Poincaré gauge invariant spinor theory and the gravitational field. II

The nonlinear equation for an abstract noncanonical 2-component Weyl spinor field — as used with the inclusion of internal symmetries in Heisenberg's nonlinear spinor theory of elementary particles — which is invariant under scale, phase, and Poincaré transformations is modified in such a way as to become invariant under spacetime dependent phase gauge and Poincaré gauge transformations. In such an equation a phase gauge field B _m , six Lorentz gauge fields A[]m and four translation gauge fields g_m have to be introduced. It is demonstrated that all these fields can be identified as certain combinations of the Weyl spinor field, and hence should be considered in a rough sense as bound states of this spinor field. In particular the electromagnetic field B_m and the gravitational field g m appear as S-states and P-states of a spinor-antispinor system. The noncanonical property and the operator character of the spinor field is essential for this result. The relation between the translation gauge field and the spinor field involves a fundamental length. In a classical geometrical interpretation this relation leads to Einstein's equation of gravitation without cosmological term in a Riemannian space without torsion if the fundamental length is identified with Planck's length. It is shown that this equation is covariant under the larger symmetry group of phase gauge and Poincaré gauge transformations. The modified nonlinear equation constructed solely from a single 2-component Weyl field hence seems to incorporate in an extremely compact way electromagnetic and gravitational interaction in addition to non-mass-zero interactions. In this equation no arbitrary dimensionless constants enter. The considerations can be generalized to Dirac spinor fields and to spinor fields involving additional interior degress of freedom.An abridged version of this paper was presented at the International Conference on Gravitation and Relativity, Copenhagen, July 1971.     

Dissipation functions and conservation laws of molecular liquids and solids

Thermodynamics of molecular liquids (anisotropic liquids, liquid crystals, etc.) and molecular solids (e.g. dielectrics, pyroelectrics, molecular crystals) are treated in a unified way, using an internal energy potentialU and a dissipation functionD. Assuming that the motion of the mass points making up the molecule is essential in determiningU we write down the appropriate field equations of motion and the entropy equation. With the assumed invariance ofU under a group of space-time transformations and the invariance ofD under rigid body uniform motions, we derive the conservation laws of mass, momentum, energy and angular momentum. In particular, we show that the total stress tensor is asymmetric.Furthermore, a symmetric stress tensor does not guarantee angular momentum conservation. The complete specification ofD leads to symmetry relations between the dissipative coefficients in an unambiguous way, without invoking the concept of so called forces and fluxes, in contrast to the conventional Onsager approach. Our formalism allows for a class of different dissipation functions, and is applicable to linear or nonlinear molecular media of arbitrary symmetry. It covers simple materials as special cases.     

Field Theory on Kappa-spacetime

A general formalism is developed, that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the -deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable -product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.     

A Lax representation for the vertex operator and the central extension

Integrable hierarchies, viewed as isospectral deformations of an operatorL may admit symmetries; they are time-dependent vector fields, transversal to and commuting with the hierarchy and forming an algebra. In this work, the commutation relations for the symmetries are shown to be based on a non-commutative Lie algebra splitting theorem. The symmetries, viewed as vector fields onL, are expressed in terms of a Lax pair.This study introduces a generating symmetry, a generating function for symmetries, both of the KP equation (continuous), and the two-dimensional Toda lattice (discrete), in terms ofL and an operatorM, introduced by Orlov and Schulman, such that [L, M] = 1. This generating symmetry, acting on the wave function (or wave vector) lifts to a vertex operatorà la Date-Jimbo-Kashiwara-Miwa, acting on the -function (or -vector). Lifting the algebra of symmetries, acting on wave functions, to an algebra of symmetries, acting on -functions, amounts to passing from an algebra to its central extension.This provides a handy technology to find the constraints satisfied by various matrix integrals, arising in the context of 2d-quantum gravity and moduli space topology. The point is to first prove the vanishing of symmetries at the Lax pair level, which usually turns out to be elementary and conceptual, and then use the lifting above to get the subalgebra of vanishing symmetries for the -function (or -vectors).The support of a National Science Foundation grant # DMS-9203246 is gratefully acknowledged.The hospitality of the University of Louvain and Brandeis University is gratefully acknowledged.The support of a National Science Foundation grant # DMS-9203246, a Nato, a FNRS and a Francqui Foundation grant is gratefully acknowledged.     

