Quantum States and Hadamard Structure

Quantum field theory in curved spacetime needs a notion of state that does not presuppose one preferred Fock space, together with an ultraviolet condition strong enough to define local observables. This chapter builds that structure in stages: positive functionals give states, the Hadamard parametrix fixes their universal singularity, microlocal analysis makes the condition coordinate-free, and symmetry or other physical data can then select within the admissible class.

Helpful background. Flat-Spacetime Quantum Field Theory provides vacuum, Fock, and positive-frequency examples; Partial Differential Equations and Green Functions supplies distributional solutions, causal propagators, and characteristic propagation; Causality and Global Structure defines time orientation, Cauchy surfaces, and stationary flows; and Quantum Fields in Curved Spacetime provides the volume-level conventions and physical scope.

Enter the state problem

On a generic globally hyperbolic spacetime there need not be a global timelike Killing field, a conserved positive Hamiltonian, or a preferred positive-frequency splitting. Beginning with a Hilbert space chosen by analogy with Minkowski spacetime would therefore insert physical structure that the geometry may not possess. The algebraic approach begins instead with local observables and identifies a state with their consistent expectation-value functional.

That freedom is not unlimited. A positive normalized functional may have pathological short-distance correlations. For a free scalar field, Hadamard regularity supplies the ultraviolet restriction: the leading singularity is fixed by local geometry and the field operator, while the state contributes a smooth remainder. This is the regularity needed before point splitting, Wick powers, or perturbative local products can be developed.

Throughout the chapter,

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

with signature $(-,+,+,+)$ and positive-frequency convention $e^{-i\omega t}$. When Synge’s world function appears, it is one half of the signed squared geodesic interval. These choices fix the normalization and orientation of every $i0$ prescription and wavefront relation.

Route through the chapter

Read the six pages in order, because each changes the status of the state question.

1. States as Positive Linear Functionals proves the GNS construction: positivity produces a Hilbert space, representation, and cyclic vector without choosing them in advance.
2. The Hadamard Parametrix derives the geometry-fixed short-distance kernel, including its transport equations, normalization, and smooth state-dependent remainder.
3. The Microlocal Spectrum Condition states Radzikowski’s paired-null-covector criterion and explains why the first covector must have the future orientation.
4. Existence and Deformation of Hadamard States uses a known reference state, controlled metric deformation, time-slice isomorphisms, and propagation of singularities to prove nonemptiness.
5. Ground, KMS, and Stationary States shows how a complete stationary automorphism group permits spectral ground states and thermal KMS states at a declared inverse temperature.
6. Adiabatic and State-Selection Criteria compares finite adiabatic order, full Hadamard regularity, symmetry, averaged-energy, instantaneous, and detector-based prescriptions.

The order prevents two common reversals. A preferred-looking mode prescription must first be shown to define a positive state, and a positive state must then pass the ultraviolet condition appropriate to the intended observables. Selection comes after both tests.

Three logically different questions

The chapter separates three layers that are often compressed into the word “vacuum.”

Layer	Mathematical test	What it establishes	What it does not establish
Algebraic state	Normalization and positivity on the observable algebra	Consistent expectation values and a GNS representation	Short-distance regularity or a preferred representation
Ultraviolet admissibility	Hadamard local form or the equivalent microlocal spectrum condition	Universal singular structure and smooth differences between admissible states	A unique state, particle interpretation, or temperature
Physical selection	Ground spectrum, KMS analyticity, symmetry, averaged energy, boundary data, or an operational protocol	A distinguished state relative to specified extra structure	A geometry-independent universal vacuum

The first layer is represented by the GNS theorem summarized in Hollands and Wald 2015, arXiv v2, §2.1, pp. 12–15, PDF. The second is sharpened by Radzikowski 1996, Theorem 5.1, p. 544: for the four-dimensional Klein–Gordon two-point distribution with the field equation and commutator hypotheses, local Hadamard form is equivalent to the future-oriented paired-null wavefront set.

