States, Hadamard Structure, and Microlocal Control

A state of a quantum field on curved spacetime must pass several logically independent tests. It must be a positive functional on the field algebra; it may or may not admit a useful Fock representation; its short-distance singularities must be controlled if local composite observables are to be defined; and additional geometric or operational data are needed to prefer it over other admissible states. This chapter separates those questions and gives a route from candidate two-point data to a defensible physical interpretation.

Helpful background. Singular Support and Wavefront Sets supplies the directional notion of singularity used in the Hadamard criterion. Distributional Kernels and Distributions on Manifolds explains why two-point functions are bidistributions. Curved Spacetimes, Cauchy Surfaces, and Global Hyperbolicity and Covariant Algebraic Quantization and Fock Realizations provide the causal and algebraic setting.

State questions are not one question

For a real Klein–Gordon field, a useful assessment proceeds in the following order.

1. Algebraic state: does the proposed two-point distribution obey the field equation, canonical commutator, reality, continuity, and positivity?
2. Representation: which GNS representation does it generate, and is a comparison with another representation global, local, unitary, or merely quasiequivalent?
3. Ultraviolet admissibility: is its singular part Hadamard, equivalently—under the standard free-field hypotheses—does its wavefront set have the required future-directed null orientation?
4. Selection: is the state distinguished by a stationary generator, a KMS condition, a sampling prescription, boundary data, or a preparation protocol?

The third test constrains the singular part of the state, not its smooth state-dependent remainder. Consequently the Hadamard condition permits many states. This is essential: it supplies a common ultraviolet class in which Wick polynomials and stress tensors can be compared without pretending that geometry supplies a universal vacuum. The distinction between admissibility and selection is central to the algebraic treatment of curved-spacetime QFT Fewster and Verch 2015, §§ 4–5 and to the Hadamard framework reviewed by Kay 2023, §§ 2–3.

Domain and failure conditions

This semantic table is the chapter’s canonical comparison. “Local comparison” refers to normality or quasiequivalence on bounded local algebras under the applicable theorem hypotheses; it never means global unitary equivalence.

State class	Existence or construction data	Uniqueness	Symmetry or selector	Ultraviolet regularity	Local comparison	Decisive failure or downgrade
Scalar quasifree	Positive covariance, field equation, reality, continuity, and antisymmetric part $-iE$	Generally many; purity is a separate condition	None required	Unrestricted until an adiabatic or Hadamard condition is added	Must be tested from restricted covariances	One negative quadratic form removes statehood
Scalar Hadamard	Positive bisolution with the future-null two-point wavefront relation	Highly nonunique; smooth bisolutions remain	None required	Full Hadamard class	Quasifree Hadamard representations are locally quasiequivalent under the standard hypotheses	Extra wavefront directions or a nonsmooth difference from a Hadamard reference
Finite-order adiabatic	Normalized WKB Cauchy data or the stated Sobolev wavefront bound	Nonunique; depends on initialization and order convention	Usually exploits spatial homogeneity	Controlled only through the declared finite order	Available only above the regularity thresholds of the relevant theorem	Treating the order label as full Hadamard form or as an observable subtraction
Ground	A complete stationary evolution with nonnegative GNS generator	Sometimes unique within a stated regular quasifree class	Invariant under the chosen normalized time flow	Hadamard under additional static or stationary spectral hypotheses	As for the resulting Hadamard representation	Zero modes, lack of a lower spectral bound, ergoregions, or superradiance
KMS	A star-automorphism flow and KMS strip analyticity at declared $\beta$	Thermal phases can be nonunique	Equilibrium relative to that flow	Hadamard under the applicable stationary free-field hypotheses	Phase and sector must be declared; Hadamard quasifree cases have the usual local comparison	Rescaling the flow, infrared divergence, or incompatible horizon conditions
Euclidean or Bunch–Davies	Analytic continuation and regular Euclidean data in the supported de Sitter field sector	Distinguished in the relevant invariant Hadamard class, not for every mass or coupling	Full de Sitter invariance	Hadamard	Locally comparable to other Hadamard states	The massless minimally coupled zero mode or replacement by a non-Euclidean alpha state
State of low energy	A Robertson–Walker trial class, observer, and smooth compact sampling function	A minimizer relative to those declared inputs, not universally	Sampled-energy selection, not generally stationary	Hadamard in Olbermann’s setting	The usual Hadamard local comparison applies	Pointlike sampling limit, changed observer, or an enlarged trial class
Fermionic Hadamard	Dirac bisolutions with CAR complementarity, $0\le C^\pm\le1$, and spin structure	Generally nonunique; zero-mode occupation may remain	Optional stationary or geometric input	Bundle-valued Hadamard relation	Requires the CAR version of the local theorem	Missing spin structure, wrong first-order constraint, or failed CAR positivity
Gauge-field Hadamard	Green-hyperbolic gauge complex, constraints or BRST quotient, physical commutator, and physical positivity	Gauge representatives are nonunique; physical sectors can also differ	Optional gauge-invariant geometric input	Bundle-valued Hadamard control on appropriate representatives or observables	Topology, boundary conditions, and superselection sector must be fixed	Treating an indefinite potential or ghost kernel as a positive physical covariance
Instantaneous proposal	Hamiltonian diagonalization on one slice in chosen canonical variables	Relative only to that slice and parametrization	Instantaneous time choice	Often insufficient unless upgraded by controlled high-order data	No general Hadamard local-comparison conclusion	Rapid background variation, negative instantaneous frequency squared, or slice dependence
de Sitter alpha proposal	Normalized antipodal Bogoliubov mixing of the Euclidean state	A nontrivial invariant family	de Sitter invariant	Non-Hadamard except at the Euclidean member	Not generally in the Hadamard local folium	Opposite-frequency and antipodal singularities
Sorkin–Johnston proposal	Positive spectral part of $i\Delta=-iE$ on a specified region and operator domain	Fixed relative to that region and spectral problem	Inherits symmetries preserved by the region	Generally non-Hadamard	Not generally locally quasiequivalent to a Hadamard state	Region, boundary, zero-mode, or continuum-limit dependence

