Event-Symmetry for Superstrings

I apply the principle of event-symmetry tosimple string models and discuss how these lead to theconviction that multiple quantization is linked todimension. It may be that string theory has to beformulated in the absence of space-time, which will thenemerge as a derived property of the dynamics. Anotherinterpretation of the event-symmetric approach whichembodies this is that instantons are fundamental. Just as solitons may be dual to fundamentalparticles, instantons may be dual to space-time events.Event-symmetry is then dual to instanton statistics. Inthat case a unification between particle statistics and gauge symmetry follows on naturally fromthe principle of event-symmetry. I build algebras whichrepresent symmetries of superstring theories extendingevent-symmetry, but which are also isomorphic to an algebra of creation and annihilationoperators for strings of fermionic partons.     

On shear free normal flows of a perfect fluid

Flows of a perfect fluid in which the flow-lines form a time-like shear-free normal congruence are investigated. The space-time is quite severely restricted by this condition on the flow: it must be of Petrov Type I and is either static or degenerate. All the degenerate fields are classified and the field equations solved completely, except in one class where one ordinary differential equation remains to be solved. This class contains the spherically symmetric non-uniform density fields and their analogues with planar or hyperbolic symmetry. The type D fields admit at least a one-parameter group of local isometries with space-like trajectories. All vacuum fields which admit a time-like shear-free normal congruence are shown to be static. Finally, shear-free perfect fluid flows which possess spherical or a related symmetry are considered, and all uniform density solutions and a few non-uniform density solutions are found. The exact solutions are tabulated in section 7.Supported by a Science Research Council Research Studentship and by a Turner and Newall Research Fellowship.     

Quantization of the Schwarzschild Black Hole: A Noether Symmetry Approach

We study the canonical formalism of a spherically symmetric space-time. In the context of the 3+1 decomposition with respect to the radial coordinate r, we set up an effective Lagrangian in which a couple of metric functions play the role of independent variables. We show that the resulting r-Hamiltonian yields the correct classical solutions which can be identified with the space-time of a Schwarzschild black hole. The Noether symmetry of the model is then investigated by utilizing the behavior of the corresponding Lagrangian under the infinitesimal generators of the desired symmetry. According to the Noether symmetry approach, we also quantize the model and show that the existence of a Noether symmetry yields a general solution to the Wheeler-DeWitt equation which exhibits a good correlation with the classical regime. We use the resulting wave function in order to (qualitatively) investigate the possibility of the avoidance of classical singularities.     

Integrability and Hopf solitons in models with explicitly broken <Emphasis Type="Italic">O</Emphasis>(3) symmetry

A wide class of models, built of the three component unit vector field living in the (3 + 1) Minkowski space-time, which explicitly break global O(3) symmetry are discussed. The symmetry breaking occurs due to the so-called dielectric function multiplying a standard symmetric term. Integrability conditions are found. Moreover, for some particular forms of the Lagrangian exact toroidal solutions with any Hopf index are obtained. It is proved that such a symmetry breaking influences the shape of the solitons whereas the energy as well as the Hopf index remain unchanged.Received: 2 September 2004, Revised: 29 September 2004, Published online: 15 November 2004     

Space-times with spherically symmetric hypersurfaces

When discussing spherically symmetric gravitational fields one usually assumes that the whole space-time is invariant under theO(3) group of transformations. In this paper, the Einstein field equations are investigated under the weaker assumption that only the 3-spacest=const areO(3) symmetric. The following further assumptions are made: (1) Thet lines are orthogonal to the spacest=const. (2) The source in the field equations in a perfect fluid, or dust, or the term, or the empty space. (3) With respect to the center of symmetry the fluid source may move only radially if at all. Under these assumptions one solution with a perfect fluid source, found previously by Stephani, is recovered and interpreted geometrically, and it is shown that it is the sole solution which is not spherically symmetric in the traditional sense. The paper ends with a general discussion of cosmological models whose 3-spacest=const are the same as in the Robertson-Walker models. No new solutions were explicitly found, but it is shown that such models exist in which the sign of curvature is not fixed in time.     

Tetrad symmetries

It is shown that ifG is a subgroup of the group of motions of a given space-time then a tetrad can be chosen which is symmetric under the subgroup if and only if the rank of the generators ofG is equal to the order ofG. As an example such a tetrad is constructed for the TypeN twisting empty space-time admitting a two-parameter group of motions.     

