Uniqueness and global Markov property for Euclidean fields: The case of general polynomial interactions

We give a general method for proving uniqueness and global Markov property for Euclidean quantum fields. The method is based on uniform continuity of local specifications (proved by using potential theoretical tools) and exploitation of a suitable FKG-order structure. We apply this method to give a proof of uniqueness and global Markov property for the Gibbs states and to study extremality of Gibbs states also in the case of non-uniqueness. In particular we prove extremality for ₂ ⁴ (also in the case of non-uniqueness), and global Markov property for weak coupling ₂ ⁴ (which solves a long-standing problem). Uniqueness and extremality holds also at any point of differentiability of the pressure with respect to the external magnetic field.     

Uniqueness and the global Markov property for Euclidean fields. The case of trigonometric interactions

We prove the global Markov property for the Euclidean measure given by weak trigonometric interactions. To obtain this result we first prove a uniqueness theorem concerning the set of regular Gibbs measures corresponding to a given interaction.     

Extremality and the global Markov property II: The global markov property for non-FKG maximal Gibbs measures

We give a condition on a Gibbs measure for an attractive Markov specification, which assures extremality and the global Markov property. As an example of application we consider the class of attractive Markov specifications defined on a compact configuration space over a two-dimensional lattice by the interaction Hamiltonians (assumed to have a finite set of periodic ground configurations) satisfying Peierl's condition. We prove that each extremal Gibbs measure for such a specification, at sufficiently low temperature, has the global Markov property.On leave of absence from the Institute of Theoretical Physics, University of Wrocaw, Poland.     

Remarks on the global Markov property

The global Markov property is established for the + state and the – state of attractive lattice systems (e.g., the ferromagnetic Ising model and most other systems for which the FKG inequalities are satisfied) and of the (continuum) Widom Rowlinson model.Supported in part by NSF Grant No. PHY 78-03816     

Some examples concerning the global Markov property

We present examples of interactions of classical lattice systems whose extremal Gibbs states fail to have the global Markov property. One of the examples is translation invariant.Research supported by Natural Sciences and Engineering Research Council grant A4015     

Euclidean electromagnetic field as Ito's random current: a simple proof of Markov property

Euclidean electromagnetic field is considered as Ito's random current, which facilitates a simple proof of its Markov property.     

The global Markov property for equilibrium states which are determined by correlations in a strip

We present a necessary and sufficient condition for the non-existence of rotational invariant circles for area-preserving twist maps of the cyclinder or annulus based on the cone-crossing and killends criteria of MacKay and Percival (1985). Given a number of technical restrictions on the implementation of these criteria, this condition leads to a proof of MacKay and Percival's Finite Computation Conjecture.     

The Markoff property of the free Euclidean electromagnetic field

We show the Markoff property for the free Euclidean electromagnetic field in terms ofF ^ij.     

From Euclidean to relativistic fields and on the notion of Markoff fields

Recently, Nelson [2] has constructed relativistic fields from Euclidean fields which satisfy the Markoff and reflection property as well as an additional domain assumption. In this paper we replace the Markoff and reflection property by a weaker condition, a very simple positivity condition (“T-positivity”) which can be very easily expressed in terms of the expectation functionalE(f)=〈ω, exp {i φ (f)} ω〉. We show that the special role of the Markoff property in Nelson's approach is entirely due to features also shared byT-positivity. The role of Nelson's domain assumption (A′) in by-passing the difficulties with the paper of Osterwalder and Schrader [4] are made transparent, and possible ways to weaken this assumption are pointed out. If the conditions of [4] should turn out to be sufficient after all, (A′) can be replaced by a simple differentiability condition onE(τf). Our approach can also be applied to Fermi fields. The notion of Markoff and reflection property is discussed and shown to implyT-positivity.     

On the markoff property for the free Euclidean electromagnetic field

We prove that the free Euclidean electromagnetic field, as constructed by L. Gross in [2], has the Markoff property.     

Comment on the surface exponential for tensor fields

Starting from an essentially commutative exponential map E(B|I) for generic tensor-valued 2-forms B, which were introduced in [10] as a direct generalization of the ordinary noncommutative P exponent for 1 forms with values in matrices (i.e., in tensors of rank 2), we suggest a nontrivial but multiparametric exponential E(B|I|t_γ), which can serve as an interesting multidirectional evolution operator in the case of higher ranks. To emphasize the most important aspects of the article, the construction is restricted to the backgrounds I _ijk , which are associated with the structure constants of the commutative associative algebras, which make it insensitive to the topology of the 2D surface. Boundary effects are also eliminated (straightforward generalization is needed to incorporate them).     

Uniqueness and exponential decay of correlations for some two-dimensional spin lattice systems

We consider a two-dimensional lattice spin system which naturally arises in dynamical systems called coupled map lattice. The configuration space of the spin system is a direct product of mixing subshifts of finite type. The potential is defined on the set of all squares in Z² and decays exponentially with the linear size of the square. Via the polymer expansion technique we prove that for sufficiently high temperatures the limit Gibbs distribution is unique and has an exponential decay of correlations.     

Analytic renormalisation of the exponential interaction and weak interaction singularities

The parity non conserving interaction of a neutral vector meson with fermions is considered as a mathematical model suitable for investigating divergence problems of the weak interactions. Through the Stückelberg formalism and a canonical transformation the interaction is converted into an exponential form. The exponential interaction is studied, in the second order of perturbation theory, through the method of analytic renormalisation. Generalised amplitudes are introduced as localizable distributions depending on auxiliary complex parameters . It is shown that the distributions possess a nonisolated singularity at the physical point ₀. A method is developed for discarding the singularity thereby obtaining the physical amplitudes as localizable distributions which display a non-analytic dependence on the coupling constant.     

Markov chains with exponentially small transition probabilities: First exit problem from a general domain. II. The general case

In this paper we consider aperiodic ergodic Markov chains with transition probabilities exponentially small in a large parameter . We extend to the general, not necessarily reversible case the analysis, started in part I of this work, of the first exit problem from a general domainQ containing many stable equilibria (attracting equilibrium points for the = dynamics). In particular we describe the tube of typical trajectories during the first excursion outsideQ.     

The connection between an Euclidean Gauss Markov Vector Field and the Real Proca Wightman Field

We construct an Euclidean Gauss Markov Vector Field which leads to the Real Proca Wightman Field describing particles of mass >0 and spin 1, for space-time dimension equal to 4.     

On the superpropagator of fields with exponential coupling

We define the vacuum expectation value of the time-ordered product of two exponentials of free fields as a distribution using minimal singularity as a criterion. The implication of this definition for an exponentially self-coupled scalar field is studied in second order of a perturbation expansion.