Quantized Abelian Principal Connections on Lorentzian Manifolds

Dappiaggi

Hack

et al. 2014

Commun. Math. Phys.

Self Cite

The aim of this work is to complete our program on the quantization of connections on arbitrary principal U (1)-bundles over globally hyperbolic Lorentzian manifolds. In particular, we show that one can assign via a covariant functor to any such bundle an algebra of observables which separates gauge equivalence classes of connections. The C * -algebra we construct generalizes the usual CCR-algebras since, contrary to the standard field-theoretic models, it is based on a presymplectic Abelian group instead of a symplectic vector space. We prove a no-go theorem according to which neither this functor, nor any of its quotients, satisfies the strict axioms of general local covariance. As a byproduct, we prove that a morphism violates the locality axiom if and only if a certain induced morphism of cohomology groups is non-injective. We then show that fixing any principal U (1)-bundle, there exists a suitable category of sub-bundles for which a quotient of our functor yields a quantum field theory in the sense of Haag and Kastler. We shall provide a physical interpretation of this feature and we obtain some new insights concerning electric charges in locally covariant quantum field theory.

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A C*-Algebra for Quantized Principal U(1)-Connections on Globally Hyperbolic Lorentzian Manifolds

Dappiaggi

Hack

et al. 2014

Commun. Math. Phys.

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Higher Structures in Algebraic Quantum Field Theory

Fortschritte der Physik

Schenkel

2019

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Models of Free Quantum Field Theories on Curved Backgrounds

Advances in Algebraic Quantum Field Theory

Dappiaggi

2015

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Abstract. Free quantum field theories on curved backgrounds are discussed via three explicit examples: the real scalar field, the Dirac field and the Proca field. The first step consists of outlining the main properties of globally hyperbolic spacetimes, that is the class of manifolds on which the classical dynamics of all physically relevant free fields can be written in terms of a Cauchy problem. The set of all smooth solutions of the latter encompasses the dynamically allowed configurations which are used to identify via a suitable pairing a collection of classical observables. As a last step we use such collection to construct a * -algebra which encodes the information on the dynamics and on the canonical commutation or anti-commutation relations depending whether the underlying field is a Fermion or a Boson. Geometric dataGoal of this section is to introduce all geometric concepts and tools which are necessary to discuss both the classical dynamics and the quantization of a free quantum field on a curved background. We assume that the reader is familiar with the basic notions of differential geometry and, to a minor extent, of general relativity. Therefore we will only sketch a few concepts and formulas, which we will use throughout this paper; a reader interested to more details should refer to [6,7,54], yet paying attention to the conventions used here, which differ from time to time from those in the cited references.Our starting point is M, a smooth manifold which is endowed with a (smooth) Lorentzian metric g of signature (+, −, . . . , −). Furthermore, although the standard generalizations to curved backgrounds of the field theories on Minkowski spacetime, on which the current models of particle physics are based, entail that M ought to be four dimensional, in this paper we shall avoid this assumption. The only exception will be Section 3.2, where we will describe Dirac spinors in four dimensions only, for the sake of simplicity. We introduce a few auxiliary, notable tensors. We employ an abstract index notation 1 and we stress that our conventions might differ from those of many textbooks, e.g. [54]. As a starting point, we introduce the Riemann tensor Riem : T M ⊗ T M → End(T M), defined using the abstract index notation by the formula 1 Notice that, in this paper, we employ the following convention for the tensor components: Latin indices, a,b,c,... , are used for abstract tensor indices, Greek ones, µ,ν,... for coordinates, while i, j,k are used for spatial components or coordinates.abc v c , where v is an arbitrary vector field and ∇ is the covariant derivative. The Ricci tensor is instead Ric : T M ⊗2 → R and its components are R ab = R d adb while the scalar curvature is simply R .= g ab R ab . For later convenience we impose a few additional technical constraints on the structure of the admissible manifolds, which we recollect in the following definition: Definition 1.1. For n ≥ 2, we call the pair (M, g) a Lorentzian manifold if M is a Hausdorff, second countable, connected, orientable, smoo...