Infinite sets of conservation laws for linear and nonlinear field equations

The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of the space-time symmetry group is established. It is shown that each symmetric element of the enveloping algebra of the space-time symmetry group of a linear field equation generates a one-parameter group of symmetries of the field equation. The cases of the Maxwell and Dirac equations are studied in detail. Then it is shown that (at least in the sense of a power series in the coupling constant) the conservation laws of the linear case can be deformed to conservation laws of a nonlinear field equation which is obtained from the linear one by adding a nonlinear term invariant under the group of space-time symmetries. As an example, our method is applied to the Korteweg-de Vries equation and to the massless Thirring model.     

A unified approach to conservation laws in general relativity,gauge theories,and elementary particle physics

A unified treatment of conservation laws in general relativity, gauge theories, and elementary particle physics is formulated in the setting of principal fiber bundles. The group AUT(P) is introduced as the general gauge transformation group that covers space-time coordinate transformations. A set of master equations is exhibited for any Lagrangian density generally covariant with respect to AUT(P). The symmetry group for elementary particle theory is shown to be the structure group of the bundle only in the special case when the gauge potential is flat and the space-time is simply connected. In the general case, the symmetry group is reduced to the symmetry group of the gauge potential. This natural mechanism for a reduction of the symmetry group is speculated on as a model for spontaneous symmetry breaking.This essay received an honorable mention from the Gravity Research Foundation for the year 1981-Ed.Partially supported by a grant from the National Science Foundation.     

Why Does the Geometric Product Simplify the Equations of Physics?

In the last decades it was observed that Clifford algebras and geometric product provide a model for different physical phenomena. We propose an explanation of this observation based on the theory of bounded symmetric domains and the algebraic structure associated with them. The invariance of physical laws is a result of symmetry of the physical world that is often expressed by the symmetry of the state space for the system implying that this state space is a symmetric domain. For example, the ball of all possible velocities is a bounded symmetric domain. The symmetry on this ball follow from the symmetry of the space-time transformations between two inertial systems, which fixes the so-called symmetric velocity between them. The Lorenz transformations acts on the ball Sof symmetric velocities by conformal transformations. The ball Sis a spin ball (type IV in Cartan's classification). The Lie algebra of this ball is defined a triple product that is closely related to geometric product. The relativistic dynamic equations in mechanics and for the Lorenz force is described by this Lie algebra and the triple product.     

Symmetries and the Einstein-Maxwell field equations the null field case

This paper deals with space-times that satisfy the Einstein-Maxwell field equations in the presence of a perfect fluid, which may be charged. The electromagnetic field is assumed to be null. It is proved that if the space-time admits a group of isometrics then the fluid velocityu ⁱ, energy density, pressurep, and charge density are invariant under the group. In addition, if the charge density is nonzero, the electromagnetic field tensorf _ij is also invariant. On the other hand, examples of exact solutions are given which establish that if = 0, thenF _ij is not necessarily invariant under the group. In the case of spherically symmetric space-times, however, in which the group of isometries acting isSO (3),f _ij is invariant, independently of whether or not is nonzero. This result leads to the conclusion that in a spherically symmetric space-time the field equations in question admit no solutions with non-trivial null electromagnetic field.     

Braid matrices and structure constants for minimal conformal models

Using the Feigin-Fuchs representation of minimal conformal models in a form introduced recently by one of us, the braid group representation matrices, describing the analytic continuation properties of conformal blocks, are computed. In a suitable normalization, their matrix elements are shown to essentially factorize into pairs of Boltzmann weights of critical RSOS models in a certain limit of the spectral parameter. These Boltzmann weights are related to quantum groupR-matrices by the vertex-SOS transformation. We show that the crossing symmetry of the four-point function in left-right symmetric models follows from a quantum group relation, also called crossing symmetry. This observation gives a simple way to evaluate the structure constants.Supported by NSF grant DMS 8610730     

Time-symmetric fluctuations in nonequilibrium systems

For nonequilibrium steady states, we identify observables whose fluctuations satisfy a general symmetry and for which a new reciprocity relation can be shown. Unlike the situation in recently discussed fluctuation theorems, these observables are time-reversal symmetric. That is essential for exploiting the fluctuation symmetry beyond linear response theory. In addition to time reversal, a crucial role is played by the reversal of the driving fields that further resolves the space-time action. In particular, the time-symmetric part in the space-time action determines the second order effects of the nonequilibrium driving.     

