Gauge group,degeneration, and degrees of freedom in the ECSK theory of gravity. II. The Moncrief decomposition

It is shown that in the ECSK theory (coupled to an arbitrary tensor matter field) the degeneration distribution of the symplectic 2-form is closely connected with the action of the gauge groupG in the space of solutions to the field equations.     

A new view of the ECSK theory with a Dirac spinor

A formulation of the ECSK (Einstein-Cartan-Sciama-Kibble) theory with a Dirac spinor is given in terms of differential forms with values in exterior vector bundles associated with a fixed principalSL(2, )-bundle over a 4-manifold. In particular, tetrad fields are represented as soldering forms. In this setting, both the scalar curvature (Einstein-Hilbert) action density and the Dirac action density are well-defined polynomial functions of the soldering form and an independentSL(2,)-connection form. Thus, these densities are defined even where the tetrad field is degenerate (e.g. when fluctuations in the gravitational field are large). A careful analysis of the initial-value problem (in terms of an evolving triad field, SU(2)-connection, second-fundamental form and spinor field) reveals a first-order hyperbolic system of 27 evolution equations (not including the 8 evolution equations for the Dirac spinor) and 16 constraints. There are 10 conservation equations (due to local Poincaré invariance) which team up with some of the evolution equations to guarantee that the 16 constraints are preserved under the evolution.     

A Hamiltonian formulation of the Einstein-Cartan-Sciama-Kibble theory of gravity

We postulate the energy-momentum functionE for the ECSK theory of gravity and formulate the functional Hamiltonian equation in terms of the energy-momentum functionE and the symplectic 2-form . The system of partial differential equations which follows from the functional Hamilton equation is equivalent to the system of variational equations of the ECSK theory. The Hamiltonian method gives rise to a natural division of these equations into 10 constraint equations and the set of dynamical equations. We discuss the geometric sense of the constraint equations and their relations to the initial value problem.     

A singularity-free cosmological model in ECSK theory

In the framework of the ECSK [Einstein-Cartan-Sciama-Kibble] theory of cosmology, a scalar field nonminimally coupled to the gravitational field is considered. For a Robertson-Walker open universe (k=0) in the radiation era, the field equations admit a singularity-free solution for the scale factor. In theory, the torsion is generated through nonminimal coupling of a scalar field to the gravitation field. The nonsingular nature of the cosmological model automatically solves the flatness problem. Further absence of event horizon and particle horizon explains the high degree of isotropy, especially of 2.7-K background radiation.     

A partly reduced system of Hamiltonian equations and the initial value problem in the Ecsk theory of gravity

It is shown that by solving 12 of the field equations with respect to the connection components and , the quantities used to describe the geometry of space-time can be divided into two sets. In the first set we have the canonical variables the time evolution of which is determined by the dynamical equations. The second set contains ten gauge variables N, N^k, , n ⁽ⁱ⁾ which can be given arbitrarily on space-time. This partial reduction of the Hamiltonian equations enabls us to discuss the initial value problem in the ECSK theory of gravity coupled to matter tensor fields. Such an analysis is performed for the phenomenological ECSK theory and for the ECSK theory coupled to: a covector matter field, the generalized Maxwell electrodynamics, and the generalized Fermi-Dirac electrodynamics. The Poisson brackets of the seven Hamiltonian constraints, which have to be satisfied by the canonical variables, are found. It is proved that they are first class.     

Photon motion in the ECSK theory

Working within the scheme of the Einstein-Cartan-Sciama-Kibble Theory (ECSK) we find the trajectory of the photon up to its third order with respect to the velocity of slow motion sources. For the general case, discrepancies from the predictions of General Relativity (GR) are found. We apply the results to a model of polarized spin and find that in this particular case ECSK and GR theories coincide. We also perform a multipole expansion of the gravitational potentials in order to find the motion of photons far away from localized sources.     

On the post-Newtonian approximation of the ECSK theory II

We complete the PNA program for the ECSK theory. We find the general post-Newtonian equations of motion for the source. The different components of the complete affine connection, the torsion, and the energy-momentum tensor, as well as the conservation theorems of the theory, are developed for the case of an ideal fluid with spin in order to find the post-Newtonian trajectories of test particles exterior to the sources distribution. The main results are compared with the corresponding ones of general relativity.Supported by a scholarship from the Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET).     