Existence does not collapse the third layer. The deformation result of Fulling, Narcowich, and Wald 1981, pp. 243–272 shows that globally hyperbolic spacetimes admit Hadamard states under the stated free-field construction, but the choices made during deformation remain in the smooth state-dependent part. There can therefore be many admissible states even when none deserves the unqualified name “the vacuum.”

Checks that travel across the chapter

Several checks recur in different language.

Positivity check. Every proposed covariance must produce nonnegative quadratic forms. Correct singular directions alone do not make a distribution a state.

Normalization and orientation check. With half-squared world function $\sigma$, the four-dimensional leading term is $1/(8\pi^2\sigma_\epsilon)$. The sign of $i\epsilon[T(x)-T(x')]$ must reproduce $e^{-i\omega(t-t')}$ in Minkowski spacetime. The same sign places the first wavefront covector in the future cone and the second in the opposite transported cone.

Propagation check. The Hadamard condition established near one Cauchy surface propagates along null bicharacteristics only because the two-point distribution is a Klein–Gordon bisolution on a globally hyperbolic spacetime. An arbitrary algebra isomorphism or a spacetime with a chronology horizon does not supply that conclusion.

Order check. Finite adiabatic order controls only a finite range of Sobolev singularities. Junker and Schrohe’s definition and thresholds make this quantitative in Junker and Schrohe 2002, Definition 3.2 and Theorem 4.7, pp. 1122 and 1131, but no fixed finite order becomes full Hadamard regularity.

What the chapter does and does not decide

The conclusions apply to the free scalar field on smooth globally hyperbolic spacetimes, with dimension and operator stated where the Hadamard coefficients are used. Gauge constraints, spin structures, timelike boundaries, and interacting higher-point functions require their own algebras and microlocal hypotheses. The deformation proof does not cross a topology-changing or chronology-violating region, and the KMS definition does not apply without a complete automorphism group.

Within scope, the hierarchy is decisive. Positivity answers whether expectation values form a state. Hadamard regularity answers whether its ultraviolet singularities are admissible. Ground, KMS, low-energy, or operational criteria answer which additional physical data favor one admissible state. This hierarchy prepares the next step: subtracting the universal Hadamard singularity to define renormalized local observables while retaining the smooth state dependence.

Review the chapter

Let $W_2$ be a positive two-point distribution for a real scalar field on a four-dimensional globally hyperbolic spacetime. It is a bisolution of $P$ and has antisymmetric part $iE$. Use it to define a centered quasifree state by Wick pairing. Near one Cauchy surface it differs smoothly from the declared Hadamard parametrix. Explain what follows globally, what GNS supplies, and what additional information would be needed to call the state ground or KMS. Finally, suppose the state is known only to have finite adiabatic order. Which conclusion must be withdrawn?

Solution

The local smooth difference gives the future-oriented Hadamard wavefront relation near the Cauchy surface. Because the two-point distribution is a bisolution and the spacetime is globally hyperbolic, propagation of singularities carries that relation along every null bicharacteristic; the fixed commutator supplies the opposite orientation under transposition. The state is therefore globally Hadamard. Positivity and normalization allow the GNS construction to produce a Hilbert space, a representation on a dense invariant domain, and a cyclic vector reproducing all expectation values. To call it ground one must specify a complete stationary automorphism group, its generator sign, continuity, and nonnegative generator spectrum. To call it KMS one must additionally specify an inverse temperature and verify the analytic strip boundary relation. If only finite adiabatic order is known, the global full-Hadamard conclusion must be withdrawn: finite Sobolev control may suffice for declared finite operations but does not exclude all unwanted high-order singular directions.

The quasifree assumption matters to the reconstruction from $W_2$: it specifies all higher moments. A two-point function alone does not determine a general non-Gaussian state, even though it still tests that state’s two-point Hadamard property.

References

• Fulling, Stephen A., Francis 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, no. 2 (1981): 243–272. doi:10.1016/0003-4916(81)90098-1.
• 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.
• Junker, Wolfgang, and Elmar Schrohe. “Adiabatic Vacuum States on General Spacetime Manifolds: Definition, Construction, and Physical Properties.” Annales Henri Poincaré 3, no. 6 (2002): 1113–1181. doi:10.1007/s000230200001. 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.