularities for hyperbolic fields supplies the characteristic flow used in the Cauchy-data exercise.
• Scaling degree and extension of distributions begins where a diagonal product test fails and renormalized extension data are required.
• Local and microcausal functionals with Peierls brackets uses these product and composition rules in the functional formalism.

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.
• Fewster, Christopher J. “A General Worldline Quantum Inequality.” Classical and Quantum Gravity 17 (2000): 1897–1911. DOI. Open PDF.
• Hörmander, Lars. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. 2nd ed. Springer, 1990. DOI.
• Khavkine, Igor, and Valter Moretti. “Algebraic QFT in Curved Spacetime and Quasifree Hadamard States: An Introduction.” In Advances in Algebraic Quantum Field Theory, edited by Romeo Brunetti, Claudio Dappiaggi, Klaus Fredenhagen, and Jakob Yngvason, 191–251. Springer, 2015. DOI. Open PDF.