Expressing entropy globally in terms of (4D) field-correlations

Understand

We express the entropy of a scalar field phi directly in terms of its spacetime correlation function W(x,y) = <phi(x) phi(y)>, assuming that the higher correlators are of &#34;Gaussian&#34; form.

• The resulting formula associates an entropy S(R) to any spacetime region R; and when R is globally hyperbolic with Cauchy surface Sigma, S(R) can be interpreted as the entropy of the reduced density-matrix belonging to Sigma.
• One acquires in particular a new expression for the entropy of entanglement across an event-horizon.
• Thanks to its spacetime character, this expression makes sense in a causal set as well as in a continuum spacetime.