Hadamard Admissibility and the Two-Point Wavefront Criterion

The microlocal Hadamard condition identifies not only where a two-point distribution is singular but also which null cotangent directions carry the singularity. For a scalar Klein–Gordon state on a time-oriented globally hyperbolic spacetime, it is equivalent to the local Hadamard form. It is an ultraviolet admissibility condition: it neither selects a unique state nor supplies positivity by itself.

Required background. Hadamard Parametrix and Short-Distance Structure gives the local singular form. Singular Support and Wavefront Sets defines directional singularity. Products, Scaling Degree, and Distribution Extensions explains why wavefront information controls products.

Helpful background. Operator Algebras and Positive Functionals keeps admissibility distinct from state positivity. Microcausality and Relativistic Compatibility supplies the causal interpretation.

The two-point wavefront set

Let $\mathcal N^+$ denote the nonzero future-directed null covectors, with “future” fixed by the spacetime time orientation and the positive-frequency convention $e^{-i\omega t}$. Write

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

when $x$ and $x'$ lie on one null geodesic and $k'$ is the parallel transport of $k$ along it. The two-point function of a Hadamard state satisfies

$$\operatorname{WF}(\omega_2) = \left\{ (x,k;x',-k')\in\dot T^*(M\times M): (x,k)\sim(x',k'),\ k\in\mathcal N^+ \right\}.$$

The dot removes the zero covectors. The minus sign on the second covector follows from the dependence on $x-x'$ in the flat-space positive-frequency kernel. Reversing the order of the fields reverses the orientation. The antisymmetric part is fixed by

$$\omega_2-\omega_2^{\mathsf T}=-iE, \qquad E=G_{\mathrm{ret}}-G_{\mathrm{adv}},$$

or, in the alternate notation $\Delta=-E$, by $\omega_2-\omega_2^{\mathsf T}=i\Delta$.

Radzikowski proved that, for Klein–Gordon two-point functions under the stated hypotheses, this microlocal spectrum condition is equivalent to the Hadamard form Radzikowski 1996, Theorem 5.1. The statement is global in phase space even though the singular coefficients are locally geometric.

A disciplined test sequence

For a proposed scalar two-point kernel, test in this order:

1. verify the bidistribution and bisolution properties;
2. verify reality, the canonical antisymmetric part, and positivity;
3. find the characteristic covectors allowed by the field equation;
4. keep only the future-directed branch in the first argument;
5. check null-geodesic pairing and parallel transport to the second argument.

The wavefront calculation does not replace the first two steps. A distribution can have the correct wavefront set and still fail positivity or the field equation; adding a smooth term never repairs such a failure automatically.

First application: orient the Minkowski singularity

For the Minkowski vacuum,

$$\omega_2(x,x')=int\frac{\mathrm d^{d-1}\mathbf k}{(2\pi)^{d-1}2\omega_{\mathbf k}} e^{-i\omega_{\mathbf k}(t-t')+i\mathbf k\cdot(\mathbf x-\mathbf x')},$$

with $\omega_{\mathbf k}>0$. High-frequency stationary phase places the first covector on the future null cone in the convention above, while translation invariance supplies the opposite second covector. Curvature replaces straight null rays by null geodesics and transports the cotangent direction; it does not introduce both time orientations into a Hadamard two-point function.

This check catches a common error: the time-reversed kernel $\omega_2(x',x)$ is also singular on null-related pairs, but its first covector is past-directed. Singular support alone cannot distinguish the two.

Same wavefront set, different states

If $S(x,x')$ is a smooth symmetric bisolution small enough to preserve positivity, then

$$\omega'_2=\omega_2+S$$

has exactly the same wavefront set as $\omega_2$. Yet its Wick-square difference is proportional to $S(x,x)$, and other local expectation values depend on derivatives of $S$ at coincidence. Thus two physically distinct Hadamard states pass the identical wavefront test. Claiming that the criterion selects a vacuum is the declared adversarial failure; the strongest justified statement is common ultraviolet admissibility.

Limits

Global hyperbolicity and normally hyperbolic dynamics are not cosmetic assumptions. Boundaries require specified boundary conditions and may generate reflected singularities; gauge systems require constraints or a subsidiary construction; non-globally-hyperbolic regions need additional data. The scalar equality above should not be copied unchanged into those settings.

Domain and failure conditions

The wavefront criterion is the ultraviolet checkpoint in the construction map. Its placement before global construction and physical selection is consequential: the future-null relation certifies the allowed singular directions but does not establish positivity, existence on every background, or a preferred vacuum.

The microlocal criterion is equivalent to local Hadamard form for the stated Klein–Gordon setting; selection remains a later, independent step. Schematic; not to scale.

The failure map explains two complementary downgrades. Extra or reversed wavefront directions trigger the regularity witness, while a correct wavefront set with a negative quadratic form triggers the positivity witness. Two kernels with the same permitted set can still represent different states.

Microlocal and algebraic controls must both pass; success of either one alone licenses only its own conclusion. Schematic; not to scale.

The full comparison appears in Domain and failure conditions.

Handoffs

Propagation of the Hadamard Property explains why the condition extends from Cauchy data. Hadamard States for Fermion and Gauge Fields adapts it to other field complexes. The proof, higher-point microlocal spectrum condition, and precise bundle-valued statements continue in Hadamard States and Wavefront Characterization.

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.
• 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.