Sorkin-Johnston vacuum for a massive scalar field in the 2D causal diamond
Bibliographic record
Abstract
We study the massive scalar field Sorkin-Johnston (SJ) Wightman function ${W}_{SJ}$ restricted to a flat 2D causal diamond $\mathcal{D}$ of linear dimension $L$. Our approach is two-pronged. In the first, we solve the central SJ eigenvalue problem explicitly in the small mass regime, up to order $(mL{)}^{4}$. This allows us to formally construct ${W}_{SJ}$ up to this order. Using a combination of analytical and numerical methods, we obtain expressions for ${W}_{SJ}$ both in the center and the corner of $\mathcal{D}$, to leading order. We find that in the center, ${W}_{SJ}$ is more like the massless Minkowski Wightman function ${W}_{0}^{\mathrm{mink}}$ than the massive one ${W}_{m}^{\mathrm{mink}}$, while in the corner it corresponds to that of the massive mirror ${W}_{m}^{\text{mirror}}$. In the second part, in order to explore larger masses, we perform numerical simulations using a causal set approximated by a flat 2D causal diamond. We find that in the center of the diamond the causal set SJ Wightman function ${W}_{SJ}^{c}$ resembles ${W}_{0}^{\mathrm{mink}}$ for small masses, as in the continuum, but beyond a critical value ${m}_{c}$ it resembles ${W}_{m}^{\mathrm{mink}}$, as expected. Our calculations suggest that unlike ${W}_{m}^{\mathrm{mink}}$, ${W}_{SJ}$ has a well-defined massless limit, which mimics the behavior of the Pauli Jordan function underlying the SJ construction. In the corner of the diamond, moreover, ${W}_{SJ}^{c}$ agrees with ${W}_{m}^{\text{mirror}}$ for all masses, and not, as might be expected, with the Rindler vacuum.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".