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

This paper calculates the binding energies of excited positronium states using the two-body Dirac equations of constraint, comparing nonperturbative and perturbative results while correcting identified literature errors and validating the model against alternative formulations.

The Gist

Imagine you are trying to predict the exact sound a specific musical instrument makes when two tiny, invisible particles (an electron and its antimatter twin, a positron) dance around each other. This dance is called Positronium.

For decades, physicists have used a very complex set of rules (Quantum Electrodynamics) to predict the notes this dance produces. Usually, they use a method called "perturbation theory," which is like trying to predict the weather by looking at small, manageable changes day by day. It works well for gentle breezes, but it can get messy when the storm gets too strong.

This paper, written by Robert Johnson, is like a master carpenter building a new, ultra-precise tool to measure that dance directly, without relying on the "day-by-day" approximations. Here is the breakdown of what he did, using simple analogies:

1. The Old Map vs. The New GPS

Most scientists use Non-Relativistic QED (a set of rules that works well for slow-moving things) to study Positronium. It's like using a paper map to navigate a city; it's good, but it misses the nuances of the terrain.

Johnson uses Two-Body Dirac Equations of Constraint. Think of this as a high-tech GPS that understands the "relativity" of the dance—how the particles move at speeds close to light and how their mass changes. This method treats the two particles as a single, unified system from the start, rather than trying to add corrections later.

2. The "Constraint" Dance Floor

In this theory, the particles aren't just floating freely; they are bound by "constraints." Imagine two dancers tied together by an elastic band. They can spin and move, but the band (the constraint) forces them to stay in a specific rhythm.

• The Problem: Calculating the exact energy of this dance is incredibly hard because the math involves infinite possibilities.
• The Solution: Johnson turned this complex math into a giant grid (like a spreadsheet) and used a computer to solve it. He didn't guess; he calculated the exact shape of the "dance floor" (the wave function) where the particles are most likely to be found.

3. Fixing the Typos in the Recipe

One of the most interesting parts of the paper is the detective work. Johnson found that the "cookbooks" (previous scientific papers) used by other researchers had typos.

• Imagine a recipe for a cake that says "add 2 cups of sugar" but the original author meant "add 1 cup." If you follow the typo, your cake is too sweet.
• Johnson found that previous formulas had missing "factors of 2" or wrong signs. He corrected these errors, showing that when you fix the recipe, the theoretical predictions finally match the computer calculations perfectly.

4. Changing the Lens (Coordinate Transformations)

To get the most accurate picture, Johnson had to look at the dance through different "lenses."

• Lens A (Standard View): Looking at the dance from a fixed distance.
• Lens B (Zoomed In): Using a special mathematical trick to zoom in incredibly close to the center of the dance (where the particles are closest).
• Lens C (Logarithmic View): Stretching the view so that the tiny details near the center are just as visible as the wide-open spaces far away.

He found that Lens C gave the clearest picture, much like how a wide-angle lens on a camera captures a better landscape than a standard zoom.

5. The Result: A Perfect Harmony

After running these massive calculations on a computer, Johnson compared his results to the "corrected" theoretical predictions.

• The Match: The numbers matched up to many decimal places.
• The Significance: This proves that the "constraint" method (the GPS) is just as accurate as the traditional methods, but it might be more robust for studying extreme conditions where the old methods fail.

The Bottom Line

Robert Johnson built a digital microscope to watch an electron and a positron dance. He found that the old instructions for how to calculate their dance had some typos. Once he fixed those typos and used his new, direct calculation method, the math and the reality agreed perfectly.

He also made his "microscope" (the computer code) available for free, inviting other scientists to look through it and verify his findings. It's a reminder that in science, even the most complex equations can sometimes just need a fresh pair of eyes to spot a missing "2" or a sign error.

Technical Summary

1. Problem Statement

The paper addresses the calculation of binding energies for the excited states of positronium (a bound state of an electron and a positron). While recent investigations have primarily relied on Non-Relativistic Quantum Electrodynamics (nrQED) for interpreting experimental data, the author argues that a fully covariant quantum formulation is necessary. The paper focuses on implementing the Two-Body Dirac Equations of Constraint (TBDE) formalism, originally developed by Dirac, Todorov, and Crater et al.

The specific challenges addressed include:

• Replicating and verifying non-perturbative numerical results against perturbative analytic formulas.
• Identifying and correcting inconsistencies and misprints in the literature regarding the analytic formulas for the positronium spectrum (specifically in reference [13]).
• Developing a robust numerical method to solve the coupled and decoupled radial equations for equal-mass QED systems without relying on perturbation theory.

2. Methodology

Theoretical Framework

The author utilizes the TBDE formalism for equal-mass systems ($m_1 = m_2$). The system is described by a set of coupled second-order radial differential equations for wave functions $u_0, u_1, u_+, u_-$.

