A Spacetime Approach to Defining Vacuum States and Entropy

Abstract

We apply a recent proposal for a distinguished ground state of a quantum field in a globally hyperbolic spacetime to the free massless scalar field in a causal diamond in two-dimensional Minkowski space. We investigate the two limits in which the Wightman function is evaluated (i) for pairs of points that lie in the centre of the diamond (i.e. far from the boundaries), and (ii) for pairs of points that are close to the left or right corner. We find that in the centre, the Minkowski vacuum state is recovered, with a definite value of the infrared cutoff. Interestingly, the ground state is not the Rindler vacuum in the corner of the diamond, as might have been expected, but is instead the vacuum of a flat space in the presence of a static mirror on that corner. We confirm these results by numerically evaluating the Wightman function of a massless scalar field on a causal set corresponding to the continuum causal diamond. We also present an independent calculation of the entanglement entropy for a free massless scalar field in 1+1 dimensions. We use a recent formula for a global expression of entropy using Green functions of the theory. This formula is independent of a density matrix with respect to a hypersurface (as the usual definition of entropy depends on), and it is applicable to both continuum spacetimes and to discrete causal sets. We compute the entanglement entropy in a spacetime of two causal diamonds in the continuum and in a causal set, in the limit that one is much larger than the other. In the continuum, we obtain the result S=1/3ln(ℓ/ɑ)+c, where ɑ is the UV cutoff of the theory, ℓ is the diameter of the smaller diamond, and c is a constant. This result is in agreement with results in CFT.