Electromagnetism, Local Covariance, the Aharonov–Bohm Effect and Gauss’ Law

Sanders, Ko; Dappiaggi, Claudio; Hack, Thomas-Paul

doi:10.1007/s00220-014-1989-x

Cited by 66 publications

(120 citation statements)

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“…This yields a satisfactory theory at the level of the algebra A , see e.g. [30,33,74] for details. But it is far from clear how to construct an analog of B I for models with interaction, such as Yang-Mills theory, by proceeding this manner.…”

Section: Yang-mills Fieldsmentioning

confidence: 96%

Quantum Fields in Curved Spacetime

Hollands¹,

Wald²

2015

General Relativity and Gravitation

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We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states. We review the Unruh and Hawking effects for free fields, as well as the behavior of free fields in deSitter spacetime and FLRW spacetimes with an exponential phase of expansion. We review how nonlinear observables of a free field, such as the stress-energy tensor, are defined, as well as timeordered-products. The "renormalization ambiguities" involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar field, but a brief discussion of gauge fields is included. We conclude with a brief discussion of a possible approach towards a nonperturbative formulation of quantum field theory in curved spacetime and some remarks on the formulation of quantum gravity. Quantum field theory in curved spacetime (QFTCS) is the theory of quantum fields propagating in a background, classical, curved spacetime (M , g). On account of its classical treatment of the metric, QFTCS cannot be a fundamental theory of nature. However, QFTCS is expected to provide an accurate description of quantum phenomena in a regime where the effects of curved spacetime may be significant, but effects of quantum gravity itself may be neglected. In particular, it is expected that QFTCS should be applicable to the description of quantum phenomena occurring in the early universe and near (and inside of) black holesprovided that one does not attempt to describe phenomena occurring so near to singularities that curvatures reach Planckian scales and the quantum nature of the spacetime metric would have to be taken into account.It should be possible to derive QFTCS by taking a suitable limit of a more fundamental theory wherein the spacetime metric is treated in accord with the principles of quantum theory. However, this has not been done-except in formal and/or heuristic ways-simply because no present quantum theory of gravity has been developed to the point where such a well defined limit can be taken. Rather, the framework of QFTCS that we shall describe in this review has been obtained by suitably merging basic principles of classical general relativity with the basic principles of quantum field theory in Minkowski spacetime. As we shall explain further below, the basic principles of classical general relativity are relatively easy to identify and adhere to, but it is far less clear what to identify as the "basic principles" of quantum field theory in Minkowski spacetime. Indeed, many of the concepts normally viewed as fundamental to quantum field theory in Minkowski spacetime, such as Poinca...

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Section: Yang-mills Fieldsmentioning

confidence: 96%

Quantum Fields in Curved Spacetime

Hollands¹,

Wald²

2015

General Relativity and Gravitation

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“…The bottom line of some of these papers is the existence of a non-trivial center in the algebra of fields, provided certain topological, or more precisely cohomological, properties of the underlying background hold true. In [SDH12], it has been advocated that the elements of the center found in that paper could be interpreted in physical terms as being related to observables measuring electric charges. However, this leads unavoidably to a violation of the locality property (injectivity of the induced morphisms between the field algebras) of locally covariant quantum field theories, as formulated in [BFV03].…”

Section: Introductionmentioning

confidence: 98%

“…This peculiar feature is extremely relevant when one employs the algebraic framework in order to quantize such a theory on curved backgrounds. The first investigations along these lines are due to Dimock [Dim92], but a thorough analysis of topological effects started only recently, from both the perspective of the Faraday tensor [DL12] and, more generally, the quantization of linear gauge theories [Pfe09,DS13,FH13,HS13,SDH12,FS13]. The bottom line of some of these papers is the existence of a non-trivial center in the algebra of fields, provided certain topological, or more precisely cohomological, properties of the underlying background hold true.…”

Section: Introductionmentioning

confidence: 99%

“…The system can be quantized in the algebraic framework following the prescription outlined in [BDS14]. This procedure is advantageous for three main reasons: First of all there is no need to fix any reference connection, as it is done (implicitly) elsewhere [Pfe09,DS13,SDH12,FS13]. As a useful consequence of this, we were able to construct in [BDS13] purely topological observables, resembling topological quantum fields, which can measure the Chern class of the underlying principal U (1)-bundle.…”

Section: Introductionmentioning

confidence: 99%

“…The source of this obstruction can be traced back to the existence of disconnected components in the gauge group in the case of spacetimes with a non-trivial first de Rham cohomology group. From a physical point of view, this entails that those observables which are measuring the configurations tied to the Aharonov-Bohm effect, as discussed in [SDH12,FS13], are not contained in the algebra of observables.…”

Section: Introductionmentioning

confidence: 99%

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

Benini

Dappiaggi

Hack

et al. 2014

Commun. Math. Phys.

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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.

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Models of Free Quantum Field Theories on Curved Backgrounds

Benini

Dappiaggi

2015

Advances in Algebraic Quantum Field Theory

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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...

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Electromagnetism, Local Covariance, the Aharonov–Bohm Effect and Gauss’ Law

Cited by 66 publications

References 50 publications

Quantum Fields in Curved Spacetime

Quantum Fields in Curved Spacetime

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

Models of Free Quantum Field Theories on Curved Backgrounds

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