Linear spin-zero quantum fields in external gravitational and scalar fields

We give mathematically rigorous results on the quantization of the covariant Klein Gordon field with an external stationary scalar interaction in a stationary curved space-time. We show how, following Segal, Weinless etc., the problem reduces to finding a “one particle structure” for the corresponding classical system. Our main result is an existence theorem for such a one-particle structure for a precisely specified class of stationary space-times. Byproducts of our approach are:

1. A discussion of when a given “equal-time” surface in a given stationary space-time is Cauchy.
2. A modification and extension of the methods of Chernoff [3] for proving the essential self-adjointness of certain partial differential operators.

     

Aspects of nucleon Compton scattering

We consider the spin-averaged nucleon forward Compton scattering amplitude in heavy baryon chiral perturbation theory including all terms to order . The chiral prediction for the spin-averaged forward Compton scattering amplitude is in good agreement with the data for photon energies110 MeV. We also evaluate the nucleon electric and magnetic Compton polarizabilities to this order and discuss the uncertainties of the various counter terms entering the chiral expansion of these quantities.     

Lower bounds on the cluster size distribution

We rigorously prove that the probabilityP _n that the origin of ad-dimensional lattice belongs to a cluster of exactlyn sites satisfiesP _n > exp(–n ^(d–1)/d ) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentrationp > p _c , as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system.     

Linear spin-zero quantum fields in external gravitational and scalar fields

The quantum theory of both linear, and interacting fields on curved space-times is discussed. It is argued that generic curved space-time situations force the adoption of the algebraic approach to quantum field theory: and a suitable formalism is presented for handling an arbitrary quasi-free state in an arbitrary globally hyperbolic space-time.For the interacting case, these quasi-free states are taken as suitable starting points, in terms of which expectation values of field operator products may be calculated to arbitrary order in perturbation theory. The formal treatment of interacting fields in perturbation theory is reduced to a treatment of free quantum fields interacting with external sources.Central to the approach is the so-called two-current operator, which characterises the effect of external sources in terms of purely algebraic (i.e. representation free) properties of the source-free theory.The paper ends with a set of Feynman rules which seems particularly appropriate to curved space-times in that it takes care of those aspects of the problem which are specific to curved space-times (and independent of interaction). Heuristically, the scheme calculates in-in rather than in-out matrix elements. Renormalization problems are discussed but not treated.Work partly supported by the Schweizerische Nationalfonds     

On the strategic origin of Brownian motion in finance

This paper is concerned with the strategic use of a private information on the stock market. A repeated auction model is used to analyze the evolution of the price system on a market with asymmetric information.  The model turns out to be a zero-sum repeated game with one-sided information, as introduced by Aumann and Maschler.  The stochastic evolution of the price system can be explicitly computed in the n times repeated case. As n grows to ∞, this process tends to a continuous time martingale related to a Brownian Motion.  This paper provides in this way an endogenous justification for the appearance of Brownian Motion in Finance theory. Received: February 2002     

Anderson localization for one- and quasi-one-dimensional systems

We prove almost-sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete Schrödinger operators on one- and quasi-one-dimensional lattices. This paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems; we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential. Our results allow us to prove that, e.g., for strong disorder, the smallest positive Lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity.