Superselection Sectors and General Covariance. I

Abstract

This paper is devoted to the analysis of charged superselection sectors in the framework of the locally covariant quantum field theories. We shall analize sharply localizable charges, and use net-cohomology of J.E. Roberts as a main tool. We show that to any 4-dimensional globally hyperbolic spacetime it is attached a unique, up to equivalence, symmetric tensor C * −category with conjugates (in case of finite statistics); to any embedding between different spacetimes, the corresponding categories can be embedded, contravariantly, in such a way that all the charged quantum numbers of sectors are preserved. This entails that to any spacetime is associated a unique gauge group, up to isomorphisms, and that to any embedding between two spacetimes there corresponds a group morphism between the related gauge groups. This form of covariance between sectors also brings to light the issue whether local and global sectors are the same. We conjecture this holds that at least on simply connected spacetimes. It is argued that the possible failure might be related to the presence of topological charges. Our analysis seems to describe theories which have a well defined short-distance asymptotic behaviour. 6 Local completeness 49 7 Conclusions and Outlook 53 A Tensor C * −categories 55 Lemma 2.3. Let M be a globally hyperbolic spacetime. If S is an open subset of M , then the sets J + (S), J − (S) and J(S) are open, hence J + (S) = I + (S), J − (S) = I − (S) and J(S) = I(S).

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

“…The authors of the present paper have recently put forward an analysis of the structure of superselection sectors [12] that provides a new perspective both by the adopted techniques and in the fact that superselection theory is now applicable to the larger setting of locally covariant quantum field theories [11]. The obtained results confirm that sectors of the kind that Doplicher, Haag and Roberts studied long ago find their most natural position in the locally covariant framework.…”

“…We refer the reader to [38,19,53,12] for a deeper introduction to some of the mathematical notions that we use.…”

Homotopy of Posets, Net-Cohomology and Superselection Sectors in Globally Hyperbolic Space-Times

“…We foresee to approach this problem in the future. Finally, we believe that this framework could be suitably generalized for applications in the context of the generally locally covariant quantum field theories [4], [7].…”

Quantum energy inequalities and local covariance II: categorical formulation

“…5 In many circumstances it would be more natural to think of E ′ as a functor to the category of topological vector spaces; however we wish to consider general subsets of testing functions in what follows, rather than subspaces.…”

“…Ideas of this type have already played an important role in the proof of a rigorous spin-statistics connection in curved spacetimes [46], the perturbative renormalisation of interacting scalar field theories in curved spacetime [32,33], and the theory of superselection sectors [6] and it seems that the complex of ideas set out by BFV will have many further applications in this area. (See also [5] for a review.) Indeed, one should say that all structural features of interest in QFT in CST should be formulated in this framework; any which are not capable of such reformulation must be either discarded as noncovariant, or (less likely, perhaps) prompt a review of the status of covariance itself.…”