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arxiv: 2510.11492 · v2 · submitted 2025-10-13 · 🧮 math-ph · math.MP

On the construction of Hadamard states from Feynman propagators

Christopher J. Fewster , Alexander Strohmaier This is my paper

Pith reviewed 2026-05-18 07:20 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords Hadamard statesFeynman propagatorsnormally hyperbolic operatorsDirac-type operatorspositivity conditionsdistinguished parametricesquantum field theory on curved spacetimetwo-point functions
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The pith

Feynman propagators for linear quantum fields on vector bundles can be chosen to satisfy positivity conditions that make them define Hadamard states.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the converse problem of starting from a Feynman propagator and recovering a Hadamard state for the associated quantum field. It shows that a recent extension of distinguished parametrices to normally hyperbolic and Dirac-type operators supplies the necessary two-point functions, but these define states only when additional positivity conditions hold. The authors supply simple domination inequalities on self-adjoint smooth kernels that guarantee the required positivity. These inequalities complete the construction for complex bosonic fields, hermitian bosonic fields, Dirac fermions, and Majorana fermions. The result applies whenever the underlying operator admits the domination properties.

Core claim

Starting from the generalised Duistermaat-Hörmander theory of distinguished parametrices for normally hyperbolic and Dirac-type operators on hermitian vector bundles, Feynman propagators can be selected so that they define Hadamard states provided the associated self-adjoint smooth kernels obey elementary domination inequalities that enforce positivity of the two-point function.

What carries the argument

Domination properties of self-adjoint smooth kernels that enforce the positivity conditions required for a Feynman propagator to define a state.

If this is right

  • Hadamard states exist for the complex bosonic field whenever the normally hyperbolic operator admits the domination properties.
  • The same construction yields states for the hermitian bosonic theory when the operator commutes with a complex conjugation.
  • Dirac fermionic fields governed by Dirac-type operators admit such states via the same kernel conditions.
  • Majorana theories follow when the Dirac-type operator commutes with a skew complex conjugation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method should extend to other first-order operators on bundles once their distinguished parametrices are known.
  • Explicit verification of the domination inequalities for concrete operators on specific spacetimes would immediately produce new families of Hadamard states.
  • The construction supplies a practical route to Hadamard states on globally hyperbolic manifolds without first solving the full Wightman axioms.

Load-bearing premise

That the self-adjoint smooth kernels associated with the operators satisfy the stated domination inequalities.

What would settle it

An explicit normally hyperbolic or Dirac-type operator on a vector bundle for which the domination inequalities fail, so that the corresponding Feynman propagator yields a two-point function that is not positive.

read the original abstract

The Wightman two-point function of any Hadamard state of a linear quantum field theory determines a corresponding Feynman propagator. Conversely, however, a Feynman propagator determines a state only if certain positivity conditions are fulfilled. Choosing a Feynman propagator to satisfy the correct positivity conditions involves a slightly subtle point that we address and resolve. Starting from a recent generalisation of the Duistermaat-H\"ormander theory of distinguished parametrices to normally hyperbolic and Dirac-type operators acting on sections of hermitian vector bundles, we complete this work by showing how Feynman propagators can be chosen so as to define Hadamard states. The theories considered are: the complex bosonic field governed by a normally hyperbolic operator; the corresponding hermitian theory if the operator commutes with a complex conjugation; the Dirac fermionic theory governed by a Dirac-type operator, and the corresponding Majorana theory in the case where the operator commutes with a skew complex conjugation. The additional key ingredients that we supply are simple domination properties of self-adjoint smooth kernels.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript shows how to select Feynman propagators from a recent generalization of the Duistermaat-Hörmander theory of distinguished parametrices for normally hyperbolic and Dirac-type operators on hermitian vector bundles so that they define Hadamard states. It treats the complex bosonic case, the hermitian bosonic case (when the operator commutes with a complex conjugation), the Dirac fermionic case, and the Majorana case (when the operator commutes with a skew complex conjugation). The central new technical step is the establishment of simple domination properties for the associated self-adjoint smooth kernels that guarantee the required positivity of the two-point functions.