• Decoupled States: For specific quantum numbers ($J=L, S=0$; $J=L, S=1$; and $J=0, L=1, S=1$), the equations decouple into single radial equations.
• Coupled States: For other states ($^3L_{L\pm1}$), the equations remain coupled, forming a $2 \times 2$ matrix eigenvalue problem.
• Potentials: The equations include a "quasipotential" $\Phi$ derived from the summation of one-particle exchange diagrams. The author explicitly derives expressions for terms like $\Phi_D, \Phi_{SS}, \Phi_{SO}, \Phi_T$, etc., noting sign corrections and factor adjustments (e.g., a missing factor of $1/2$ in previous literature) required for the equal-mass case.

Numerical Implementation

The author implements the solution using GNU Octave with the ARPACK library for eigenvalue analysis.

• Coordinate Transformations: To handle the singularities at $r=0$ and the infinite domain, three coordinate systems are tested:
  1. $y = rw/2\alpha$ (linear scaling).
  2. $z = \log y$ (logarithmic scaling).
  3. $x = (1 + s/y)^{-1}$ (compactified mapping to $[0,1]$).
• Discretization: The kinetic term is approximated using finite differences with a 7-point stencil (3rd order). Special edge corrections are applied at the boundaries ($r=0$ and $r \to \infty$) to maintain accuracy, utilizing algorithms from C.R. Taylor.
• Iterative Improvement: Since the eigenvalue $w$ (total energy) appears in the potential terms, the problem is non-linear. The author employs an iterative scheme:
  1. Assume an initial mass $w \approx 2m_1$.
  2. Solve the eigenvalue problem to get an estimate $\tilde{w}$.
  3. Update the Hamiltonian with the new $w$ and re-solve.
  4. Two iterations are found sufficient for convergence.

3. Key Contributions

1. Correction of Literature Misprints: The author identifies significant errors in the analytic perturbative formulas presented in reference [13] (Crater et al., 1992).
   • Specifically, for states where $J=L \ge 1$, the formula for the parameter $\eta_{\pm}$ requires an additional factor of $1/2$ to match numerical results.
   • The author provides corrected analytic expressions (Eq. 54–58) that align with the non-perturbative numerical data.
2. Open Source Implementation: The paper releases a complete open-source toolbox for GNU Octave, including the derivations in wxMaxima format, allowing for reproducibility.
3. Comparison with Improved Breit Equation: The author compares the TBDE results with an "Improved Breit Equation" (singularity-free form).
   • It is shown that solving the Breit equation in its standard form (Eq. 63) yields incorrect spectra (missing ground states, wrong energy scaling).
   • However, moving a factor of $w$ from the LHS to the RHS (Eq. 64) recovers the correct spectrum, suggesting a subtle but critical difference in how the energy eigenvalue is treated in the matrix inversion.
4. Validation of Coordinate Systems: The study demonstrates that while linear coordinates ($y$) are intuitive, logarithmic ($z$) and compactified ($x$) coordinates provide superior numerical stability and accuracy, particularly for states with $L=0$ where the wavefunction behavior near the origin is critical.

4. Results

• Binding Energies: The paper presents a comprehensive table (Table VI) of non-perturbative binding energies for positronium excited states (up to $n=3$) across various quantum numbers ($J, S, L$).
• Agreement with Perturbation Theory: After correcting the analytic formulas, the non-perturbative numerical results agree with perturbative estimates to approximately 8–10 decimal places (depending on the coordinate system used).
  • Coordinate $z$ (logarithmic) generally provided the most robust results.
  • Coordinate $y$ (linear) tended to overestimate confidence, especially for $L=0$ states.
• Large $\alpha$ Behavior: In Appendix B, the author explores the regime of large coupling constants ($\alpha$). For $\alpha \approx 0.74$, the binding energy reaches ~81% of the constituent mass, consistent with previous unpublished results by Eliahu Comay, validating the method's ability to handle strong coupling.

5. Significance

• Covariant Bound State Physics: The work reinforces the TBDE formalism as a viable, covariant alternative to nrQED for bound state problems, demonstrating that it can reproduce high-precision results without relying on non-relativistic expansions.
• Resolution of Discrepancies: By identifying and fixing the factor-of-2 errors in the literature, the paper resolves long-standing inconsistencies between analytic perturbative calculations and numerical non-perturbative solutions for positronium.
• Methodological Rigor: The detailed discussion on finite difference edge corrections and coordinate transformations provides a valuable guide for future numerical studies of relativistic two-body systems.
• Reproducibility: The release of the code and derivations ensures that the community can verify these results and extend the methodology to other equal-mass QED or QCD systems.

In conclusion, this paper successfully validates the Two-Body Dirac Equations of Constraint for positronium, corrects historical errors in the analytic formulas, and provides a high-precision numerical framework for calculating excited state binding energies.