Higher-Point Microlocal Spectrum Conditions

The higher-point microlocal spectrum condition replaces flat-space joint momentum support by graph-labeled cones in $T^*M^n$. Directed edges carry future-oriented causal covectors, and the covector at each vertex is the signed sum of incident edge covectors. For quasifree Hadamard fields, Wick’s theorem embeds every $n$-point wavefront set in this cone.

Required background. Microlocal calculus for quantum fields supplies oriented cones; propagation of singularities for hyperbolic fields supplies the causal transport.

Helpful background. Wightman functions and spectral support gives the Minkowski antecedent; wavefront-set products, pullbacks, and pushforwards supplies the tensor-product calculus.

Graph cones on curved spacetime

Fix ordered points $x=(x_1,\ldots,x_n)$. Take a finite directed graph with vertices $1,\ldots,n$, immerse each edge as a causal curve between its endpoint points, and attach a parallel covector field $k_e$ along it. Reverse orientation reverses the covector. For an edge directed from a lower-numbered to a higher-numbered vertex, require the attached covector to be future directed. Define

$$k_i=\sum_{e:s(e)=i}k_e(x_i)-\sum_{e:t(e)=i}k_e(x_i).$$

The union of all nonzero configurations $(x_1,k_1;\ldots;x_n,k_n)$ obtained this way is the cone $\Gamma_n$. A state satisfies the microlocal spectrum condition when $\operatorname{WF}(\omega_n)\subset\Gamma_n$ for every $n$. The precise graph immersion and orientation rules are Brunetti, Fredenhagen, and Köhler 1996, Definition 4.1, pp. 10–11.

The condition is local and covariant: embeddings push forward points and cotangent data without choosing a global energy generator. It is stable under tensor products and the contractions allowed by the cone calculus. It is stronger than symmetry of singular support because it remembers edge orientation.

Quasifree Hadamard n-point functions

For a centered quasifree scalar state, odd distributions vanish and

$$\omega_{2r}(x_1,\ldots,x_{2r})= \sum_{\text{pairings }P}\prod_{(i,j)\in P}\omega_2(x_i,x_j).$$

Each factor has the Hadamard wavefront relation: a null edge between $x_i$ and $x_j$, with future orientation determined by their ordering. A pairing is therefore a disjoint graph whose vertices have degree one. The tensor-product theorem combines its covectors without generating a zero total tuple, and the finite sum stays within the union of the same graph cones. Hence every quasifree Hadamard hierarchy satisfies $\mu$SC; this is Brunetti, Fredenhagen, and Köhler 1996, Proposition 4.3, pp. 10–11.

For example, a four-point term $\omega_2(x_1,x_3)\omega_2(x_2,x_4)$ is represented by two oriented edges, $1\to3$ and $2\to4$. At vertex one the covector is future oriented along the first null curve; at vertex three it is its transported negative. Summing the three pairings gives the full four-point distribution and stays inside $\Gamma_4$.

This derivation supplies the higher-point control used when applying Hadamard admissibility and the two-point wavefront criterion to composite observables. For quasifree states the two-point condition and Wick factorization do the work. In a nonquasifree theory, a good two-point function does not determine higher truncated distributions; they need independent bounds.

An independent flat-space check sums vertex covectors. Every internal edge contributes once with each sign, so $\sum_i k_i=0$, the local analogue of translation-invariant momentum conservation. At the earliest nonzero vertex in the ordering, all outgoing edge covectors are future directed, reproducing the spectrum-condition orientation.

Truncated functions expose the content beyond quasifreeness. Writing $\omega_n$ as sums of products of truncated distributions partitions the vertex set into connected blocks. If each truncated block obeys its corresponding graph cone and the product transversality conditions hold, joining the block graphs gives the bound for $\omega_n$. The implication is one-way: the full hierarchy can satisfy the cone bound without being determined by $\omega_2$, because connected three- and higher-point distributions may carry additional allowed graphs. In particular, the quasifree proof above is an existence proof for one large class, not a reconstruction theorem for every state satisfying the microlocal spectrum condition.

Adversarial edge reversal

Reverse one edge in the four-point graph while leaving its endpoints and singular support unchanged. The earliest incident vertex now receives a past-directed covector. The configuration can no longer belong to $\Gamma_4$, even though every pair of base points is still null related and the distribution may remain permutation symmetric at the level of singular support. This is exactly why an unoriented “singular only on causal graphs” rule is too weak.

The converse boundary is substantial. Membership in every $\Gamma_n$ does not by itself prove positivity, the field equation, locality, or existence of a state. Moreover, the graph condition bounds possible singularities; it need not say that every allowed direction actually occurs.

Exercises

1. Three-point function. What does quasifreeness imply for a centered scalar state’s three-point distribution?

Solution

It vanishes. Its wavefront set is therefore empty and trivially contained in $\Gamma_3$; this says nothing about a nonquasifree state’s three-point function.

2. Vertex conservation. Prove $\sum_i k_i=0$ for any immersed graph.

Solution

Each edge covector appears with a plus sign at its source and, after parallel transport, with a minus sign at its target. Summing over all vertices cancels every edge contribution.

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.
• 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.