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The quantum null energy condition (QNEC) is a quantum generalization of the null energy condition which gives a lower bound on the null energy in terms of the second derivative of the von Neumann entropy or entanglement entropy of some region with respect to a null direction. The QNEC states that T kk p ≥ limA→0 2πA S out where Sout is the entanglement entropy restricted to one side of a codimension-2 surface Σ which is deformed in the null direction about a neighborhood of point p with area A. A proof of QNEC has been given which applies to free and super-renormalizable bosonic field theories, and to any points that lie on a stationary null surface. Using similar assumptions and method, we prove the QNEC for fermionic field theories.

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A central finite difference approximation for the radial contribution Δr to the Laplacian ∇2=Δr+Δr⊥(r) is considered in a three-dimensional cylindrical coordinate system (r,θ,z). A free-particle Schrödinger time evolution operator is constructed by exponentiation, e(i/2)ξ∇2=⋯e−(1/2)ξ2[Δr,Δr⊥(r)] e(i/2)ξΔr⊥(r) e(i/2)ξΔr→⋯e(i/2)ξΔr. Denoting the central finite difference approximation of Δr by (1/Δr2) T, the matrix S≡e(i/2)λT, with λ=ξ/Δr2, is shown to be similar to a particular unitary representation UVK of the group of motions on Euclidean three-space that has been described by Vilenkin and Klimyk. The matrix elements of UVK generalize the Bessel function and provide an approximation of the leading term in the radial contribution to the evolution operator.

Various proposals for gravitational entropy densities have been constructed from the Weyl tensor. In almost all cases, though, these studies have been restricted to general relativity, and little has been done in modified theories of gravity. However, in this paper, we investigate the simplest proposal for an entropy density constructed from the Weyl tensor in five-dimensional Gauss–Bonnet gravity and find that it fails to reproduce the expected entropy of a black hole.