The Rainich conditions for neutrino fields

The algebraic Rainich conditions for neutrino fields are obtained in terms of an algebraic classification of symmetric second-rank tensors. The differential Rainich conditions are also obtained for ail but one class of neutrino fields. The algebraic classification of symmetric second-rank tensors used here is compared to two previously published classifications. The Weyl square of a symmetric second-rank tensor is introduced and the Petrov type of the Weyl square is used to give a method for determining the class of the symmetric tensor.     

Collective hamiltonians with Kac-Moody algebraic conditions

We describe the general framework for constructing collective-theory hamiltonians whose hermiticity requirements imply a Kac-Moody algebra of constraints on the associated jacobian. We give explicit examples for the algebras sl(2)_k and sl(3)_k. The reduction to W_n-constraints, relevant to n-matrix models, is described for the jacobians.     

On the Rainich geometrization of scalar meson fields

Similarly as in the Rainich geometrization of an electromagnetic field, the author finds a system of differential equations for the metric tensor, equivalent to the equations of the gravitational and scalar meson field, and shows how to find the wave function of the meson field if the Ricci tensor is known.     

A contribution to the Rainich theory of the neutrino field

The neutrino-gravitational field is divided into fourteen distinct classes. This classification is shown also to determine the class of the space-time. Then for four particular classes of space-time, necessary and sufficient conditions on the concomitants of the Ricci tensor are given for it to admit a neutrino field. It is further shown that for three of these classes the neutrino field is determined uniquely by the metric of space-time.     

Rainich theory for type D aligned Einstein–Maxwell solutions

The original Rainich theory for the non-null Einstein–Maxwell solutions consists of a set of algebraic conditions and the Rainich (differential) equation. We show here that the subclass of type D aligned solutions can be characterized just by algebraic restrictions.     

The algebraic Bethe ansatz for the Izergin–Korepin model with open boundary conditions

We present the procedure of exactly solving the Izergin–Korepin model with open boundary conditions by using the algebraic Bethe ansatz, which include constructing the multi-particle state and achieving the eigenvalue of the transfer matrix and corresponding Bethe equations. We give a proof about our conclusions on the multi-particle state based on an assumption. When the model is U_q(su(2)) quantum invariant, our results agree with that obtained by analytic Bethe ansatz method.     

The algebraic Bethe ansatz for the Osp(2|2) model with open boundary conditions

We obtain four different diagonal reflecting matrices by solving the reflection equation of the Osp(2|2) model. At the same time, we solve the model with open boundary condition by using the algebraic Bethe ansatz. The procedure of constructing the multi-particle state and achieving the eigenvalue of the transfer matrix and corresponding Bethe equations is presented in detail.     

On the Rainich geometrization of a vector meson field in the Kibble theory

The variables of a vector meson field are determined within the framework of the Kibble theory as the functions of the metric tensor, affine connection and their derivatives and a system of differential equations is found for the metric tensor and affine connection which is equivalent to the equations of motion of gravitational and vector meson fields.     

The logical quantization of algebraic groups

In a previous paper we introduced a highly abstract framework within which the theory of manuals initiated by Foulis and Randall is to be developed. The framework enabled us in a subsequent paper to quantize the notion of a set. Following these lines, this paper is devoted to quantizing algebraic groups viewed from Grothendieck's functorial standpoint.     

The algebraic structure of BRST quantization

It is shown that a system of first-class bosonic constraints obeying a Lie algebra has associated with it a natural superalgebra. BRST quantization arises as a non-linear representation of this superalgebra. Two distinct superalgebras are explicitly constructed and their associated BRST quantizations presented. The first BRST quantization is the canonical one with the BRST charge a grassmannian scalar. The second is new — the BRST charge is a grassmannian spinor transforming in the fundamental representation of the appropriate superalgebra. Generalizations are briefly discussed.     

