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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26268_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mi>λ</mi> <mi>ℏ</mi> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{\lambda }{\hslash } $</EquationSource> </InlineEquation>, consider the set of states where K takes a value not bigger than λ and let V denote the phase space volume of this set. One has then ln(V) ≤ <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26268_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mi>λ</mi> <mi>ℏ</mi> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{\lambda }{\hslash } $</EquationSource> </InlineEquation>. Especially, we show for the Einstein-Hilbert Lagrangian in any dimension with cosmological constant and arbitrary minimally coupled matter, one has K = <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13130_2025_26268_Article_IEq3.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mi>A</mi> <mrow> <mn>4</mn> <mi>G</mi> </mrow> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{A}{4G} $</EquationSource> </InlineEquation>. Hereby, A denotes the area of B in a particular state.

Formulation and proof of the gravitational entropy bound

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2025年5月22日

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Artem Averin

摘要

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 $\frac{λ}{ℏ}$ $ \frac{\lambda }{\hslash } $ , 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) ≤ $\frac{λ}{ℏ}$ $ \frac{\lambda }{\hslash } $ . Especially, we show for the Einstein-Hilbert Lagrangian in any dimension with cosmological constant and arbitrary minimally coupled matter, one has K = $\frac{A}{4 G}$ $ \frac{A}{4G} $ . Hereby, A denotes the area of B in a particular state.