On the relativistic quantum mechanics of a photon between two electrons in $$1+1$$ dimensions
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Bibliographic record
Abstract
Abstract A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions $$\Psi ({\textbf {x}}_{\text {ph}},{\textbf {x}}_{\text {e}_1},{\textbf {x}}_{\text {e}_2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ψ</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mtext>ph</mml:mtext> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:msub> <mml:mtext>e</mml:mtext> <mml:mn>1</mml:mn> </mml:msub> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:msub> <mml:mtext>e</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> where $${\textbf {x}}_{\text {ph}},{\textbf {x}}_{\text {e}_1},{\textbf {x}}_{\text {e}_2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mtext>ph</mml:mtext> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:msub> <mml:mtext>e</mml:mtext> <mml:mn>1</mml:mn> </mml:msub> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:msub> <mml:mtext>e</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:msub> </mml:mrow> </mml:math> are generic spacetime events of the photon and two electrons, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds $$\{{\textbf {x}}_{\text {ph}}={\textbf {x}}_{\text {e}_1}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mtext>ph</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:msub> <mml:mtext>e</mml:mtext> <mml:mn>1</mml:mn> </mml:msub> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> and $$\{{\textbf {x}}_{\text {ph}}={\textbf {x}}_{\text {e}_2}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mtext>ph</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:msub> <mml:mtext>e</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it