Choose a route

• For a first encounter, move from vacuum ambiguity to complex structures and quasifree states, then study the Hadamard parametrix and wavefront criterion.
• To compare particle descriptions, add Bogoliubov implementability and then local quasiequivalence; global inequivalence need not obstruct bounded-region physics.
• To construct states, follow Hadamard propagation into deformation and gluing, then compare adiabatic, ground/KMS, and low-energy prescriptions.
• For spinor or gauge fields, establish the scalar microlocal logic first and then replace CCR positivity by the appropriate CAR or constrained physical-algebra condition.
• To evaluate a proposed “vacuum,” end with the comparative failure tests. A label is accepted only after positivity, singularity, auxiliary-choice, and operational checks survive.

Chapter guide

The pages below appear in their canonical order.

1. Vacuum Ambiguity, Time Flow, and Observer Dependence explains why a generic background does not determine positive frequency and identifies the extra structure behind a vacuum claim.
2. Complex Structures and One-Particle Spaces derives a one-particle Hilbert space from a positive symplectic-compatible complex structure.
3. Quasifree States and Two-Point Functions gives the field-equation, commutator, positivity, reality, and continuity tests on two-point data.
4. Bogoliubov Transformations and Unitary Implementability separates canonical mode mixing from its stronger Hilbert–Schmidt implementability condition.
5. GNS Representations, Local Normality, and Local Quasiequivalence explains why globally inequivalent representations can describe the same local normal states.
6. Hadamard Parametrix and Short-Distance Structure separates universal geometric singular terms from the smooth state-dependent remainder.
7. Hadamard Admissibility and the Two-Point Wavefront Criterion states the future-directed null wavefront condition and its limits.
8. Propagation of the Hadamard Property states the hyperbolic theorem that carries suitable local Cauchy data through the full development.
9. Constructing Hadamard States by Deformation and Gluing turns that theorem into a state-construction method while retaining positivity and nonuniqueness checks.
10. Adiabatic States, WKB Order, and Regularity relates high-momentum WKB order to regularity without confusing state choice with subtraction.
11. Ground, KMS, and Symmetry-Selected States identifies when a normalized time flow or symmetry distinguishes a state and when obstructions prevent it.
12. States of Low Energy and Smeared-Energy Selection explains observer- and sampling-dependent minimization in cosmological settings.
13. Hadamard States for Fermion and Gauge Fields adapts microlocal and positivity tests to CAR systems and constrained gauge fields.
14. Instantaneous Vacua, Alpha Vacua, Sorkin–Johnston States, and Failure Tests compares prominent prescriptions against ultraviolet, positivity, locality, and regulator-removal requirements.

Review the chapter

A complete state specification should answer six questions: What algebra is represented? What two-point data define the state? Which positivity statement holds? What is its singularity class? Which global or boundary assumptions enter? What physical criterion, if any, selects it? If “Hadamard” is offered as an answer to the final question, the specification is incomplete.

As a compact check, take two quasifree Hadamard states whose two-point functions differ by a smooth bisolution. They have the same allowed wavefront set, but their expectation values of a Wick square differ by the coincidence limit of that smooth term. Ultraviolet equivalence therefore coexists with physical state dependence. Radzikowski’s characterization makes the common singular class precise Radzikowski 1996, pp. 529–553; it does not erase the smooth remainder.

Construction and failure maps

The construction map summarizes the chapter’s logical order. Read it from left to right: positivity and the algebraic relations make a state, Hadamard structure controls its ultraviolet singularities, propagation or another construction makes it global, and only extra physical hypotheses can select one admissible state over another.

The controlled path keeps algebraic statehood, Hadamard admissibility, global construction, and physical selection as separate conclusions. Schematic; not to scale.

The companion map shows how to limit a claim. Inspect the lower branches: a failure of positivity, regularity, zero-mode control, or a uniqueness inference identifies which hypothesis is missing and which conclusion must be withdrawn.

A conclusion is licensed only for the specified observable, state, geometry, and approximation; any displayed witness stops or narrows it. Schematic; not to scale.

Handoffs

Particles, Detectors, and Nonadiabatic Production owns operational particle and detector questions. Local Observables, Stress Tensors, and Anomalies owns point-split composite observables. Hadamard States and the Wavefront-Set Characterization gives the proof-first treatment. Any numerical reconstruction must declare its discretization, cutoff, convergence test, and error control.

References

• Fewster, Christopher J., and Rainer Verch. “Algebraic Quantum Field Theory in Curved Spacetimes.” In Advances in Algebraic Quantum Field Theory, edited by Romeo Brunetti, Claudio Dappiaggi, Klaus Fredenhagen, and Jakob Yngvason, 125–189. Cham: Springer, 2015. DOI. Open PDF.
• Kay, Bernard S. “Quantum Field Theory in Curved Spacetime.” 2nd ed., 2023. arXiv:2308.14517.
• 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.