Quantum charges and spacetime topology: the emergence of new superselection sectors
Over the years, the language and theory of topology has been successfully applied to specific effects in quantum physics and general relativity. To name just two of them recall the Aharonov-Bohm effect and the Berry phase. The language of the present paper is more abstract since it tries to give a complete model independent description using the algebraic structure of local quantum field theory in the sense of Doplicher, Haag and Roberts (DHR). Within this setting one finds charged superselection sectors for the net of algebras of local observables in a Minkowski space-time of dimension \(d\geq 3\). Recently, the authors of the present paper provided a new perspective of the structure of superselection sectors and general covariance applicable to a wider setting. They associate a unique symmetric tensorial \(C^*\)-category with any 4-dimensional globally hyperbolic space-time and show that the local covariance of sectors results from the analysis of 1-cocycles associated with the cohomology of posets as introduced and studied in detail by John Roberts. The main part of the present paper, however, is not yet based on local covariance. It is hopefully done in the future. Its main ingredients are (1) a generalized notion for representations of a net of local algebras of observables, (2) connecting these with 1-cocyles in the sense of Roberts, (3) a new selection criterion generalizing that of DHR, and (4) the definition of a von Neumann algebra as the group algebra obtained from each 1-cocycle. It is proven that any such cocycle may be split into two two parts: one that carries information on the charge content of the sector and one that carries information on the topology of space-time (perhaps a hint to geometric phases as they are called in ordinary quantum mechanics).
scientific article