Symplectic approach to the theory of quantized fields. I

Our aim in this paper the first one of a series concerned with the problem of field quantization starting from the symplectic structure underlying the classical theory, is to build up the variational theory necessary to all further constructions. The basic notions are the vertical bundle and thestructure 1-form used to define thegeneralized infinitesimal contact transformation which allows us to state and solve the variational problem related to field physics.Giving a system of modulevalued differential forms of different degree on the vertical bundle which solutions are the stationary cross sections is the main result in the paper. In this scheme the Euler-Lagrange classical equations are the expressions induced by such a system of differential forms on any cross section of the vertical bundle. This gives us a complete linearization of the Euler-Lagrange equations and, starting from it, a natural globalization of these equations. Finally, the notion of variational problem invariant by a Lie group is defined in this scheme, Noether's theorem related to such invariant problem is formulated and an intrinsic version of the so-called Noether invariants of classical variational calculus is obtained.This work has been realized in the Seminar of Mathematical Physics, directed by ProfessorJ. Sancho, in the Faculty of Science at the University of Barcelona (Spain).     

Gravitational,electromagnetic, and nuclear theory

Through an examination of neutron lifetimes we shall propose a new theory of gravitational, electromagnetic, and nuclear fields. The fundamental assumption of our theory is that the motion of a particle in a combination of gravitational, electromagnetic, and nuclear fields is determined from a variational principle of the form _A ^B d=0. The form of the physical time is determined from an examination of classical equations of motion. The field equations are determined from an examination of Maxwell-Einstein theory and Yukawa theory. Utilizing the standard elementary model of the deuteron, the theory predicts that at a proton-neutron separationr 10^–13 cm the neutron lifetime is infinite and that nucleons have a repulsive core. These predictions have been experimentally verified.     

Remarks on the quantum field theory in lattice space. II

We consider a strong-coupling approach to ⁴-meson theory as formulated in a lattice space which is of simple cubic type having lattice constanta and total volumeV. Self-adjointness and regularity of the Hamiltonian are established. The strong-coupling perturbation series are examined for the cases with and without mass renormalization. The series for ground state and one-particle state as well as for their energies are shown to converge when the coupling constant is sufficiently large, say > _c (sufficient condition). The bounds _c we have found increase with the total volumeV and/or the cut-off momentuma ^–1. Some other features of the strong-coupling perturbation theory are also discussed.     

The foundation of quantum theory and noncommutative spectral theory. Part I

The present paper is the first part of a work which follows up on H. Kummer: A constructive approach to the foundations of quantum mechanics,Found. Phys. 17, 1–63 (1987). In that paper we deduced the JB-algebra structure of the space of observables (=detector space) of quantum mechanics within an axiomatic theory which uses the concept of a filter as primitive under the restrictive assumption that the detector space is finite-dimensional. This additional hypothesis will be dropped in the present paper.It turns out that the relevant mathematics for our approach to a quantum mechanical system with infinite-dimensional detector space is the noncommutative spectral theory of Alfsen and Shultz.We start off with the same situation as in the previous paper (cf. Sects. 1 and 2 of the present paper). By postulating four axioms (Axioms S, DP, R, and SP of Sec. 3), we arrive in a natural way at the mathematical setting of Alfsen and Shultz, which consists of a dual pair of real ordered linear spaces Y, M: A base norm space, called the strong source space (which, however, in slight contrast to the setting of Alfsen and Shultz, is not 1-additive) and an order unit space, called the weak detector space, which is the norm and order dual space of Y. The last section of part I contains the guiding example suggested by orthodox quantum mechanics. We observe that our axioms are satisfied in this example. In the second part of this work (which will appear in the next issue of this journal) we shall postulate three further axioms and derive the JB-algebra structure of quantum mechanics.     

Effective-field theory of spin glasses and the coherent-anomaly method. I

A new cluster-effective-field theory of spin glasses is formulated. Basic formulas for the spin-glass transition point and the spin-glass susceptibility in the high-temperature phase are obtained. The present theory combined with the coherent-anomaly method is shown to be useful to estimate the true critical point and the nonclassical critical exponent of a spin-glass transition. Concerning the two-dimensional ±J model, we have _s =5.2(1) forT _SG=0, which agrees well with the data by some other authors. As for the threedimensional±J model, the present tentative analysis givesT _SG=1.2(1)(J/k _B) and _s =4(1), but more extensive calculations are needed.     

On the tetrad theory of gravity

After reviewing problems which appear in the theory of teleparallelism (tetrad theory) based on a Lagrangian quadratic in the torsion, the possibility of adding higher-order terms is discussed. For a test Lagrangian,O(3)-symmetric vacuum solutions and spatially homogeneous and isotropic solutions are found. The latter contain nonsingular cosmological models.     