On the algebraic consistency conditions in canonical λφ4 theory

The algebraic consistency requirements for the short distance behaviour of the operator product expansions φφ, φJ and JJ where (□ ? m²) φ = λJ, and J is local and trilinear (λφ⁴ theory) are studied. It is shown that in four dimensions (D = 4) under the assumption of a canonical scheme we cannot use “naive” Wilson expansions, abstracted from the short-distance behaviour of the products of free Wick powers. The inconsistency is also studied in an arbitrary and continuous number of dimensions and it disappears for D< 4.     

The algebraic structure of the fermionic string

The local structure of the product expansion algebra of the covariant NSR string is analyzed. An “on-shell” Kac-Moody like algebra is found to generate the BRST invariant part of the covariant lattice Γ_5,1. This algebra is a local version of ten-dimensional SUSY.     

The algebraic structure of the Onsager algebra

We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal theory.     

The algebraic structure of cohomological field theory

The algebraic foundation of cohomological field theory is presented. It is shown that these theories are based upon realizations of an algebra which contains operators for both BRST and vector supersymmetry. Through a localization of this algebra, we construct a theory of cohomological gravity in arbitrary dimensions. The observables in the theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, different moduli spaces are obtained by introducing curvature singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occuring differential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum field theory, and the discrete Regge calculus.     

The operational approach to algebraic quantum theory I

Recent work of Davies and Lewis has suggested a mathematical framework in which the notion of repeated measurements on statistical physical systems can be examined. This paper is concerned with an examination of their formulation in the abstract and its application to theC*-algebra model for quantum mechanics. In particular, a study is made of the notion of the restriction of a physical system and a definition, which coincides with the usual definition in theC*-algebra model, is formulated.     

The program ORTOCARTAN for algebraic calculations in relativity

The program ORTOCARTAN can calculate the curvature tensors (Riemann, Ricci, Einstein and Weyl) from a given orthonormal tetrad representation of the metric tensor. It was first announced in 1981, but since then has undergone several extensions and transplants onto other computers. This article reviews the current status of the program from the point of view of a user. The following topics are discussed: the problems that the program can be applied to, the special features of the algorithms that make the program powerful, the technical requirements to run the program and two simple examples of applications.     

The use of algebraic computing in general relativity

In the past ten years various computer systems have been developed able to perform algebraic calculations. Unfortunately, the fact that there are ready to use, mostly easily attainable, computer languages and programs for manipulation of non-numerical algebraic data is often overlooked by potential users. Several investigations in general relativity have been performed using such systems in the past few years, and in many cases the calculations were of such a length that it would have been prohibitive to complete them without help from a computer. In the first part of the paper we discuss the type of calculations that can be performed by algebraic systems, and several of these relativistic calculations are very briefly reviewed by way of example. In the second and main part of the paper we present a comparative review of most of the leading algebraic systems. To make the comparison more concrete we have taken two calculations from relativity and programed them, as closely as possible, in the same way for all these systems. It is not necessary for a future user who wants to do the same kind of calculations for other metrics to learn the complete syntax of one of these languages. He can make a slight modification to one of our programs, which we are prepared to distribute.     

Empirical algebraic geometry

In a previous paper we introduced the notion of an orthogonal category and generalized the notion of a sheaf of sets on a complete Boolean algebraB to that of a sheaf on the complete Boolean algebraB with values in an orthogonal categoryA. By properly replacing the complete Boolean algebraB by a manualM of Boolean locales, we get a notion of a sheaf onM with values inA, which can be regarded as a quantum generalization of a sheaf onB. TakingA to be the category of sheaves of Abelian groups or that of schemes à la Grothendieck, we will discuss some fundamental aspects of the quantum generalizations of sheaves and schemes.     

The principle of relativity, kinematics and algebraic relations

Based on the relativistic principle and the postulate of universal invariant constants (c, l), all kinematic symmetries can be set up as the subsets of the Umov-Weyl-Fock-Hua transformations for the inertial motions. These symmetries are connected to each other via combinations rather than via contractions and deformations.