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arxiv: 2606.00311 · v2 · pith:3ZB2CBE6new · submitted 2026-05-29 · 🌀 gr-qc · hep-th· math-ph· math.MP· math.SP

Spectral Density of the Causal Propagator

Joshua Y. L. Jones , Yasaman K. Yazdi This is my paper

Pith reviewed 2026-06-28 21:02 UTC · model grok-4.3

classification 🌀 gr-qc hep-thmath-phmath.MPmath.SP
keywords causal propagatorPauli-Jordan functionspectral densityasymptotic scalingLorentzian spectral geometrycausal set theoryfree scalar field theoryWeyl's law
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The pith

The causal propagator in free scalar QFT has a conjectured asymptotic spectral density scaling supported by examples.

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

The paper examines the causal propagator, also called the Pauli-Jordan function, which encodes the spacetime commutator of a scalar field in quantum field theory. It discusses the propagator's role in covariant QFT formulations and in causal set theory. The authors then state a conjecture for the asymptotic spectral density of this object in free theories and supply examples that support the scaling. If the conjecture holds, it would supply a Lorentzian version of Weyl's asymptotic law from Riemannian spectral geometry.

Core claim

We present a conjecture for its asymptotic spectral density in a free theory, and give examples that lend evidence to the conjectured scaling. Our work has implications for Lorentzian spectral geometry in much the same way as Weyl's asymptotic law has for Riemannian spectral geometry.

What carries the argument

The causal propagator (Pauli-Jordan function) and the conjecture for the asymptotic density of its spectrum.

If this is right

  • The conjecture supplies a tool for defining quantum field theory in a more explicitly covariant manner.
  • It offers concrete input relevant to constructions in causal set theory.
  • It opens a route to Lorentzian spectral geometry that mirrors the established Riemannian case.

Where Pith is reading between the lines

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

  • The scaling could be checked by explicit diagonalization of the propagator on simple manifolds such as Minkowski or de Sitter space.
  • If the free-theory result survives, similar spectral analysis might be attempted for interacting fields or discrete causal-set discretizations.
  • The conjecture may link to other volume-dependent spectral counts already studied in quantum gravity models.

Load-bearing premise

Examples computed in free theories are taken to indicate that the same spectral-density scaling holds for the causal propagator in general.

What would settle it

A direct calculation of the spectrum of the causal propagator on a curved background such as Schwarzschild spacetime that fails to match the conjectured asymptotic scaling.

Figures

Figures reproduced from arXiv: 2606.00311 by Joshua Y. L. Jones, Yasaman K. Yazdi.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

The causal propagator (or Pauli-Jordan function), which multiplied by $i$ is the spacetime commutator of the field $[\phi(x),\phi(x')]$, plays an essential role in scalar quantum field theory. We discuss the role of the causal propagator and its spectrum in recent developments in defining quantum field theory in a more explicitly covariant manner, as well as in causal set theory. We then present a conjecture for its asymptotic spectral density in a free theory, and give examples that lend evidence to the conjectured scaling. Our work has implications for Lorentzian spectral geometry in much the same way as Weyl's asymptotic law has for Riemannian spectral geometry.

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 / 2 minor

Summary. The manuscript discusses the role of the causal propagator (Pauli-Jordan function) in scalar QFT, particularly in covariant formulations and causal set theory. It presents a conjecture for the asymptotic spectral density of this propagator in free theories, supported by examples that lend evidence to a conjectured scaling, and draws an analogy to Weyl's law for implications in Lorentzian spectral geometry.

Significance. If the conjecture holds with greater generality, it would establish a foundational result in Lorentzian spectral geometry analogous to Weyl's asymptotic law, potentially aiding covariant definitions of QFT and discrete quantum gravity models such as causal sets. The examples provide initial supporting evidence, strengthening the exploratory value of the work.

major comments (1)
  1. Abstract and main text: The central claim is explicitly framed as a conjecture supported by examples in free theory, but no derivation, general argument, or error analysis is provided to establish the scaling beyond the specific cases considered; this makes the evidence for generality limited, as noted in the abstract's description of the claim.
minor comments (2)
  1. Clarify the precise class of spacetimes or field theories to which the conjecture is intended to apply, as the abstract does not specify this.
  2. Consider adding quantitative measures (e.g., fit errors or scaling exponents with uncertainties) to the examples to strengthen the evidence presentation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [—] Abstract and main text: The central claim is explicitly framed as a conjecture supported by examples in free theory, but no derivation, general argument, or error analysis is provided to establish the scaling beyond the specific cases considered; this makes the evidence for generality limited, as noted in the abstract's description of the claim.

    Authors: We agree that the central claim is presented as a conjecture supported by examples in free scalar QFT, without a general derivation, argument, or error analysis. This is by design: the manuscript proposes the conjecture and supplies concrete supporting cases as initial evidence, rather than claiming a general result. The abstract and main text already describe the claim in precisely these terms, so the limitation on generality is explicitly acknowledged. No revision is required on this point. revision: no

Circularity Check

0 steps flagged

Conjecture with supporting examples; no circular derivation chain

full rationale

The paper explicitly frames its central result as a conjecture for the asymptotic spectral density of the causal propagator in free theory, backed by examples that provide evidence rather than a closed derivation. No load-bearing step reduces by construction to a self-definition, fitted input renamed as prediction, or self-citation chain. The work positions itself as exploratory (analogous to Weyl's law) without claiming a forced uniqueness theorem or smuggling an ansatz. The derivation chain is self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified from the provided text.

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discussion (0)

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Reference graph

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    We choose −L≤u, v≤L; see Figure 1 for the setup. The causal propagator is [5, 18] ∆(2d)(u, v;u ′, v′) = (1−Θ(u−u ′)−Θ(v−v ′))/2, (10) andi∆ (2d) has two families of eigenfunctions: fk(u, v) :=e−iku −e −ikv;k= nπ L , n=±1,±2, ..., (11) gk(u, v) :=e−iku +e −ikv −2 cos(kL);k∈ K,(12) whereK={k∈R|tan(kL) = 2kL}.The eigenvalues for both families of eigenfunctio...

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