Formulation and proof of the gravitational entropy bound

A bstract We provide a formulation and proof of the gravitational entropy bound. We use a recently given framework which expresses the measurable quantities of a quantum theory as a weighted sum over paths in the theory’s phase space. If this framework is applied to a field theory on a spacetime foliated by a hypersurface Σ, the choice of a codimension-2 surface B without boundary contained in Σ specifies a submanifold in the phase space. We show here that this submanifold is naturally restricted to obey an entropy bound if the field theory is diffeomorphism-invariant. We prove this restriction to arise by considering the quantum-mechanical sum of paths in phase space and exploiting the interplay of the commutativity of the sum with diffeomorphism-invariance. The formulation of the entropy bound, which we state and derive in detail, involves a functional K on the submanifold associated to B . We give an explicit construction of K in terms of the Lagrangian. The gravitational entropy bound then states: for any real $$ \frac{\lambda }{\hslash } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mi>λ</mml:mi> <mml:mi>ℏ</mml:mi> </mml:mfrac> </mml:math> , consider the set of states where K takes a value not bigger than λ and let V denote the phase space volume of this set. One has then ln( V ) ≤ $$ \frac{\lambda }{\hslash } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mi>λ</mml:mi> <mml:mi>ℏ</mml:mi> </mml:mfrac> </mml:math> . Especially, we show for the Einstein-Hilbert Lagrangian in any dimension with cosmological constant and arbitrary minimally coupled matter, one has K = $$ \frac{A}{4G} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mi>A</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>G</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> . Hereby, A denotes the area of B in a particular state.