Hanno Gottschalk

CONVOLUTED GENERALIZED WHITE NOISE, SCHWINGER FUNCTIONS AND THEIR ANALYTIC CONTINUATION TO WIGHTMAN FUNCTIONS

We construct Euclidean random fields X over by convoluting generalized white noise F with some integral kernels G, as X=G*F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general equivalent formulation of the cluster property in terms of truncated Schwinger functions which we then apply to the above fields. We present a partial negative result on the reflection positivity of convoluted generalized white noise. Furthermore, by representing the kernels Gα of the pseudo-differential operators for α∈(0, 1) and m0>0 as Laplace transforms we perform the analytic continuation of the (truncated) Schwinger functions of X=Gα*F, obtaining the corresponding (truncated) Wightman distributions on Minkowski space which satisfy the relativistic postulates on invariance, spectral property, locality and cluster property. Finally we give some remarks on scattering theory for these models.

Models of Local Relativistic Quantum Fields with Indefinite Metric (in All Dimensions)

Abstract.A condition on a set of truncated Wightman functions is formulated and shown to permit the construction of the Hilbert space structure included in the Morchio--Strocchi modified Wightman axioms. The truncated Wightman functions which are obtained by analytic continuation of the (truncated) Schwinger functions of Euclidean scalar random fields and covariant vector (quaternionic) random fields constructed via convoluted generalized white noise, are then shown to satisfy this condition. As a consequence such random fields provide relativistic models for indefinite metric quantum field theory, in dimension 4 (vector case), respectively in all dimensions (scalar case).

Scattering Theory for Quantum Fields¶with Indefinite Metric

Abstract: In this work, we discuss the scattering theory of local, relativistic quantum fields with indefinite metric. Since the results of Haag–Ruelle theory do not carry over to the case of indefinite metric [4], we propose an axiomatic framework for the construction of in- and out-states, such that the LSZ asymptotic condition can be derived from the assumptions. The central mathematical object for this construction is the collection of mixed vacuum expectation values of local, in- and out-fields, called the “form factor functional”, which is required to fulfill a Hilbert space structure condition. Given a scattering matrix with polynomial transfer functions, we then construct interpolating, local, relativistic quantum fields with indefinite metric, which fit into the given scattering framework.

Dynamical Backreaction in Robertson-Walker Spacetime

The treatment of a quantized field in a curved spacetime requires the study of backreaction of the field on the spacetime via the semiclassical Einstein equation. We consider a free scalar field in spatially flat Robertson–Walker spacetime. We require the state of the field to allow for a renormalized semiclassical stress tensor. We calculate the singularities of the stress tensor restricted to equal times in agreement with the usual renormalization prescription for Hadamard states to perform an explicit renormalization. The dynamical system for the Robertson–Walker scale parameter a(t) coupled to the scalar field is finally derived for the case of conformal and also general coupling.

A comment on the infra-red problem in the AdS/CFT correspondence

In this note we report on some recent progress in proving the AdS/CFT correspondence for quantum fields using rigorously defined Euclidean path integrals. We also comment on the infra-red problem in the AdS/CFT correspondence and argue that it is different from the usual IR problem in constructive quantum field theory. To illustrate this, a triviality proof based on hypercontractivity estimates is given for the case of an ultraviolet regularized potential of type: φ4:. We also give a brief discussion on possible renormalization strategies and the specific problems that arise in this context.

Nontrivial scattering amplitudes for some local relativistic quantum field models with indefinite metric

Abstract We study models of self-interacting massless spin 1 local relativistic quantum fields with indefinite metric in space-time dimension four. We prove that these models for large times converge to free fields and we derive explicit formulae for their (nontrivial, gauge invariant) scattering amplitudes. These scattering amplitudes have properties expected for S-matrix theory.

Representing Euclidean quantum fields as scaling limit of particle systems

We give a new representation of Euclidean quantum fields as scaling limits of systems of interacting, continuous, classical particles in the grand canonical ensemble.

Scattering behaviour of quantum vector fields obtained from Euclidean covariant SPDEs

We discuss Euclidean covariant vector random fields as the solution of stochastic partial differential equations of the form DA = η, where D is a covariant (w.r.t. a representation τ of SO(d)) differential operator with “positive mass spectrum” and η is a non-Gaussian white noise. We obtain explicit formulae for the Fourier transformed truncated Wightman functions, using the analytic continuation of Schwinger functions discussed by Becker, Gielerak and Lugewicz. Based on these formulae we give necessary and sufficient conditions on the mass spectrum of D which imply nontrivial scattering behaviour of relativistic quantum vector fields associated to the given sequence of Wightman functions. We compute the scattering amplitudes explicitly and we find that the masses of particles in the obtained theory are determined by the mass spectrum of D.

Optimal Reliability in Design for Fatigue Life

The failure of a component often is the result of a degradation process that originates with the formation of a crack. Fatigue describes the crack formation in the material under cyclic loading. Activation and deactivation operations of technical units are important examples in engineering where fatigue and especially low-cycle fatigue (LCF) play an essential role. A significant scatter in fatigue life for many materials results in the necessity of advanced probabilistic models for fatigue. Moreover, optimization of reliability is of vital interest in engineering, where with respect to fatigue the cost functionals are motivated by the predicted probability for the integrity of the component after a certain number of load cycles. The natural mathematical language to model failure, here understood as crack initiation, is the language of spatio-temporal point processes and their first failure times. This translates the problem of optimal reliability in the framework of shape optimization. The cost functionals derived in this way for realistic optimal reliability problems are too singular to be

Risk Estimation for LCF Crack Initiation

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