An Algebra of Observables for de Sitter Space

Abstract

We describe an algebra of observables for a static patch in de Sitter space, with operators gravitationally dressed to the worldline of an observer. The algebra is a von Neumann algebra of Type II${}_1$. There is a natural notion of entropy for a state of such an algebra. There is a maximum entropy state, which corresponds to empty de Sitter space, and the entropy of any semiclassical state of the Type II${}_1$ algebras agrees, up to an additive constant independent of the state, with the expected generalized entropy $S_{\text{gen}}=(A/4G_N)+S_{\text{out}}$. An arbitrary additive constant is present because of the renormalization that is involved in defining entropy for a Type II${}_1$ algebra.

54 pages, v2: added references, v3,v4: minor corrections, v5: correction at end of section 2.4

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