Sorkin-Johnston vacuum for a massive scalar field in the 2D causal diamond

• Published in: Physical review. D 2019-08, Vol.100 (4), p.1, Article 045007
• Main Authors: Mathur, Abhishek, Surya, Sumati
• Format: Article
• Language: English
• Subjects: Computer simulation; Diamonds; Eigenvalues; Numerical methods
• ISSN: 2470-0010; 2470-0029
• DOI: 10.1103/PhysRevD.100.045007

We study the massive scalar field Sorkin-Johnston (SJ) Wightman function WSJ restricted to a flat 2D causal diamond 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 WSJ up to this order. Using a combination of analytical and numerical methods, we obtain expressions for WSJ both in the center and the corner of D, to leading order. We find that in the center, WSJ is more like the massless Minkowski Wightman function W0mink than the massive one Wmmink, while in the corner it corresponds to that of the massive mirror Wmmirror. 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 WSJc resembles W0mink for small masses, as in the continuum, but beyond a critical value mc it resembles Wmmink, as expected. Our calculations suggest that unlike Wmmink, WSJ 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, WSJc agrees with Wmmirror for all masses, and not, as might be expected, with the Rindler vacuum.