Microlocal Calculus for Quantum Fields

The wavefront set records not only where a distribution is singular but also the nonzero cotangent directions in which localized Fourier decay fails. In QFT that extra direction is decisive: it distinguishes a positive-frequency two-point function from its reversal, provides robust sufficient criteria for canonically multiplying or restricting singular kernels, and replaces global momentum support by local geometric data on a curved spacetime.

Required background. Domains, signatures, supports, and regularity fixes distributional domains; Wightman functions and spectral support supplies the flat-space spectrum condition.

Helpful background. Local and microcausal functionals with Peierls brackets shows where the cones are used; analytic continuation between Euclidean and Lorentzian domains explains the frequency boundary values.

What a wavefront point means

Let $u\in\mathcal D'(M)$ and choose local coordinates near $x$. First multiply by a smooth cutoff $\chi$ supported in the coordinate patch, with $\chi(x)\ne0$. The product $\chi u$ isolates the behavior near $x$. A pair $(x,k)\in T^*M\setminus 0$ is absent from $\operatorname{WF}(u)$ when there are such a $\chi$ and a conic neighborhood $\Gamma$ of $k$ for which, for every $N\geq0$,

$$\bigl|\widehat{\chi u}(\xi)\bigr| \le C_N(1+\lvert\xi\rvert)^{-N}, \qquad \xi\in\Gamma.$$

Here the hat uses the site’s forward Fourier phase $e^{+i\xi\cdot x}$. Hörmander 1990, §8.1 and Brunetti, Fredenhagen, and Köhler 1996, Definitions 2.2–2.4, pp. 4–5 (Open PDF) use the opposite phase; results imported from those conventions have fiber label $-\xi$ relative to ours and are reflected below. Khavkine and Moretti’s forward-transform convention already matches the one used here, so their Wightman cone needs no reflection. The zero-sum product and pullback tests are unchanged when all fiber covectors are reflected together.

A conic neighborhood contains every positive rescaling of its covectors, so a wavefront set records directions rather than a preferred frequency magnitude. The zero covector is excluded: it carries no directional information.

The complement of the rapidly decreasing directions is $\operatorname{WF}(u)$. Three facts make the definition useful:

• $\operatorname{WF}(u)$ is closed and conic in each cotangent fiber;
• its projection to $M$ is the singular support $\operatorname{singsupp}u$; and
• multiplication by a smooth function and application of a differential operator cannot add wavefront directions.

Thus singular support answers where?, whereas the wavefront set answers where and in which covector directions? The local Fourier definition and its intrinsic manifold version are stated in Brunetti, Fredenhagen, and Köhler 1996, Definitions 2.2–2.4, pp. 4–5 (Open PDF).

The Dirac delta gives the simplest example:

$$\operatorname{WF}(\delta_0) =\{(0,\tau):\tau\ne0\}.$$

It is singular only at the origin and in both nonzero cotangent directions. Boundary values show that the same singular point can instead carry only one orientation. With the forward Fourier phase,

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

Both distributions have singular support $\{0\}$, but their frequency orientations are opposite. This elementary pair is the local model for the distinction between a Wightman function and its transpose.

For kernels on $M\times M$, a wavefront element has the form $(x,k;x',k')$, and the total covector must be nonzero; one slot is allowed to carry the zero covector. These partial-zero elements matter in tensor products and kernel composition. We write

$$(x,k)\sim(x',k')$$

only when $k$ is a nonzero null covector, there exists a null geodesic from $x$ to $x'$, the metric-dual vector $k^\sharp$ is tangent to that geodesic at $x$, and $k'$ is the parallel transport of $k$ along it to $x'$. At coincidence, the relation includes $(x,k)\sim(x,k)$ for every nonzero null $k$. This definition is geometric; it needs neither a Killing field nor a global Fourier transform.

Computing the Minkowski vacuum cone

For a real free scalar of mass $m\geq0$ in four-dimensional Minkowski space, put $z=x-x'$ and $p=(\omega_{\mathbf p},\mathbf p)$, where $\omega_{\mathbf p}=\sqrt{\lvert\mathbf p\rvert^2+m^2}$. The vacuum two-point distribution is

$$W_2(x,x') =\int\frac{\mathrm d^3\mathbf p}{(2\pi)^3\,2\omega_{\mathbf p}} e^{-ip\cdot(x-x')}.$$

The integral is understood distributionally; it is not an absolutely convergent ordinary integral. Its four-momentum measure is proportional to $\theta(p^0)\delta(p^2-m^2)\,\mathrm d^4p$, so only the future mass shell occurs. Four checks unpack the geometry and constrain the possible wavefront directions:

1. Field equation. $(\Box_x+m^2)W_2=(\Box_{x'}+m^2)W_2=0$. The principal symbol retains the highest-derivative part of the operator; a nonzero covector at which it vanishes is characteristic. Microlocal elliptic regularity therefore confines each nonzero slot covector to the characteristic cone $g^{ab}k_a k_b=0$.
2. High-frequency limit. Although each integration momentum is timelike when $m>0$, $\omega_{\mathbf p}=\lvert\mathbf p\rvert+O(\lvert\mathbf p\rvert^{-1})$. Its conic directions approach the null cone. The mass changes lower-frequency and tail behavior, not the ultraviolet directions.
3. Translation invariance. Dependence only on $x-x'$ forces the two covectors to be $(k,-k)$ after identifying the two Minkowski cotangent spaces. Geometrically this becomes $(k,-k')$, with $k'$ obtained by parallel transport.
4. Positive frequency and propagation. With the site’s forward Fourier transform, the factor $e^{-ip\cdot z}$ places the first covector on the future branch. Hamilton propagation carries that covector along the joining null geodesic.

These checks constrain the wavefront set to the following cone and explain each piece of its geometry. They do not, by themselves, prove the reverse inclusion. For separated null-related points, localized boundary-value and stationary-phase analysis of the mass-shell integral shows that every listed future null conormal actually occurs. At coincidence, the unbounded future mass shell approaches every future null ray, supplying all of the stated diagonal directions. The exact result is

$$\operatorname{WF}(W_2)= \left\{ (x,k;x',-k'): (x,k)\sim(x',k'),\quad k\triangleright0 \right\},$$

where $k\triangleright0$ means nonzero and future directed. Here $k^\sharp$ is the metric-dual vector, which is tangent to the joining null geodesic; $k'$ is obtained by parallel transport. At distinct points in Minkowski space, $k^\sharp$ is parallel to the null displacement $x-x'$; at coincidence every future-directed nonzero null covector occurs. The mass-shell representation and the exact cone are given in Khavkine and Moretti 2015, §3.4, Eqs. (74)–(76), pp. 47–49 (Open PDF).

In four spacetime dimensions, for Klein–Gordon two-point distributions satisfying the field equation, commutator, and state hypotheses, this geometric wavefront relation is equivalent to the Hadamard condition. Khavkine and Moretti 2015, Theorem 9, pp. 48–49, and Remark 23, pp. 51–52 (Open PDF) give a modern statement and explain the repair of a gap in the original equivalence proof; Radzikowski 1996, Theorem 5.1 is the historical source.

Thus the equality is a theorem about both exclusion and occurrence, not merely a dimensional guess. The four checks above explain why its ingredients are necessary; the localized nondecay analysis supplies the lower inclusion that turns the constraint into an equality.

Comparing the two-point function with the commutator

Define $W_2^{\mathrm{op}}(x,x')=W_2(x',x)$. Under the site’s causal convention,

$$E=G_{\mathrm{ret}}-G_{\mathrm{adv}}, \qquad W_2-W_2^{\mathrm{op}}=-iE.$$

The factor $-i$ is essential for agreement with $[\Phi(f),\Phi(h)]=-iE(f,h)\mathbf1$, although multiplication by a nonzero constant does not change a wavefront set. Transposition reverses the two-point orientation. Hence

$$\operatorname{WF}(E) =\mathcal C^+\cup\mathcal C^-,$$

where $\mathcal C^+$ is the future-first relation above and $\mathcal C^-$ is its reversal. In Minkowski space, $W_2$, $W_2^{\mathrm{op}}$, and $E$ all have the same singular support—the set of null-related pairs—but only their wavefront sets distinguish the two frequency orientations. The canonical Hadamard-pairing schematic and state criterion visualize the same orientation before applying it to curved-spacetime state selection.

There is a direct Fourier check. Localizing $W_2$ convolves its future-shell measure with a rapidly decreasing function. The convolution produces small tails in other directions but cannot produce a non-rapidly-decaying past cone. Transposition reverses the cone, while the causal propagator contains both branches. This is exactly the information that singular support discards.

The operations that need cone checks

Smooth algebra is not a reliable guide to singular distributions. The basic microlocal operations can be organized as follows. For a submanifold $S\subset M$, its conormal bundle $N^*S$ consists of covectors that annihilate tangent vectors in $TS$. A map $F$ is proper on $\operatorname{supp}u$ when the inverse image of every compact set, intersected with $\operatorname{supp}u$, is compact.

