Dynamical locality of the nonminimally coupled scalar field and enlarged algebra of Wick polynomials

We discuss dynamical locality in two locally covariant quantum field theories, the nonminimally coupled scalar field and the enlarged algebra of Wick polynomials. We calculate the relative Cauchy evolution of the enlarged algebra, before demonstrating that dynamical locality holds in the nonminimally coupled scalar field theory. We also establish dynamical locality in the enlarged algebra for the minimally coupled massive case and the conformally coupled massive case.Comment: 39p

Local covariant quantum field theory over spectral geometries

A framework which combines ideas from Connes' noncommutative geometry, or spectral geometry, with recent ideas on generally covariant quantum field theory, is proposed in the present work. A certain type of spectral geometries modelling (possibly noncommutative) globally hyperbolic spacetimes is introduced in terms of so-called globally hyperbolic spectral triples. The concept is further generalized to a category of globally hyperbolic spectral geometries whose morphisms describe the generalization of isometric embeddings. Then a local generally covariant quantum field theory is introduced as a covariant functor between such a category of globally hyperbolic spectral geometries and the category of involutive algebras (or *-algebras). Thus, a local covariant quantum field theory over spectral geometries assigns quantum fields not just to a single noncommutative geometry (or noncommutative spacetime), but simultaneously to ``all'' spectral geometries, while respecting the covariance principle demanding that quantum field theories over isomorphic spectral geometries should also be isomorphic. It is suggested that in a quantum theory of gravity a particular class of globally hyperbolic spectral geometries is selected through a dynamical coupling of geometry and matter compatible with the covariance principle.Comment: 21 pages, 2 figure

Dirac field on Moyal-Minkowski spacetime and non-commutative potential scattering

The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzian spectral geometries of which some basic aspects will be sketched. The scattering transformation will be shown to be unitarily implementable in the canonical vacuum representation of the Dirac field. Furthermore, it will be indicated how the functional derivatives of the ensuing unitary scattering operators with respect to the strength of the non-commutative potential induce, in the spirit of Bogoliubov's formula, quantum field operators (corresponding to observables) depending on the elements of the non-commutative algebra of Moyal-Minkowski spacetime.Comment: 60 pages, 1 figur

Cosmological horizons and reconstruction of quantum field theories

As a starting point, we state some relevant geometrical properties enjoyed by the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds. Those properties are generalised to a larger class of expanding spacetimes $M$ admitting a geodesically complete cosmological horizon \scrim common to all co-moving observers. This structure is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat at null infinity. Under suitable hypotheses on $M$, encompassing both the cosmological de Sitter background and a large class of other FRW spacetimes, the algebra of observables for a Klein-Gordon field is mapped into a subalgebra of the algebra of observables \cW(\scrim) constructed on the cosmological horizon. There is exactly one pure quasifree state $\lambda$ on \cW(\scrim) which fulfils a suitable energy-positivity condition with respect to a generator related with the cosmological time displacements. Furthermore $\lambda$ induces a preferred physically meaningful quantum state $\lambda_M$ for the quantum theory in the bulk. If $M$ admits a timelike Killing generator preserving \scrim, then the associated self-adjoint generator in the GNS representation of $\lambda_M$ has positive spectrum (i.e. energy). Moreover $\lambda_M$ turns out to be invariant under every symmetry of the bulk metric which preserves the cosmological horizon. In the case of an expanding de Sitter spacetime, $\lambda_M$ coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case. Remarks on the validity of the Hadamard property for $\lambda_M$ in more general spacetimes are presented.Comment: 32 pages, 1 figure, to appear on Comm. Math. Phys., dedicated to Professor Klaus Fredenhagen on the occasion of his 60th birthda