Classical symmetries of some two-dimensional models

It is well-known that principal chiral models and symmetric space models in two-dimensional Minkowski space have an infinite-dimensional algebra of hidden symmetries. Because of the relevance of symmetric space models to duality symmetries in string theory, the hidden symmetries of these models are explored in some detail. The string theory application requires including coupling to gravity, supersymmetrization, and quantum effects. However, as a first step, this paper only considers classical bosonic theories in flat space-time. Even though the algebra of hidden symmetries of principal chiral models is confirmed to include a Kac-Moody algebra (or a current algebra on a circle), it is argued that a better interpretation is provided by a doubled current algebra on a semi-circle (or line segment). Neither the circle nor the semi-circle bears any apparent relationship to the physical space. For symmetric space models the line segment viewpoint is shown to be essential, and special boundary conditions need to be imposed at the ends. The algebra of hidden symmetries also includes Virasoro-like generators. For both principal chiral models and symmetric space models, the hidden symmetry stress tensor is singular at the ends of the line segment.     

An analysis of Fierz identities,factorization and inversion theorems

We show that the full set of Fierz identities which are used to compute electro-weak interactions reported by Y. Takahashi can be considered as particular cases of the Clifford product between multivector Cartan maps. Moreover, we think that our approach can be generalized to higher-dimensional models.We discuss the factorization and inversion theorems for the recovery of the spinor from its multivectorial Cartan map.A new classification given by P. Lounesto is applied to the recovered spinors for Cl_1,3 space-time symmetry and SU(2)×U(1) isotopic group.Dedicated to David Hestenes on his 60th birthday.     

Geodesics in a Quash-Spherical Spacetime: A Case of Gravitational Repulsion

No Heading Geodecis are studied in one of the Weyl metrics, referred to as the M-Q solution. First, arguments are provided, supporting our belief that this space-time is the more suitable (among the known solutions of the Weyl family) for discussing the properties of strong quasi-spherical gravitational fields. Then, the behaviour of geodesics is compared with the spherically symmetric situation, bringing out the sensitivity of the trajectories to deviations from spherical symmetry. Particular attention deserves the change of sign in proper radial acceleration of test particles moving radially along symmetry axis, close to the r = 2M surface, and related to the quadrupole moment of the source.     

Spherical symmetry of generalized EYMH fields

The possible actions of symmetry groups on generalized Higgs fields coupled to an Einstein–Yang–Mills field are studied with differential geometrical techniques involving principal and associated bundles. A classification of conjugacy classes of these actions and the form of the corresponding invariant Einstein–Yang–Mills–Higgs (EYMH) fields is obtained and then applied to the case of static spherically symmetric fields over four-dimensional space-time. We identify the representations of the gauge group for which spherically symmetric Higgs fields exist. Then the set of all field equations for the independent functions that describe these fields is analyzed and the corresponding ordinary system of differential equations is derived and shown to be consistent.     

Cosmological Models in Barber's Second Self-Creation Theory

An attempt has been taken to solve the field equations of Barber's second self creation theory with a perfect fluid in an inhomogeneous anisotropic Locally rotationally symmetric Bianchi type I space-time where the metric potentials are arbitrary functions of x and t. Because of the mathematical complexities, for particular forms of metric potentials vacuum, Zeldovich and radiation models are determined. It is shown that -vacuum model does not exist with the above choice of metric potentials. Even though the geometrical structure of Zeldovich and radiation models are the same and reveal same physical behaviour but they are governed by different Barber's scalar.     

Unified gravitational and Yang-Mills fields

We unify the gravitational and Yang-Mills fields by extending the diffeomorphisms in (N=4+n)-dimensional space-time to a larger group, called the conservation group. This is the largest group of coordinate transformations under which conservation laws are covariant statements. We present two theories that are invariant under the conservation group. Both theories have field equations that imply the validity of Einstein's equations for general relativity with the stress-energy tensor of a non-Abelian Yang-Mills field (with massive quanta) and associated currents. Both provide a geometrical foundation for string theory and admit solutions that describe the direct product of a compactn-dimensional space and flat four-dimensional space-time. One of the theories requires that the cosmological constant shall vanish. The conservation group symmetry is so large that there is reason to believe the theories are finite or renormalizable.     

Radiating black hole solutions in arbitrary dimensions

We prove a theorem that characterizes a large family of non-static solutions to Einstein equations in N-dimensional space-time, representing, in general, spherically symmetric Type II fluid. It is shown that the best known Vaidya-based (radiating) black hole solutions to Einstein equations, in both four dimensions (4D) and higher dimensions (HD), are particular cases from this family. The spherically symmetric static black hole solutions for Type I fluid can also be retrieved. A brief discussion on the energy conditions, singularities and horizons is provided.