• Tensor product $u\otimes v$. Put independent distributions on a product manifold. It is always defined, but its wavefront bound must retain possible zero covectors in either slot.
• Product $uv$. Pull $u\otimes v$ back to the diagonal. A sufficient check is that no covector of $u$ is the negative of one of $v$ at the same point.
• Restriction to $S$. Pull back by the inclusion $S\hookrightarrow M$. A sufficient check is $\operatorname{WF}(u)\cap N^*S=\varnothing$.
• Pushforward $F_*u$. Integrate along the fibers of $F$. Properness on $\operatorname{supp}u$ controls support, while covectors generated at critical points must remain in the output bound.
• Kernel composition. Multiply on a shared variable and then push forward. The covector-matching and proper-support checks together give the standard sufficient criterion.

The product bound and kernel-composition bookkeeping appear in Brunetti, Fredenhagen, and Köhler 1996, Theorems 2.6–2.7 (Open PDF); the general pullback and pushforward results are Hörmander 1990, Theorems 8.2.4 and 8.2.12. These are sufficient conditions and bounds, not converses.

For later bookkeeping, adjoin the zero covector over the support,

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

The tensor-product theorem then gives the compact bound

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

A zero second-slot covector does not mean that $v$ is singular; it records its support while $u$ supplies the singular direction. The next page derives the product, pullback, pushforward, and composition bounds rather than treating this summary as a proof.

Why the statement is coordinate independent

If $F:U\to V$ is a diffeomorphism, localization followed by a change of variables sends a high-frequency covector $\eta$ on $V$ to $(dF_x)^{\mathsf T}\eta$ on $U$. Hence

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

Under a passive coordinate change, the metric and time-orientation data transform together with the covector, so nullness and future direction are invariant. For an active map between spacetimes, those properties are preserved only when the map also preserves the metric and time orientation, as a time-orientation-preserving isometry does. A rule phrased only as a particular coordinate component $k_0>0$ would fail this test. The geometric rule asks whether $k^\sharp$ is future directed, so it remains meaningful without a preferred time coordinate.

This equality is special to a diffeomorphism. Pullback by a general smooth map can send a nonzero target covector to zero; that is why restriction requires a transversality hypothesis.

What the cone does not prove

Keep only $\operatorname{singsupp}W_2$. Then $W_2$, $W_2^{\mathrm{op}}$, and $E$ all look singular at the same null-related pairs. Yet $W_2^{\mathrm{op}}$ has past-directed first covectors and is not the same positive-frequency boundary value. Singular support can still say where a detector kernel might fail to be smooth, but it cannot certify the Hadamard orientation, positivity, or the product criterion.

Nor does an acceptable wavefront cone prove that a state exists: positivity, normalization, the field equation, and the commutation relation are separate conditions. When a standard microlocal compatibility criterion is satisfied, it licenses the named operation; failure of a merely sufficient criterion is not an impossibility theorem, and no cone condition by itself constructs a positive functional.

Common pitfalls

Calling the massive shell the wavefront cone. The integration measure is supported at $p^2=m^2$, but wavefront sets retain high-frequency directions. Those conic directions are null because the mass is lower order relative to $\lvert p\rvert^2$.

Dropping the minus sign in the second slot. A translation-invariant kernel depends on $x-x'$. Differentiating that difference with respect to the second argument reverses the covector, so the pair is $(k,-k)$, not $(k,k)$.

Treating proper support as a cone condition. Properness controls whether a fiber integral acts on compactly supported test functions. It does not erase critical-point covectors or repair a forbidden distributional product.

Reading a cone as a state-existence theorem. The Hadamard cone controls ultraviolet singularities. Positivity and the field algebra remain independent requirements.

Exercises

1. Smooth functions and derivatives. Show that $\operatorname{WF}(f)=\varnothing$ for $f\in C^\infty_c(M)$, and explain why $\operatorname{WF}(Pu)\subset\operatorname{WF}(u)$ for a differential operator $P$ with smooth coefficients.

Solution

Every localized product $\chi f$ is smooth and compactly supported. Repeated integration by parts in its Fourier transform gives decay faster than every inverse power in every cone, so no directed singular point remains.

