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 ${W}_{SJ}$ 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.