States, Hadamard Structure, and Microlocal Control

A physically useful quantum state must be positive and have suitable short-distance behavior, but these conditions rarely select it uniquely. This chapter separates state construction, representation, ultraviolet admissibility, and physical selection.

Enter this chapter

Begin with Vacuum Ambiguity, Time Flow, and Observer Dependence to identify the time-flow assumptions behind positive frequency and a vacuum.

Helpful background. Singular Support and Wavefront Sets and Distributional Kernels and Distributions on Manifolds develop the definitions used by the chapter’s opening pages.

Topics

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.

How the ideas fit together

For a quasifree field, two-point data determine higher correlations, subject to positivity and the canonical algebra. The Hadamard condition further restricts short-distance singularities so that local composite observables can be renormalized; its wavefront-set formulation exposes the causal directions of those singularities Radzikowski 1996, pp. 529–553.

Hadamard admissibility does not minimize energy or establish thermality. Ground, KMS, adiabatic, and low-energy constructions use different additional information. Comparing them requires keeping the representation and limiting procedure fixed rather than treating every acceptable state as the same vacuum.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

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

Where to go next

Continue to Interacting Fields and Local Covariant Renormalization to understand why ultraviolet subtraction must respect local geometry.

References

• 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.