Geometric phases characterise operator algebras and missing information

Abstract

We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.

Crossed product algebras and generalized entropy for subregions

“…The choice (41) simply shifts the entropy by the total energy, which scales like N . 33 In other words, one may choose whether to absorb this factor into the renormalization of U, or into the result for the entropy; but the former removes the area contribution of interest, so we choose the latter.…”

“…Additional support for this idea was subsequently given in [29,30], which conjectured that the change in character of the boundary algebra from type I to type III arises from the Hawking-Page transition, in which black holes become canonically favourable in the bulk; see [22,31] for more explanation. The nature of the algebras was finally proven only recently in [32] (see also [33] for further discussion of missing information in this context). In a remarkable paper [27], it was shown that one can deform the boundary algebra perturbatively in 1/N from type III 1 to type II ∞ by adjoining the generator of the modular automorphism group, which is dual to the ADM mass in the bulk, via the crossed product [4] (reviewed below, not to be confused with the more familiar cross product).…”