  We derive a general relation between the ground state entanglement
Hamiltonian and the physical stress tensor within the path integral formalism.
For spherical entangling surfaces in a CFT, we reproduce the \emph{local}
ground state entanglement Hamiltonian derived by Casini, Huerta and Myers. The
resulting reduced density matrix can be characterized by a spatially varying
"entanglement temperature." Using the entanglement Hamiltonian, we calculate
the first order change in the entanglement entropy due to changes in conserved
charges of the ground state, and find a local first law-like relation for the
entanglement entropy. Our approach provides a field theory derivation and
generalization of recent results obtained by holographic techniques. However,
we note a discrepancy between our field theoretically derived results for the
entanglement entropy of excited states with a non-uniform energy density and
current holographic results in the literature. Finally, we give a CFT
derivation of a set of constraint equations obeyed by the entanglement entropy
of excited states in any dimension. Previously, these equations were derived in
the context of holography.