Combinatorial quantization of the Hamiltonian Chern-Simons theory II

This paper further develops the combinatorial approach to quantization of the Hamiltonian Chern Simons theory advertised in [1]. Using the theory of quantum Wilson lines, we show how the Verlinde algebra appears within the context of quantum group gauge theory. This allows to discuss flatness of quantum connections so that we can give a mathematically rigorous definition of the algebra of observablesA _CS of the Chern Simons model. It is a *-algebra of functions on the quantum moduli space of flat connections and comes equipped with a positive functional (integration). We prove that this data does not depend on the particular choices which have been made in the construction. Following ideas of Fock and Rosly [2], the algebraA _CS provides a deformation quantization of the algebra of functions on the moduli space along the natural Poisson bracket induced by the Chern Simons action. We evaluate a volume of the quantized moduli space and prove that it coincides with the Verlinde number. This answer is also interpreted as a partition partition function of the lattice Yang-Mills theory corresponding to a quantum gauge group.Supported by Swedish Natural Science Research Council (NFR) under the contract F-FU 06821304 and by the Federal Ministry of Science and Research, Austria.Part of project P8916-PHY of the Fonds zur Förderung der wissenschaftlichen Forschung in ÖsterreichSupported in part by DOE Grant No DE-FG02-88ER25065     

Existence and renormalization of the Sine-Gordon field theory in all dimensions

The Euclidean field theory of the Sine-Gordon model (SG) is investigated by using methods of statistical physics. The SG model is shown to be the continuum limit of the ferromagnetic Harmonic Rotator model (HR) which itself is equivalent to a classical Coulomb plasma of unit charges on a lattice. Using our recent results for the latter models in all Euclidean dimensionsD we determine the existence of the SG field theory in terms of the HR parameters, temperaturet and magnetic fieldB. The following results are derived: 1. InD=1 the SG exists neart=B=0; the quantum theory in zero space dimensions is just the Mathieu equation as Schrödinger equation. 2. ForD>2 the SG field theory exists for allt nearB=0. The theory is actually constructed and is equivalent to a free massive scalar theory. 3. In the most interesting case ofD=2 the SG field theory exists for allt<t=8 nearB=0, it does not exist fortt. All necessary renormalizations are performed and all necessary subtractions are obtained in closed form which proves that the theory is superrenormalizable when it exists. We also discuss the relations between the structure of the particle spectrum of the SG, the phase transitions of the HR, and the binding properties of the classical Coulomb charges.     

The foundation of quantum theory and noncommutative spectral theory: Part II

The present paper comprises Sects. 5–8 of a work which proposes an axiomatic approach to quantum mechanics in which the concept of a filter is the central primitive concept. Having layed down the foundations in the first part of this work (which appeared in the last issue of this journal and comprises Sects. 0–4), we arrived at a dual pair Y, M consisting of abase norm space Y and anorder unit space M, being in order and norm duality with respect to each other. This is precisely the setting of noncommutative spectral theory, a theory which has been developed during the late nineteen seventies by Alfsen and Shultz. ^(2,3) In this part we add to the four axioms (Axioms S, DP, R, SP) of Sect. 3 three further axioms (Axioms E, O, L). These axioms are suggested by the work of Alfsen and Shultz and enable us to derive the JB-algebra structure of quantum mechanics (cf. Theorem 8.9).     

A non-local extension of the Gaspari-Gyorffy theory for superconductors

We present a theory which extends the Gaspari-Gyorffy theory to include non-local corrections within the framework of the spherical rigid muffin tin approximation. We thereby construct an on-Fermi-sphere pseudopotential similar to that of the simple free electron metals. We apply our theory to Al and NbC for which ²() andT _c are calculated. It turns out that for NbC the local approximation works well whereas for Al non-local corrections are important.     

On the physical interpretation of Wesson's gravitational theory

Wesson has proposed a five-dimensional theory of gravity where the fifth dimension is a parametrization of rest mass. In this paper we investigate the spacetime projection of this theory and point out its scalar-vector-tensor nature. A Brans-Dicke theory with=0 is shown to appear as a special case of Wesson's theory. It is pointed out that the task of constructing a realistic gravitational theory according to the scheme suggested by Wesson meets with serious difficulties.     

Kerr-type solution in Brans-Dicke theory

The Kerr-type solution in the Brans-Dicke theory should contain three parameters: a massm, a rotational parametera ₀, and a coupling parameter It goes over to the Kerr solution in Einstein's theory of general relativity in the limit 8. Using these conditions, we construct a special solution from Bruckman's solutions which can be regarded as a Kerr-type solution in the Brans-Dicke theory.     

Quantum theory of Ur-objects as a theory of information

The quantum theory of ur-objects proposed by C. F. von Weizsäcker has to be interpreted as a quantum theory of information. Ur-objects, or urs, are thought to be the simplest objects in quantum theory. Thus an ur is represented by a two-dimensional Hilbert space with the universal symmetry groupSU(2), and can only be characterized asone bit of potential information. In this sense it is not a spatial but aninformation atom. The physical structure of the ur theory is reviewed, and the philosophical consequences of its interpretation as an information theory are demonstrated by means of some important concepts of physics such as time, space, entropy, energy, and matter, which in ur theory appear to be directly connected with information as the fundamental substance. This hopefully will help to provide a new understanding of the concept of information.