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Quantum Charges and Spacetime Topology: The Emergence of New Superselection Sectors

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Abstract

A new form of superselection sectors of topological origin is developed. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C*–algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts’ cohomological analysis to the case where 1–cocycles bear non-trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in the case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1–cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I. All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1–cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part much resembles what in literature is known as geometric phases. Indeed, by the very geometrical origin of the 1–cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Trento, Via Sommarive 14, I-38050, Povo (TN), Italy

    Romeo Brunetti

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata,”, Via della Ricerca Scientifica, I-00133, Roma, Italy

    Giuseppe Ruzzi

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  1. Romeo Brunetti
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  2. Giuseppe Ruzzi
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Correspondence to Romeo Brunetti.

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Communicated by Y. Kawahigashi

Dedicated to Klaus Fredenhagen on the occasion of his sixtieth birthday

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Brunetti, R., Ruzzi, G. Quantum Charges and Spacetime Topology: The Emergence of New Superselection Sectors. Commun. Math. Phys. 287, 523–563 (2009). https://doi.org/10.1007/s00220-008-0671-6

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  • DOI: https://doi.org/10.1007/s00220-008-0671-6

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