Entanglement entropy of scalar fields in causal set theory

Abstract

Entanglement entropy is now widely accepted as having deep connections with quantum gravity. It is therefore desirable to understand it in the context of causal sets, especially since they provide in a natural and covariant manner the UV cutoff needed to render entanglement entropy finite. Defining entropy in a causal set is not straightforward because the usual hypersurface data on which definitions of entanglement typically rely is not available. Instead, we appeal to a more global expression which, for a gaussian scalar field, expresses the entropy of a spacetime region in terms of the field’s correlation function within that region. In this talk I will present results from evaluating this entropy for causal sets sprinkled into a 1 + 1-dimensional causal diamond in flat spacetime, and specifically for a smaller causal diamond within a larger concentric one. In the first instance we find an entropy that obeys a (spacetime) volume law instead of the expected (spatial) area law. We find, however, that one can obtain the expected area law by following a prescription for truncating the eigenvalues of a certain “Pauli-Jordan” operator and the projections of their eigenfunctions on the Wightman function that enters into the entropy formula.