Classical Statistical Field Theory Research Articles

FROM EUCLIDEAN FIELD THEORY TO QUANTUM FIELD THEORY

In order to construct examples for interacting quantum field theory models, the methods of Euclidean field theory turned out to be powerful tools since they make use of the techniques of classical statistical mechanics. Starting from an appropriate set of Euclidean n-point functions (Schwinger distributions), a Wightman theory can be reconstructed by an application of the famous Osterwalder–Schrader reconstruction theorem. This procedure (Wick rotation), which relates classical statistical mechanics and quantum field theory, is, however, somewhat subtle. It relies on the analytic properties of the Euclidean n-point functions. We shall present here a C*-algebraic version of the Osterwalder–Schrader reconstruction theorem. We shall see that, via our reconstruction scheme, a Haag–Kastler net of bounded operators can directly be reconstructed. Our considerations also include objects, like Wilson loop variables, which are not point-like localized objects like distributions. This point of view may also be helpful for constructing gauge theories.

Classical limit for scalar fields at high temperature

We study real-time correlation functions in scalar quantum field theories at temperature T = 1/ β. We show that the behaviour of soft, long-wave-length modes is determined by classical statistical field theory. The loss of quantum coherence is due to interactions with the soft modes of the thermal bath. The soft modes are separated from the hard modes by an infrared cutoff Λ ⪡ 1/( h ̷ β) . Integrating out the hard modes yields an effective theory for the soft modes. The infrared cutoff Λ controls corrections to the classical limit which are O (ħβΛ). As an application, the plasmon damping rate is calculated.

C*-ALGEBRAIC METHODS IN STATISTICAL MECHANICS AND FIELD THEORY

We survey the recent work in non-commutative operator algebras (especially AF-algebras, those which are inductive limits of finite dimensional C*-algebras) and which arise in studying critical phenomena in classical statistical mechanics and conformal field theory, from a C*- or topological viewpoint, rather than a von Neumann algebra/measure theoretic one.

A statistical dynamical theory of strongly nonlinear internal gravity waves

Abstract A statistical dynamical closure theory describing the interaction of strongly (and weakly) nonlinear two-dimensional internal waves in the presence of viscous dissipation and thermal conduction is derived. By applying renormalization methods originally formulated for quantum and classical statistical field theory, closures similar to the Direct Interaction and eddy-damped quasi-normal procedures of turbulence are derived. These methods are applied directly to the strongly nonlinear primitive field equations in Eulerian variables, thus avoiding the small amplitude assumptions inherent in the resonant interaction formalism. Propagator renormalization techniques provide formulas for the nonlinear internal wave frequency and spectral width in terms of the energy spectrum. The commonly used multiple time and space scale analysis is replaced by an analysis of the two-point correlation functions in terms of sum and difference variables. This permits the systematic development of a Landau equation. This ...

Field theoretical techniques in statistical fluid dynamics: With application to nonlinear wave dynamics

Abstract A derivation of two-point Markovian closure is presented in classical statistical field theory formalism. It is emphasized that the procedures used in this derivation are equivalent to those employed in the quantum statistical field theory derivation of the Boltzmann equation. Application of these techniques to the study of two-dimensional flow on a β-plane yields a quasi-homogeneous, quasi-stationary transport equation and a renormalized dispersion relation for Rossby waves

The use of projective limits in classical statistical mechanics and Euclidean quantum field theory

The Gibbs states of an infinite classical statistical system correspond to the states of “reservoir at infinity”. It is shown that its configuration space can be thought of as a generalized projective limit of configuration spaces of remote reservoirs. This notion of projective limit is defined and it is noted that it can also be used e.g. for proofs of the existence of Gibbs states in the thermodynamic limit and their decomposition into pure phase. A similar approach to (nonperturbative) Euclidean quantum field theory is suggested and connections with the concept of renormalizability are found.