Significance. If the domination properties hold as stated, the work supplies a systematic, parameter-free route from distinguished parametrices to Hadamard states for a broad class of linear theories on vector bundles. This resolves the positivity obstruction that had prevented direct use of such propagators as two-point functions and extends the geometric construction to fermionic and Majorana statistics under the stated commutation conditions. The approach is grounded in the analytic properties of the operators and therefore strengthens the foundations for QFT on curved spacetimes with internal degrees of freedom.

major comments (1)
  1. [§4] §4 (Domination properties for the Majorana case): The proof that the skew-conjugation condition implies the required domination inequality for the self-adjoint kernel must be checked for general hermitian vector bundles; if the argument tacitly assumes a trivial bundle or a flat metric, the positivity claim for non-trivial bundles would not follow and the scope of the main theorem would need to be restricted.
minor comments (3)
  1. [§1] The introduction should explicitly list the four operator classes together with the precise commutation hypotheses under which each is treated, so that the scope is immediately clear to readers.
  2. [Notation] Notation for the Feynman propagator versus the two-point function should be made uniform across sections; currently the same symbol appears to be used for both objects in different contexts.
  3. [§1] A brief remark on how the new domination inequalities relate to earlier positivity conditions in the literature (e.g., those of Kay-Wald or Radzikowski) would help situate the contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment and for the constructive comment on the scope of the argument in §4. We address the point directly below.

read point-by-point responses

  1. Referee: [§4] §4 (Domination properties for the Majorana case): The proof that the skew-conjugation condition implies the required domination inequality for the self-adjoint kernel must be checked for general hermitian vector bundles; if the argument tacitly assumes a trivial bundle or a flat metric, the positivity claim for non-trivial bundles would not follow and the scope of the main theorem would need to be restricted.

    Authors: The proof in §4 is carried out intrinsically with respect to the hermitian metric on the vector bundle E and treats the skew complex conjugation as a bundle automorphism. The self-adjoint kernel is defined using this metric, and the domination inequality follows from the positive-definiteness of the metric together with the algebraic properties of the conjugation. Local trivializations are employed only as a computational device; the resulting pointwise estimates are tensorial and therefore global. No assumption of bundle triviality or flatness of the metric enters the argument, and curvature terms do not appear in the orders relevant to the domination. Consequently the positivity claim holds for general hermitian vector bundles and the scope of the main theorem requires no restriction. A short clarifying paragraph will be added to §4 in the revised version to make this explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external generalization plus newly supplied domination properties

full rationale

The paper's derivation begins with a cited recent generalization of the Duistermaat-Hörmander theory of distinguished parametrices for normally hyperbolic and Dirac-type operators on hermitian vector bundles, then supplies independent domination inequalities for self-adjoint smooth kernels to enforce the required positivity conditions on the two-point functions. These domination properties are presented as additional key ingredients that must be verified separately for a given operator; they are not derived from the target Hadamard state or fitted to data within the paper. No step reduces the final construction of the state to a self-definition, a renamed input, or a load-bearing self-citation chain. The result remains self-contained against the stated assumptions on the operator and bundle, with explicit case distinctions (bosonic, hermitian, fermionic, Majorana) handled by the supplied inequalities rather than by tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on a prior generalization of distinguished parametrices and introduces domination properties of kernels as the new technical step; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Recent generalisation of the Duistermaat-Hörmander theory to normally hyperbolic and Dirac-type operators on hermitian vector bundles
    Invoked as the starting point for the construction (abstract).

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    math-ph 2026-04 unverdicted novelty 6.0

    A new complete gauge fixing at initial data via Hodge decomposition on complete Riemannian manifolds enables existence proofs for Hadamard states in the quantization of Maxwell theory on globally hyperbolic Lorentzian...

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