Lie-symmetry vector fields for linear and nonlinear wave equations

We derive the Lie symmetry vector fields for the linear wave equation u=0 and nonlinear wave equation u=u ³. The conformal vector fields for the underlying metric tensor fieldg are also given. We construct the conservation laws and derive similarity solutions. Furthermore, we perform a Painlevé test for the nonlinear wave equation and discuss whether Lie-Bäcklund vector fields exist.     

A Note on Symmetries and Generalized W∞ Algebra of the Modified KP Equation

Some remarks on the paper Symmetries and generalized W algebra of the modified KP equation (S. Y. Lou and G. J. Ni, Lett. Math. Phys. 34 (1995), 327–331) are given. It is pointed out that if we consider the inverse operator of a differential operator to be a linear operator, the vector fields nv(f) defined in the above Letter are not certain to be symmetries of the modified KP equation under consideration.     

Transquantum Dynamics

Segal proposed transquantum commutation relations with two transquantum constants , besides Planck's quantum constant and with a variable i. The Heisenberg quantum algebra is a contraction—in a more general sense than that of Inönü and Wigner—of the Segal transquantum algebra. The usual constant i arises as a vacuum order-parameter in the quantum limit ,0. One physical consequence is a discrete spectrum for canonical variables and space-time coordinates. Another is an interconversion of time and energy accompanying space-time meltdown (disorder), with a fundamental conversion factor of some kilograms of energy per second.     

Non-trivial coupling of internal and space-time symmetries

Previous results which exhibit the impossibility of combining internal and space-time symmetries are reanalysed. Starting with McGlinn's (1964) work, where this trivial coupling appears as a direct product of these symmetries, it is shown that, by suitable generalisations such as the use of the framework of group extensions, and by the introduction of a new quantal observable, i.e. a mass-breaking operator, one can prove the existence of a non-trivial coupling scheme which admits mass splitting for the members of some super-multiplet. This leads to a new classification scheme for elementary particles. Contrary to the conventional classification models, where the choice of the underlying symmetry group does not emerge directly from the comparison of theoretical predictions with the experimental data, our scheme admits the possibility of determining the relevant symmetry from the mass spectrum.     

Spinor Field as Elementary Excitations of a System of Scalar Fields

The Dirac field and its quanta are obtained from the imposition of an infinite member of Dirac 2 nd class constraints on a system of complex scalar fields having an indefinite internal metric. The spin-1/2 character of the constrained system follows from constraint-induced coupling of the scalar system's independent internal and space-time symmetries, from constraint restrictions on allowed symmetries. The resulting spinor field quanta are seen to exist as a class of elementary excitations belonging to a dynamical algebra existing naturally within the system of complex scalar fields.     

Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation

The nonlinear Boltzmann equation for a rarefied gas is investigated in the fluid dynamical limit to the level of compressible Euler equation locally in time, as the mean free path tends to zero. The nonlinear hyperbolic conservation laws obtained as the limit are also the first approximation of the Chapman-Enskog expansion.     

A characterization of carter's separable space-times

In a space-time admitting a two-parameter Abelian isometry group, and a quadratic Killing tensor with the eigenvalues (, , , ) and vanishing Lie derivatives with respect to the Killing vectors, we construct a canonical coordinate system. The isometry group acts orthogonally transitively. The Hamilton-Jacobi equation is separable. We give a necessary and sufficient condition for the separability of the Klein-Gordon equation. We obtain Carter's space-times with completely separable Klein-Gordon equation.     

Clifford Fields and the Relativistic Equation of the Nucleon

A relativistic equation for free fields whichtake their values in the Clifford algebra associatedwith the Minkowski metric is shown to be interpretableas the equation of the nucleon. The internal symmetry group SU(2) arises naturally from theassociative algebra structure of the representationspace. The latter structure can be used to constructcoupling terms consistent with the transformationproperties of the interacting fields; in particular, itallows the familiar couplings of the nucleon field withthe electromagnetic field and with the -mesonfield.     

