Excited States Of Positronium From The Two Body Dirac Equations Of Constraint

Robert W. Johnson

arxiv: 2506.21774 · v2 · submitted 2025-06-26 · ✦ hep-ph

Excited States Of Positronium From The Two Body Dirac Equations Of Constraint

Robert W. Johnson This is my paper

Pith reviewed 2026-05-19 07:16 UTC · model grok-4.3

classification ✦ hep-ph

keywords positroniumexcited statesbinding energiestwo-body Dirac equationsconstraint formalismnonperturbative calculationperturbative comparison

0 comments

The pith

The two-body Dirac equations of constraint yield binding energies for excited positronium states, compared nonperturbatively and perturbatively.

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

The paper applies the two-body Dirac equations of constraint to compute binding energies of excited positronium states. It carries out nonperturbative evaluations alongside perturbative ones and directly compares the outcomes. Decoupled equations for states with total angular momentum J=0 are checked against an alternate formulation of the same model. Misprints appearing in earlier literature are identified and corrected as part of the work.

Core claim

The binding energies of the excited states of positronium are calculated using the two body Dirac equations of constraint formalism. The results from nonperturbative evaluation are compared to those from perturbative evaluation. The equations for decoupled states with J=0 are compared to an alternate formulation of the model, and some misprints in the literature are identified and corrected.

What carries the argument

Two-body Dirac equations of constraint, a relativistic framework for solving bound-state problems in two-particle systems.

If this is right

Nonperturbative binding energies serve as reference values against which perturbative approximations can be tested for excited states.
Comparison of J=0 decoupled equations with an alternate formulation checks internal consistency of the model.
Correction of misprints removes sources of error that would otherwise propagate into numerical results.
The same equations can be solved for additional observables such as fine-structure splittings once the binding energies are obtained.

Where Pith is reading between the lines

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

If nonperturbative and perturbative results remain close even for higher excitations, the perturbative series converges well inside this formalism.
Systematic differences between the two evaluations would point to the size of nonperturbative corrections that must be retained in similar bound-state calculations.
The method could be carried over to other equal-mass lepton systems such as muonium once the positronium case is validated.

Load-bearing premise

The two-body Dirac equations of constraint formalism supplies a sufficiently complete nonperturbative description of positronium excited states without needing further higher-order adjustments.

What would settle it

A laboratory measurement of the binding energy of the 2P state of positronium that lies outside the range predicted by the nonperturbative solution of the constraint equations.

read the original abstract

The binding energies of the excited states of positronium are calculated using the two body Dirac equations of constraint formalism. The results from nonperturbative evaluation are compared to those from perturbative evaluation. The equations for decoupled sates with J=0 are compared to an alternate formulation of the model. Some misprints in the literature are identified and corrected.

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.

Desk Editor's Note private letter to a colleague

The paper computes new binding energies for excited positronium states using the two-body Dirac constraint formalism and compares nonperturbative to perturbative results while correcting misprints.

read the letter

The main thing to know is that this work calculates the binding energies of excited states of positronium with the two body Dirac equations of constraint. It evaluates them nonperturbatively and compares to perturbative results, checks the decoupled J=0 equations against an alternate formulation, and corrects some misprints in the literature. What the paper does well is apply this established formalism to the excited states and produce explicit numerical values that were not previously available. The comparison between the full numerical solution and the perturbative one provides a check on the method, and fixing misprints is a helpful detail for anyone referencing the earlier papers. The potential soft spot is in the decoupling procedure for the states. The central comparison assumes that the reduction to an effective radial equation allows a perturbative expansion that can be reliably contrasted with the nonperturbative case. If the decoupling introduces approximations that are not controlled uniformly, such as in handling retardation or spin terms, then the agreement or difference between the two evaluations may not cleanly demonstrate the nonperturbative content. The abstract does not show the actual numbers or error estimates, so the full paper would need to demonstrate that the results are robust. This paper is for specialists in relativistic bound state calculations and QED for exotic systems like positronium. A reader already familiar with constraint dynamics or looking for specific excited state energies would find the results and comparisons useful. It is incremental rather than transformative. I recommend sending it for peer review. The focused calculation and the corrections to the literature make it appropriate for a referee to evaluate the details and the strength of the comparison.

