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 orientation distinguishes positive-frequency two-point functions from their reversals, controls products and restrictions, and replaces global momentum support by local geometric data.

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.

Singular points with cotangent directions

Let $u\in\mathcal D'(M)$ and choose local coordinates near $x$. A pair $(x,k)\in T^*M\setminus0$ is absent from $\operatorname{WF}(u)$ when there are a cutoff $\chi$ with $\chi(x)\ne0$ and a conic neighborhood $\Gamma$ of $k$ such that, for every $N$,

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

The complement is the wavefront set. It is closed and conic in each fiber, its projection to $M$ is the singular support, smooth multiplication does not enlarge it, and differentiation can only preserve or remove directions. The coordinate definition is invariant because changes of coordinates act on covectors by the cotangent map. Brunetti, Fredenhagen, and Köhler 1996, Definitions 2.2–2.4, pp. 4–5 gives both the local Fourier and intrinsic manifold formulations.

For kernels on $M\times M$, signs in the second covector matter. It is convenient to say $(x,k)\sim(x',k')$ when $x,x'$ lie on one null geodesic and $k'$ is the parallel transport of $k$ along it. This relation is geometric; it does not assume a Killing field or global Fourier transform.

The Minkowski vacuum two-point distribution

For the massive scalar vacuum in Minkowski space,

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

Its momentum support lies on the future mass shell, but its high-frequency characteristic directions are null because the principal symbol of $\Box+m^2$ is $g^{ab}k_ak_b$. Consequently

$$\operatorname{WF}(W_2)= \{(x,k;x',-k'):(x,k)\sim(x',k'),\ k\text{ future directed}\}.$$

The set includes the coincident limit with nonzero null $k$. The mass changes the smooth infrared and tail behavior but not this ultraviolet cone. Radzikowski’s characterization makes this oriented null relation the two-point Hadamard condition; Brunetti, Fredenhagen, and Köhler 1996, Definition 3.2 and Eq. (3), pp. 8–9.

The Pauli–Jordan commutator $E=E^- -E^+$ is the antisymmetric difference $W_2-W_2^{\mathrm{op}}$. Its singular support is again the null relation, but its wavefront set contains both time orientations. Thus projection to singular support loses exactly the spectral information that distinguishes $W_2$ from its transpose. This calculation is the microlocal input to Hadamard admissibility and the two-point wavefront criterion.

An independent check comes directly from momentum support. Localizing $W_2$ convolves its future-shell Fourier measure with a rapidly decreasing function; it can create small tails but cannot create a nondecaying past-directed cone. Transposing the arguments reverses the cone. Adding the two orientations gives the commutator’s unoriented characteristic relation.

There is also a coordinate-invariance check that is useful in curved spacetime. If $F:U\to V$ is a diffeomorphism, localization followed by a change of variables sends a high-frequency covector $\eta$ on $V$ to ${}^tdF\,\eta$ on $U$. Hence

$$\operatorname{WF}(F^*u)= \{(x,{}^tdF_x\eta):(F(x),\eta)\in\operatorname{WF}(u)\}.$$

The null character and future orientation in the Hadamard relation are therefore invariant under orientation- and time-orientation-preserving coordinate changes. A rule phrased in terms of a particular coordinate frequency $k_0>0$ would fail this check; the geometric rule instead evaluates whether the metric dual of $k$ is future directed. This is why the cotangent formulation survives on a spacetime without a preferred time coordinate.

It also makes every subsequent product and restriction criterion intrinsically meaningful on overlapping coordinate patches.

Adversarial loss of orientation

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. Microlocal admissibility is a necessary structural test for the named operation, not a construction of a positive functional.

Exercises

1. Smooth functions. Show that $\operatorname{WF}(f)=\varnothing$ for $f\in C^\infty_c(M)$.

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.

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

Solution

Pullback by the swap map sends $(x,k;x',k')$ to $(x',k';x,k)$. Applied to the Hadamard set, this places a past-directed covector in the first slot, demonstrating why singular support alone is insufficient.

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.
• 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.