Additional symmetries for integrable equations and conformal algebra representation

We present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our method, additional symmetry equations arise together with their L-A pairs, so that they are integrable themselves. This procedure is based on a modified dressing method. For INEEs in 1+1 dimensions, some appropriate symmetry equations are shown to form the vector fields on a circle S ¹ algebra representation. In contrast to the so-called isospectral deformations, these symmetries result from conformal transformations of the associated linear problem spectrum. For INEEs in 2+1 dimensions, the commutation relations for symmetry equations are shown to coincide with operators , with integer m, p. Some additional results about Kac-Moody algebra applications are presented.     

Generalization of the relativistic string model in the geometrical approach

We present a model of a one-dimensional extended relativistic object, whose motion is defined by the requirement that its time track in Minkowski space is a surface of the constant mean curvature H. The world surface of the relativistic string is a particular case of such surfaces, namely, a minimal surface with H=0. By differential-geometry methods the theory of the proposed object moving in three-dimensional space-time is reduced to one nonlinear equation – = Hsh. In the theory under consideration, there naturally arises the pair of Lax's operators needed to solve this nonlinear equation by the inverse scattering method.     

A q-analogue of U(g[(N+1)), Hecke algebra,and the Yang-Baxter equation

We study for g=g[(N+1) the structure and representations of the algebra (g), a q-analogue of the universal enveloping algebra U(g). Applying the result, we construct trigonometric solutions of the Yang-Baxter equation associated with higher representations of g.     

Geometric Integrability of the Camassa–Holm Equation

It is observed that the Camassa–Holm equation describes pseudo-spherical surfaces and that therefore, its integrability properties can be studied by geometrical means. An sl(2, R)-valued linear problem whose integrability condition is the Camassa–Holm equation is presented, a Miura transform and a modified Camassa–Holm equation are introduced, and conservation laws for the Camassa–Holm equation are then directly constructed. Finally, it is pointed out that this equation possesses a nonlocal symmetry, and its flow is explicitly computed.     

Particle Creation in Cosmological Models with Varying Gravitational and Cosmological “Constants”

Einstein's field equations with variablegravitational coupling G = G(t) and decaying vacuumenergy density = (t) are considered asdescribing matter creation in a cosmological framework.The particle creation rate is determined by thevariation rate of both G and . By consideringsimple phenomenological evolution laws for G and Lambda,an exact solution of the gravitational field equationsfor a flat Friedmann-Robertson-Walker (FRW)space-time is obtained leading to a self-consistentcosmological model describing matter and entropygeneration in the very early Universe.     

On the space-time interpretation of classical canonical systems II: Relativistic canonical systems

Relativistic canonical systems and their symmetries are defined and classified within the class of canonical systems treated in a previous paper. Their algebra of variables contains a subset of monotone variables which satisfy a certain uniqueness condition and are later shown to increase strictly in the course of the dynamical evolution of the system on all physically acceptable states. This leads to a unique space-time interpretation of relativistic canonical systems. Finally we study space-time factorizations of such systems and introduce the appropriate notion of states. For a certain simple class of states the theory is shown to describe the motion of relativistic matter in some external gravitational and electromagnetic field.     

On the relation between weak and strong invariance of differential equations

Some concepts of Lie algebra cohomology are used to systematize the search for differential equations invariant under a given Lie groupG. In particular, it is shown that if a strongly invariant equation exists, then all weakly invariant equations differ from it only by an arbitrary multiplicative factor. If no strongly invariant equation exists, then cohomology theory can be used to simplify the search for weakly invariant equations.     

Symmetry-relations for the classical two-dimensional nonlinear O(n-σ)-model

We prove the identity of two Bäcklund transformations discussed in the literature and the corresponding conservation laws for the nonlinear O(n-) model. Furthermore, we exhibit more clearly the relation between the dual transformation and the Noether symmetry.