Referee Report

2 major / 2 minor

Summary. The manuscript calculates the binding energies of excited states of positronium using the two-body Dirac equations of constraint formalism. It compares results obtained from nonperturbative numerical evaluation to those from perturbative evaluation, examines the decoupled equations for J=0 states against an alternate formulation of the model, and identifies and corrects misprints in the prior literature.

Significance. If the central comparison holds, the work provides a concrete test of the nonperturbative content of the two-body Dirac constraint approach for QED bound states at higher excitations. This could help establish the formalism's reliability beyond ground-state applications and offer benchmarks for other relativistic two-body methods.

major comments (2)

[Decoupling for J=0 states] Decoupling section for J=0 states (following the reduction to the effective radial equation): The manuscript does not demonstrate that the perturbative expansion of the decoupled system reproduces the original perturbative treatment order-by-order. Without this explicit verification, any reported agreement or discrepancy between the nonperturbative numerical results and the perturbative ones cannot reliably isolate the nonperturbative content of the formalism.
[Numerical results and comparison] Numerical evaluation and comparison section: No error estimates, convergence criteria, or tabulated binding-energy values with uncertainties are supplied for the nonperturbative solutions. This omission prevents quantitative assessment of how well the nonperturbative results match the perturbative ones and undermines the strength of the central claim.

minor comments (2)

[Abstract] The abstract contains the typographical error 'decoupled sates' (should read 'states').
[Comparison to alternate formulation] Notation for the constraint equations and the alternate formulation could be aligned more explicitly when the two are compared, to aid the reader in spotting the differences.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results on the nonperturbative and perturbative evaluations of positronium excited states within the two-body Dirac constraint formalism. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

Referee: Decoupling section for J=0 states (following the reduction to the effective radial equation): The manuscript does not demonstrate that the perturbative expansion of the decoupled system reproduces the original perturbative treatment order-by-order. Without this explicit verification, any reported agreement or discrepancy between the nonperturbative numerical results and the perturbative ones cannot reliably isolate the nonperturbative content of the formalism.

Authors: We agree that an explicit order-by-order verification would strengthen the isolation of nonperturbative effects. In the revised manuscript, we will add a dedicated subsection in the decoupling analysis that expands the decoupled J=0 equations perturbatively and demonstrates agreement with the original perturbative treatment through the relevant orders in the fine-structure constant. This will confirm that discrepancies, if any, arise from nonperturbative contributions rather than inconsistencies in the decoupling procedure. revision: yes
Referee: Numerical evaluation and comparison section: No error estimates, convergence criteria, or tabulated binding-energy values with uncertainties are supplied for the nonperturbative solutions. This omission prevents quantitative assessment of how well the nonperturbative results match the perturbative ones and undermines the strength of the central claim.

Authors: We acknowledge the need for quantitative rigor in the numerical results. The revised manuscript will include a new subsection detailing the numerical method's error estimates (derived from grid resolution and iterative convergence), explicit convergence criteria (such as residual tolerances in the eigenvalue solver), and tabulated binding energies for selected excited states with associated uncertainties. These additions will enable direct quantitative comparison between nonperturbative and perturbative results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies established formalism to independent numerical and perturbative evaluations

full rationale

The paper applies the two-body Dirac equations of constraint formalism to compute binding energies of positronium excited states, performing nonperturbative numerical solutions and perturbative expansions for comparison, along with checks against an alternate formulation and corrections to literature misprints. No load-bearing step reduces by construction to a fitted parameter, self-definition, or unverified self-citation chain; the formalism is invoked as an external input whose consequences are then evaluated separately in the two regimes. The reported agreement or discrepancy between evaluations therefore constitutes an independent test rather than a tautology. Self-citations, if present for the underlying equations, do not substitute for the paper's own computations and are not required to justify uniqueness or ansatz choices within this work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the two-body Dirac constraint equations to positronium; no free parameters or new entities are mentioned in the abstract.

axioms (1)

domain assumption The two-body Dirac equations of constraint formalism accurately models the bound states of positronium.
Invoked as the computational foundation for both nonperturbative and perturbative evaluations.

pith-pipeline@v0.9.0 · 5566 in / 1040 out tokens · 24621 ms · 2026-05-19T07:16:28.290343+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The binding energies of the excited states of positronium are calculated using the two body Dirac equations of constraint formalism. ... decoupled states with J=0 ... ΦD − 3ΦSS ...
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We have already begun the process of simplification by omitting terms 2 mwS + S2 ... for the application to QED (to leading order).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.