Now suppose $(x_0,\xi_0)\notin\operatorname{WF}(u)$. Choose nested cutoffs $\chi\prec\widetilde\chi$, meaning that $\widetilde\chi=1$ on a neighborhood of $\operatorname{supp}\chi$, and choose a cone $\Gamma$ on which $\widehat{\widetilde\chi u}$ decreases rapidly. Locality gives $\chi Pu=\chi P(\widetilde\chi u)$; contributions involving $1-\widetilde\chi$ are supported away from $\operatorname{supp}\chi$. In coordinates, the Fourier transform of the right-hand side is a finite sum of convolutions of Schwartz functions with polynomial multiples of $\widehat{\widetilde\chi u}$. The standard conic separation estimate preserves rapid decay in a slightly smaller cone around $\xi_0$. Therefore $(x_0,\xi_0)\notin\operatorname{WF}(Pu)$ and $\operatorname{WF}(Pu)\subset\operatorname{WF}(u)$.

2. Boundary-value orientation. Using the forward Fourier phase, verify the wavefront sets of $(t-i0)^{-1}$ and $(t+i0)^{-1}$ stated above.

Solution

The Sokhotski–Plemelj identities are

$$\frac1{t-i0}=\operatorname{PV}\frac1t+i\pi\delta_0, \qquad \frac1{t+i0}=\operatorname{PV}\frac1t-i\pi\delta_0.$$

With $\widehat f(\tau)=\int e^{+i\tau t}f(t)\,\mathrm dt$, they give, before localization,

$$\widehat{(t-i0)^{-1}}=2\pi i\,\theta(\tau), \qquad \widehat{(t+i0)^{-1}}=-2\pi i\,\theta(-\tau).$$

Equivalently, for $\tau>0$ the contour for $e^{+i\tau t}/(t-i0)$ encloses the upper pole and contributes $2\pi i$, while for $\tau<0$ it contributes zero; the other boundary value reverses the roles. Multiplication by a cutoff equal to one near $0$ convolves these step functions with a Schwartz function. The positive tail of the first and negative tail of the second remain nondecaying, while the opposite tails decrease rapidly. Hence the stated one-sided wavefront sets follow.

3. Transposition. If $u^{\mathsf T}(x,x')=u(x',x)$, determine $\operatorname{WF}(u^{\mathsf T})$ from $\operatorname{WF}(u)$.

Solution

Pullback by the swap map gives

$$\operatorname{WF}(u^{\mathsf T}) =\{(x',k';x,k):(x,k;x',k')\in\operatorname{WF}(u)\}.$$

Applied to $\operatorname{WF}(W_2)=\mathcal C^+$, this yields $\operatorname{WF}(W_2^{\mathsf T})=\mathcal C^-$ and places a past-directed covector in the first slot, demonstrating why singular support alone is insufficient.

4. Two products of two-point functions. Use the product criterion to compare $W_2^2$ with $W_2W_2^{\mathrm{op}}$ as distributions on $M\times M$.

Solution

Every element of $\operatorname{WF}(W_2)$ has a future-directed first covector. Two such elements cannot sum to the total zero covector, so the canonical product theorem defines $W_2^2$. The transpose has the opposite cone. At any Hadamard-related pair, $\operatorname{WF}(W_2^{\mathrm{op}})$ therefore contains the negative of a wavefront covector of $W_2$, and the sufficient product criterion fails for $W_2W_2^{\mathrm{op}}$. The general wavefront-set product theorem therefore supplies no canonical product; any extension or prescription requires additional input.

5. A partial-zero tensor covector. Let $u$ be singular at $x$ and let $v$ be smooth but nonzero at $y$. Which wavefront directions may occur in $u\otimes v$ over $(x,y)$?

Solution

Because $v(y)\ne0$, it stays nonzero on a sufficiently small neighborhood of $y$. There, multiplication by $v$ is invertible with smooth inverse, so $u\otimes v$ has the same local wavefront directions as $u\otimes1$. Thus every $(x,k)\in\operatorname{WF}(u)$ gives an actual direction $(x,k;y,0)\in\operatorname{WF}(u\otimes v)$, not merely one permitted by the upper bound. The total product-space covector is nonzero even though its second slot vanishes. Forgetting these terms creates gaps in later kernel-composition and two-point propagation arguments.

Handoffs

A wavefront set is therefore the coordinate-invariant set of directions in which localized Fourier decay fails. It refines singular support enough to retain frequency orientation and to provide practical sufficient checks for when basic operations on singular distributions have a canonical meaning.

Wavefront-set products, pullbacks, and pushforwards turns the operation summary into exact theorems and applies the pullback test to timelike and null curves. Propagation of singularities for hyperbolic fields derives the Hamilton flow behind the null relation. The physical state-selection question remains with Hadamard admissibility and the two-point wavefront criterion.

The chapter’s dependency map, hypothesis table, and failure map place these operations in the full construction of renormalized local fields.

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. Berlin: 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, 191–251. Cham: Springer, 2015. DOI. Open PDF.
• 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.