The Microlocal Spectrum Condition

The microlocal spectrum condition replaces a global positive-energy spectrum by a local statement about the singular cotangent directions of a two-point distribution. For a Klein–Gordon state, the Hadamard condition is equivalent to a precise pairing of null covectors: the first covector is future-directed, the second is its negative after parallel transport along a null geodesic. This theorem records both where singularities occur and their frequency orientation.

Required background. Partial Differential Equations and Green Functions supplies distributional bisolutions, characteristic covectors, and propagation of singularities. The bounded local bridge used here is the real-principal-type fact that singularities of $P$ travel along null bicharacteristics; the full pseudodifferential calculus is not redeveloped.

Helpful background. Flat-Spacetime Quantum Field Theory provides the positive-energy cone and the $e^{-i\omega t}$ frequency convention, Causality and Global Structure defines time orientation, null geodesics, and Cauchy surfaces, and Quantum Fields in Curved Spacetime provides the volume-level field and sign conventions.

What a wavefront set records

For a distribution $u$ on a manifold, $(x,k)$ is absent from $\operatorname{WF}(u)$ if, after localizing near $x$, its Fourier transform decays faster than every inverse power in an open cone around the nonzero covector $k$. The wavefront set therefore refines singular support:

$$\operatorname{sing\,supp}u =\pi\big(\operatorname{WF}(u)\big),$$

but it also retains the directions in which rapid Fourier decay fails. For this definition use the localized Fourier transform $\widehat u(k)=\int e^{-ik_\mu x^\mu}u(x)\,d^dx$ in a coordinate chart. A distribution may be singular at one point while being regular in some cotangent directions there. For example, $\delta(x^1)$ on $\mathbb R^2$ is singular along the line $x^1=0$, but its wavefront covectors are normal to that line: $(k_1,0)$ with $k_1\ne0$. It is smooth in the tangential direction. The null relation below is the spacetime analogue of retaining this directional information.

This localization-and-conic-decay definition, together with the product criterion used later in curved-spacetime field theory, is reviewed in Hollands and Wald 2015, arXiv v2, Appendix A, PDF.

For a two-point distribution $W_2\in\mathcal D'(M\times M)$, a wavefront element is a pair

$$(x,k;x',k')\in T^*(M\times M)\setminus\{0\}.$$

The distinction between $k'$ and $-k'$ will be decisive. Ordinary singular support cannot see it, which is why a statement merely saying “singular on the light cone” is weaker than the Hadamard condition.

Geometric relation between null covectors

Write $(x,k)\sim(x',q)$ when $x$ and $x'$ lie on one null geodesic and the nonzero null covector $q$ is obtained by parallel-transporting $k$ along that geodesic. If more than one null geodesic joins the points, the relation includes every corresponding transported pair.

A causal covector $k$ is called future-directed here when its metric dual $k^\sharp$ is future-directed. With signature $(-,+,+,+)$, a Minkowski positive-frequency phase

$$e^{-i\omega(t-t')+i\boldsymbol p\cdot(\boldsymbol x-\boldsymbol x')}$$

has first covector $k=(-\omega,\boldsymbol p)$, whose metric dual is future-directed, and second covector $-k$. This fixes the orientation convention without relying on an ambiguous sign for the coordinate component $k_0$.

Define

$$\mathcal C^+ =\left\{ (x,k;x',-q): (x,k)\sim(x',q),\ k^\sharp\text{ future-directed} \right\}.$$

Both $k$ and $q$ are null. The minus sign belongs to the covector in the second slot of $W_2$; it is not part of the parallel-transport relation itself.

Radzikowski’s equivalence theorem

Let $(M,g)$ be a four-dimensional globally hyperbolic spacetime, let

$$P=\Box-m^2-\xi R,$$

and let $W_2$ be the two-point distribution of a scalar-field state. The relevant hypotheses are

$$P_xW_2=P_{x'}W_2=0, \qquad W_2(x,x')-W_2(x',x)=iE(x,x'),$$

where the inherited convention is $E=G_{\mathrm{ret}}-G_{\mathrm{adv}}$ and its bidistribution is oriented so that $E(f,g)=\int_M f(Eg)\,d\mathrm{vol}_g$. Positivity is already contained in the assumption that $W_2$ comes from a state.

Under these hypotheses, the state is Hadamard if and only if

$$\boxed{\operatorname{WF}(W_2)=\mathcal C^+.}$$

This is the scalar specialization of Radzikowski 1996, Theorem 5.1, p. 544, translated from that paper’s operator signs to the convention above. A concise restatement with the paired-covector formula appears in Brunetti, Fredenhagen, and Köhler 1996, Definition 3.2, p. 640.