A geometrical origin for the covariant entropy bound

Causal diamond-shaped subsets of space-time are naturally associated with operator algebras in quantum field theory, and they are also related to the Bousso covariant entropy bound. In this work we argue that the net of these causal sets to which are assigned the local operator algebras of quantum theories should be taken to be non orthomodular if there is some lowest scale for the description of space-time as a manifold. This geometry can be related to a reduction in the degrees of freedom of the holographic type under certain natural conditions for the local algebras. A non orthomodular net of causal sets that implements the cutoff in a covariant manner is constructed. It gives an explanation, in a simple example, of the non positive expansion condition for light-sheet selection in the covariant entropy bound. It also suggests a different covariant formulation of entropy bound.Comment: 20 pages, 8 figures, final versio

Conformal generally covariant quantum field theory: The scalar field and its Wick products

In this paper we generalize the construction of generally covariant quantum theories given in the work of Brunetti, Fredenhagen and Verch to encompass the conformal covariant case. After introducing the abstract framework, we discuss the massless conformally coupled Klein Gordon field theory, showing that its quantization corresponds to a functor between two certain categories. At the abstract level, the ordinary fields, could be thought as natural transformations in the sense of category theory. We show that, the Wick monomials without derivatives (Wick powers), can be interpreted as fields in this generalized sense, provided a non trivial choice of the renormalization constants is given. A careful analysis shows that the transformation law of Wick powers is characterized by a weight, and it turns out that the sum of fields with different weights breaks the conformal covariance. At this point there is a difference between the previously given picture due to the presence of a bigger group of covariance. It is furthermore shown that the construction does not depend upon the scale mu appearing in the Hadamard parametrix, used to regularize the fields. Finally, we briefly discuss some further examples of more involved fields.Comment: 21 pages, comments added, to appear on Commun. Math. Phy

Functional Integral Construction of the Thirring model: axioms verification and massless limit

We construct a QFT for the Thirring model for any value of the mass in a functional integral approach, by proving that a set of Grassmann integrals converges, as the cutoffs are removed and for a proper choice of the bare parameters, to a set of Schwinger functions verifying the Osterwalder-Schrader axioms. The corresponding Ward Identities have anomalies which are not linear in the coupling and which violate the anomaly non-renormalization property. Additional anomalies are present in the closed equation for the interacting propagator, obtained by combining a Schwinger-Dyson equation with Ward Identities.Comment: 55 pages, 9 figure

Quantum Inequalities for the Electromagnetic Field

A quantum inequality for the quantized electromagnetic field is developed for observers in static curved spacetimes. The quantum inequality derived is a generalized expression given by a mode function expansion of the four-vector potential, and the sampling function used to weight the energy integrals is left arbitrary up to the constraints that it be a positive, continuous function of unit area and that it decays at infinity. Examples of the quantum inequality are developed for Minkowski spacetime, Rindler spacetime and the Einstein closed universe.Comment: 19 pages, 1 table and 1 figure. RevTex styl

Deformations of quantum field theories on spacetimes with Killing vector fields

The recent construction and analysis of deformations of quantum field theories by warped convolutions is extended to a class of curved spacetimes. These spacetimes carry a family of wedge-like regions which share the essential causal properties of the Poincare transforms of the Rindler wedge in Minkowski space. In the setting of deformed quantum field theories, they play the role of typical localization regions of quantum fields and observables. As a concrete example of such a procedure, the deformation of the free Dirac field is studied.Comment: 35 pages, 3 figure

The Asymptotic Safety Scenario in Quantum Gravity -- An Introduction

The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. All presently known evidence is surveyed: (a) from the 2+\eps expansion, (b) from the perturbation theory of higher derivative gravity theories and a `large N' expansion in the number of matter fields, (c) from the 2-Killing vector reduction, and (d) from truncated flow equations for the effective average action. Special emphasis is given to the role of perturbation theory as a guide to `asymptotic safety'. Further it is argued that as a consequence of the scenario the selfinteractions appear two-dimensional in the extreme ultraviolet. Two appendices discuss the distinct roles of the ultraviolet renormalization in perturbation theory and in the flow equation formalism.Comment: 77p, 1 figure; v2: revised and updated; discussion of perturbation theory in higher derivative theories extended. To appear as topical review in CQ