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

Abstract

We study the massive scalar field Sorkin-Johnston (SJ) Wightman function restricted to a flat 2D causal diamond 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, upto order (mL)^4. This allows us to formally construct the SJ Wightman function up to this order. Using a combination of analytic and numerical methods, we obtain expressions for the SJ Wightman function both in the center and the corner of the diamond, to leading order. We find that in the center, it is more like the massless Minkowski Wightman function than the massive one, while in the corner it corresponds to that of the massive 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 resembles the massless Minkowski Wightman function for small masses, as in the continuum, but beyond a critical value it resembles the massive Minkowski Wightman function as expected. Our calculations suggest that unlike the massive Minkowski vacuum, the SJ vacuum 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, it agrees with the mirror vacuum for all masses, and not, as might be expected, with the Rindler vacuum.