Hadamard States and the Wavefront-Set Characterization

For a free Klein–Gordon field on a globally hyperbolic spacetime, the Hadamard short-distance expansion is equivalent to one oriented null wavefront set for the two-point distribution. The equivalence uses the field equation, the fixed commutator, propagation of singularities, and global hyperbolicity; the cone condition alone does not construct a positive state.

Required background. Green-hyperbolic operators and causal propagators supplies the field equation and commutator; propagation of singularities for hyperbolic fields supplies global transport.

Helpful background. Higher-point microlocal spectrum conditions gives the hierarchy; no-natural-state results and covariant state spaces explains why no universal preferred state is expected; Hadamard parametrix and short-distance structure, Hadamard admissibility and the two-point wavefront criterion, adiabatic states and WKB order, and Hadamard states for fermion and gauge fields supply the physical variants.

Local parametrix and global wavefront set

Let $P=\Box_g+m^2+\xi R$ on a four-dimensional globally hyperbolic spacetime. A quasifree state has two-point distribution $\omega_2$ satisfying $P_x\omega_2=P_y\omega_2=0$, positivity, and

$$\omega_2(x,y)-\omega_2(y,x)=iE(x,y).$$

In a geodesically convex neighborhood, Hadamard form means

$$\omega_2(x,y)=\frac{1}{8\pi^2} \left(\frac{U(x,y)}{\sigma_\epsilon(x,y)} +V(x,y)\log\frac{\sigma_\epsilon(x,y)}{\ell^2}\right)+W(x,y),$$

where $U,V$ are determined recursively by the geometry and $P$, $W$ is smooth and state dependent, and the $i0$ prescription in $\sigma_\epsilon$ fixes time orientation. The arbitrary length $\ell$ changes only a smooth local term.

Radzikowski’s theorem says this local form is equivalent to

$$\operatorname{WF}(\omega_2)= \{(x,k;y,-k'):(x,k)\sim(y,k'),\ k\text{ future directed}\}.$$

The equivalence of the global Hadamard condition, the distinguished Feynman parametrix, and this wavefront condition is Radzikowski 1996, Theorem 5.1. Locally the parametrix fixes the diagonal singularity. The bisolution equation and propagation theorem extend it along null geodesics. The commutator fixes the difference between the two orientations, while positivity selects the state rather than a formal bisolution.

Ultrastatic ground state

Let $M=\mathbb R\times\Sigma$ be ultrastatic with compact $\Sigma$ and positive spatial operator $A=-\Delta_h+m^2+\xi R$. Spectral calculus defines

$$\omega_2(t,x;t',y)= \left\langle\delta_x, \frac{e^{-iA^{1/2}(t-t')}}{2A^{1/2}}\delta_y\right\rangle.$$

Positivity follows because $A^{1/2}$ is positive; the antisymmetric part is $iE$ by the sine functional calculus. A pseudodifferential high-frequency expansion of $A^{1/2}$ gives the local Hadamard parametrix, and propagation carries the oriented null cone globally. The same orientation is visible from the positive spectral factor $e^{-i\omega(t-t')}$.

This is the model calculation underlying constructing Hadamard states by deformation and gluing: construct a known Hadamard state in a controlled region, transfer Cauchy data, and use propagation. The present theorem certifies the singularity class; the deformation argument supplies existence on the target spacetime.

An independent check compares two Hadamard states. Their two-point functions have the same universal oriented singular part. The wavefront set of the difference is empty after using the field equation and commutator, so the difference is smooth. This is why local Wick expectations have state differences without ultraviolet singularities.

The proof has a useful local-to-global structure. In a convex normal neighborhood, the transport equations determine $U$ and the coefficients of $V$ from $P$ and the geometry, while the $i0$ boundary value determines the directed covectors. Subtracting this local parametrix from a candidate two-point function gives a kernel smooth near the diagonal. The bisolution equation then propagates that smoothness along every null bicharacteristic in each argument. A Cauchy surface meets every inextendible causal curve, so global hyperbolicity prevents an uncontrolled branch from entering from an unseen boundary. Finally, the fixed antisymmetric part rules out adding the opposite orientation without changing the commutator. Each hypothesis has a distinct role: local geometry fixes the singular model, propagation globalizes it, and positivity remains a separate state axiom.

Adversarial image singularity

Add to $\omega_2$ a bisolution whose singularity follows an image null geodesic with the wrong first-slot time orientation, together with its transpose chosen so the antisymmetric commutator remains $iE$. The field equation and commutator can still hold, but the new past-directed component violates the Hadamard wavefront set. Point splitting against the local parametrix will not remove it.

Conversely, a distribution with the correct cone and field equation is not automatically a state: it may fail positive type. The theorem characterizes Hadamard singularity structure among valid two-point functions; it is not a positivity or state-existence theorem. Finite adiabatic order can also give a weaker Sobolev wavefront condition without the full smooth Hadamard remainder.

Exercises

1. Smooth state differences. Why does changing the Hadamard length scale $\ell$ not change the wavefront set?

Solution

Changing $\ell$ adds a multiple of the smooth coefficient $V(x,y)$. Adding a smooth kernel does not change a wavefront set.

2. Positivity check. Show that the ultrastatic ground-state kernel is of positive type.

Solution

After Fourier transformation in time and spectral decomposition of $A$, $\omega_2(\bar f,f)$ is an integral of absolute squares weighted by the positive measure $(2\omega)^{-1}\delta(p_0-\omega)$. Hence it is nonnegative.

References

• Fulling, Stephen A., Frank J. Narcowich, and Robert M. Wald. “Singularity Structure of the Two-Point Function in Quantum Field Theory in Curved Spacetime, II.” Annals of Physics 136 (1981): 243–272. 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.