One sometimes sees inclusion $\operatorname{WF}(W_2)\subset\mathcal C^+$ used as the microlocal condition. For a bisolution with the fixed antisymmetric part $iE$, it strengthens to equality: the causal propagator has both null orientations, while transposition exchanges $\mathcal C^+$ and $\mathcal C^-$. The commutator therefore prevents the required branch from being absent.

Why local Hadamard form implies the condition

The proof begins in a convex normal neighborhood. The boundary values

$$(\sigma+i0\,\tau)^{-1}, \qquad \log(\sigma+i0\,\tau)$$

in the Hadamard parametrix have wavefront covectors normal to the null cone, with the first one future-directed because of the declared $i0$ sign. Multiplication by the smooth transport coefficients $U$ and $v_n$ does not create new wavefront directions, and addition of the state-dependent smooth remainder changes none. This computes $\mathcal C^+$ locally.

Next, the propagation-of-singularities theorem carries each characteristic covector along the null Hamilton flow. Applying it in both variables extends the local relation along every null geodesic. Global hyperbolicity supplies Cauchy surfaces and excludes causal pathologies that would invalidate this propagation argument. The detailed implication and its correction of an earlier flat-only step are given in Radzikowski 1996, proof of Theorem 5.1, pp. 544–550.

Why the wavefront condition implies Hadamard form

For the converse, combine $W_2$ with the advanced and retarded fundamental solutions to form its time-ordered, or Feynman, distribution. The assumed wavefront set gives exactly the distinguished Feynman relation of Duistermaat and Hörmander. Distinguished parametrices for the same normally hyperbolic operator are unique modulo a smooth kernel. Comparing this distribution with a locally constructed Hadamard parametrix therefore shows that their difference is smooth, which recovers the local Hadamard expansion.

This proof architecture explains why each hypothesis matters. The Klein–Gordon equation confines singularities to the characteristic set and propagates them. The commutator fixes the antisymmetric part and hence both available orientations. Global hyperbolicity provides the Cauchy development used to pass between local and global statements. Positivity turns the admissible two-point distribution into a state rather than merely a parametrix.

The theorem has a limitation: it characterizes ultraviolet admissibility, not a preferred vacuum. Any two Hadamard two-point functions differ smoothly, yet their smooth parts can encode different temperatures, particle interpretations, or long-range correlations. It also does not automatically extend the two-point formula to interacting higher-point functions; their microlocal spectrum conditions require a graph-based generalization.

Orientation check and adverse case

In Minkowski spacetime, the Fourier representation

$$W_2(x,x') =\int\frac{d^3\boldsymbol p}{(2\pi)^3\,2\omega_{\boldsymbol p}} e^{-i\omega_{\boldsymbol p}(t-t') +i\boldsymbol p\cdot(\boldsymbol x-\boldsymbol x')}$$

has $\omega_{\boldsymbol p}=\sqrt{\lvert\boldsymbol p\rvert^2+m^2}$. Its Fourier support lies on the future mass shell, while its wavefront directions are the null high-frequency limits of that shell; a massive field does not have a timelike Hadamard wavefront set. The second covector is the negative of the first. Curvature changes parallel transport and focusing but not this local orientation.

If one reverses the cone and uses $\mathcal C^-$, the transpose $W_2(x',x)$ is the model. It still solves the Klein–Gordon equation, but its antisymmetric part is $-iE$, so it represents the opposite canonical commutator. Keeping the original commutator while reversing only the wavefront orientation is inconsistent. This adverse case shows why “one future and one past covector” is incomplete unless their order is stated.

Check your understanding

Assume a bisolution $W_2$ has $\operatorname{WF}(W_2)\subset\mathcal C^+$ and $W_2-W_2^T=iE$. Explain why its wavefront set cannot be empty on an open segment of a null bicharacteristic that occurs in $\operatorname{WF}(E)$.

Solution

Transposition interchanges the positive and negative frequency relations. If the future-oriented branch of the two-point distribution were absent on that segment, the corresponding past-oriented branch of its transpose would also be absent after exchanging the arguments. Their difference would consequently be microlocally smooth in both orientations there. The commutator identity says that this difference is the causal propagator multiplied by the imaginary unit. The causal propagator, however, has both null orientations in its wavefront set, and propagation of singularities carries them along the entire bicharacteristic. The proposed gap therefore contradicts the commutator identity. This forces the allowed future-oriented branch to be present, after which the propagation theorem carries it along the null ray. Notice that the argument uses three ingredients together: the one-sided inclusion, transposition, and the fixed antisymmetric part. The inclusion alone would not prove equality, because a smoother bisolution could have a strictly smaller wavefront set.

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, no. 3 (1996): 633–652. doi:10.1007/BF02099626. Open PDF.
• Hollands, Stefan, and Robert M. Wald. “Quantum Fields in Curved Spacetime.” Physics Reports 574 (2015): 1–35. doi:10.1016/j.physrep.2015.02.001. 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, no. 3 (1996): 529–553. doi:10.1007/BF02100096.