Thermodynamics and area in Minkowski space: Heat capacity of entanglement

Tracing over the degrees of freedom inside (or outside) a subvolume V of Minkowski space in a given quantum state |ψ〉 results in a statistical ensemble described by a density matrix ρ. This enables one to relate quantum fluctuations in V when in the state |ψ〉 to statistical fluctuations in the ensemble described by ρ. These fluctuations scale linearly with the surface area of V. If V is half of space, then ρ is the density matrix of a canonical ensemble in Rindler space. This enables us to “derive” the area scaling of thermodynamic quantities in Rindler space from the area scaling of quantum fluctuations in half of Minkowski space. When considering shapes other than half of Minkowski space, even though area scaling persists, ρ does not have an interpretation as a density matrix of a canonical ensemble in a curved, or geometrically nontrivial, background.