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 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 W mink 0 than the massive one W mink m , while in the corner it corresponds to that of the massive mirror W mirror m . 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 c SJ resembles W mink 0 for small masses, as in the continuum, but beyond a critical value mc it resembles W mink m , as expected. Our calculations suggest that unlike W mink m , 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, W c SJ agrees with W mirror m for all masses, and not, as might be expected, with the Rindler vacuum. * abhishekmathur@rri.res.in arXiv:1906.07952v2 [hep-th]

The causal set approach to quantum gravity

“…In the continuum, the SJ vacuum was constructed for the massless scalar field in the d = 2 causal diamond (Afshordi et al 2012) and recently extended to the small mass case (Mathur and Surya 2019). It has also been obtained for the trousers topology and shown to produce a divergent energy along both the future and the past light cones associated with the Morse point singularity (Buck et al 2017).…”

Studies on the SJ vacuum in de Sitter spacetime

“…While there is progress on finding the SJ modes via an ab initio calculation in some 2d as well as higher dimensional examples [10,11], the calculation in global de Sitter is considerably more difficult. In the absence of this, we can still carry out numerical calculations 11 using causal sets to study the two-point function.…”

“…We begin this section with laying out some basic properties of CST. 11 The bulk of the simulations for this work were done using Mathematica [26].…”

“…The ab initio calculation is however computationally challenging since it is non-trivial to calculate the spectral (or eigen) decomposition of i ∆ explicitly. Indeed, the spectral decomposition of i ∆ is known in very few examples other than the 2d causal diamond [8,9,10,11].…”

A Spacetime Calculation of the Calabrese-Cardy Entanglement Entropy

“…Finally, the effect of mass on these calculations would be interesting to explore, since this would eliminate the IR divergence. In [16] the small mass approximation of the SJ modes in the causal diamond was calculated upto second order in perturbation theory which can be used as a starting point. Figure 6: A plot comparing the eigenvalues µ of the entropy equation for one choice of complementary regions with α = 0.2, 0.8 for n max /α = 2600 and different choices of γ.…”