On the equivalence of the Euclidean and Wightman formulation of field theory

A mistake in the paper [1] on the “Axioms for Euclidean Green's Functions” is corrected in the following sense: thanks to these axioms the Euclidean Schwinger functionsS _n can be analytically continued to the corresponding Wightman functionsW _n possessing all the correct analyticity properties and satisfying a generalized positivity condition in the complex domain. It is however suggested by the proof that their tempered behaviour near the Minkowski points cannot be guaranteed without additional assumptions.     

Once again on the equivalence theorem

The equivalence theorem in quantum field theory is proven on the basis of the field-antifield formalism. It is shown that the equivalence theorem does not contradict the well-known fact that, in quantum theory, some symmetries of the classical action functional are broken (anomalous). By way of example, a model is considered where natural finite counterterms can be chosen in different ways leading to physically nonequivalent quantum theories, but where the equivalence theorem remains valid.     

The equivalence theorem is subtle

We show that the equivalence theorem as is usually stated is valid only within the scheme of perturbation theory. New conditions have to be imposed on the theorem to make it valid in a nonperturbative context. Implications for a nonperturbative quantization of gravitation as also quantization of gauge theories in the unitary gauge are discussed.     

The equivalence theorem and effective Lagrangians

We point out that the equivalence theorem, which relates the amplitude for a process with external longitudinally polarized vector bosons to the amplitude in which the longitudinal vector bosons are replaced by the corresponding pseudo-Goldstone bosons, is not valid for effective Lagrangians. However, a more general formulation of this theorem also holds for effective interactions. The generalized theorem can be utilized to determine the high-energy behaviour of scattering processes just by power counting and to simplify the calculation of the corresponding amplitudes. We apply this theorem to the phenomenologically most interesting terms describing effective interactions of the electroweak vector and Higgs bosons in order to examine their effects on vector-boson scattering and on vector-boson-pair production in $f\bar f$ annihilation. The use of the equivalence theorem in the literature is examined.     

Generalization of boson-fermion equivalence and Fay's addition theorem

Generalizations of Fay's addition theorem for Abel functions are obtained by using generalized boson-fermion equivalence of off-shell string amplitudes. A simple example of such generalizations is presented explicitly which relates derivatives of a Riemann -function to its determinant.     

Multipole moments for stationary systems: The equivalence of the Geroch-Hansen formulation and the Thorne formulation

It is proved that the multipole moments of a stationary, asymptotically flat system in general relativity theory as defined by Thorne are identical, aside from normalization, to those defined by Geroch and Hansen: Here is Thorne's mass moment of orderl, is the Geroch-Hansen mass moment, is Thorne's current moment of orderl, and is Hansen's current moment. The mathematical techniques of Thorne are combined with those of Geroch and Hansen to prove several new theorems about multipole moments, and to give new proofs to some of the old theorems.     

Adler-Bardeen theorem in path integral formulation

The higher order contributions to Jacobian in Fujikawa’s path integral framework is considered and the form of anomaly equation in higher orders is established. An argument for the Adler-Bardeen theorem in this formulation is given.     

The central limit theorem and the problem of equivalence of ensembles

In this paper we show that the local limit theorem is a consequence of the integral central limit theorem in the case of a Gibbs random field _t ,tZ corresponding to a finite range potential.We apply this theorem to show that the equivalence between Gibbs and canonical ensemble is a consequence of the integral central limit theorem and of very weak conditions on decrease of correlations.Research supported by a C.N.R.-Ak. Nauk. U.R.S.S. fellowship     

An equivalence theorem for string solutions of the Einstein matter-gauge equations

String-like static solutions of the Einstein matter-gauge equations have interesting implications in cosmology. It has been shown recently that, at a critical coupling phase, this system of equations allows a reduction into a coupled Einstein-Bogomol'nyi system. In this Letter, we prove that, in the important case where the underlying two-dimensional Riemannian manifold is either compact or asymptotically Euclidean, the two systems are actually equivalent. Moreover, we show that the standard assumption that the strings reside in a conformally Euclidean surface will give us a metric which fails to be asymptotically Euclidean. In particular, in the radially symmetric case, we establish under the finite energy condition the boundary behavior of the metric. These results may indicate that a string solution will inevitably lead to nonflatness of the space at infinity even on the cross-section.     

On the equivalence of boundary conditions

We show that ifb andb are two boundary conditions (b.c.) for general spin systems on ^d such that the difference in the energies of a spin configuration in ^d is uniformly bounded, |H _,b ()–H _,b()|C < , then any infinite-volume Gibbs states and obtained with these b.c. have the same measure-zero sets. This implies that the decompositions of and into extremal Gibbs states are equivalent (mutually absolutely continuous). In particular, if is extremal,=. Application of this observation yields in an easy way (among other things) (a) the uniqueness of the Gibbs states for one-dimensional systems with forces that are not too long-range; (b) the fact that various b.c. that are natural candidates for producing non-translation-invariant Gibbs states cannot lead to such an extremal Gibbs state in two dimensions.Supported in part by NSF Grant PHY 78–15920 and by the Swiss National Foundation For Scientific Research.