Symmetric Spaces of Matter and Real Fermion Manifolds

The geometric algebra Cl_3,1 generated by the Minkowski spacetime with signature {+++− } possesses a natural ternary partition which provides the Lie algebra of the standard model symmetry in an improved form. The symmetric spaces of matter embed a differentiable manifold of primitive idempotents which represents a real valued fermion space as an 8-dimensional real unitsphere in a 10-dimensional subspace with positive definite signature. The algebraic properties of the present theory of spacetime-matter are developed, beginning with the definiteness of the stabilizer algebra of neutrinos, investigating the orthogonality between fermions and neutrinos and ending with the curvature of the symmetric spaces of the strong force. The model brings together the quantum theory and relativity, as we conceive it at present, such that the standard model turns out to be a definite property of the spacetime algebra.     

Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Hyperbolic ends with particles and grafting on singular surfaces

We prove that any 3-dimensional hyperbolic end with particles (cone singularities along infinite curves of angles less than π) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with particles and convex globally hyperbolic maximal (GHM) de Sitter spacetime with particles, it follows that any 3-dimensional convex GHM de Sitter spacetime with particles also admits a unique foliation by constant Gauss curvature surfaces. We prove that the grafting map from the product of Teichmüller space with the space of measured laminations to the space of complex projective structures is a homeomorphism for surfaces with cone singularities of angles less than π, as well as an analogue when grafting is replaced by “smooth grafting”.     

On the Sporadic 196884-Dimensional Group,Strings and e (∞) Spacetime

Some arguments are presented using the dimension of the Golay code and the center density of intersecting spheres in 24-dimensional lattices to show that the 196884 dimensions of the so-called monster group can be regarded as hierarchical and reduced by clustering to a quasi expectation value of four, in analogy to the E ^(∞) spacetime dimensional reduction. The analysis suggests that the conjectured DNA-like Cantorian spacetime may resemble a giant error correction code.     

Deriving the curvature of fractal-Cantorian spacetime from first principles

The paper gives various exact derivations of the curvature of spacetime manifold at different energy scales within the frame work of a fractal-Cantorian theory. It is argued that at a Hausdorff spacetime dimensionality equal 4 + ³ = 4.236067977 the unification fractal spacetime Cantorian manifold possesses a curvature equal to K = 26 + k = 26.18033989.     

Dynamical symmetry breaking in hyperbolic 4D spacetime and in extra dimensions

We study the dynamical symmetry breaking in quark matter within two different models. First, we consider the effect of gravitational catalysis of chiral and color symmetries breaking in strong gravitational field of ultrastatic hyperbolic spacetime ℝ ⊗ H ³ in the framework of an extended Nambu-Jona-Lasinio model. Second, we discuss the dynamical fermion mass generation in the flat 4-dimensional brane situated in the 5D spacetime with one extra dimension compactified on a circle. In the model, bulk fermions interact with fermions on the brane in the presence of a constant abelian gauge field A ₅ in the bulk. The influence of the A ₅-gauge field on the symmetry breaking is considered both when this field is a background parameter and a dynamical variable.     

Hints of a new relativity principle from p-brane quantum mechanics

This report is an extension of a previous one hep-th/9812189. Several quantum mechanical wave equations for p-branes are proposed. The most relevant p-brane quantum mechanical wave equations determine the quantum dynamics involving the creation/destruction of p-dimensional loops of topology S^p, moving in a D-dimensional spacetime background, in the quantum state Φ. To implement full covariance we are forced to enlarge the ordinary relativity principle to a new relativity principle, suggested earlier by the author based on the construction of C-space, and also by Pezzaglia's poly-dimensional relativity, where all dimensions and signatures of spacetime should be included on the same footing.     

Gravitational instanton in Hilbert space and the mass of high energy elementary particles

While the theory of relativity was formulated in real spacetime geometry, the exact formulation of quantum mechanics is in a mathematical construction called Hilbert space. For this reason transferring a solution of Einstein’s field equation to a quantum gravity Hilbert space is far of being a trivial problem.

On the other hand ^(∞) spacetime which is assumed to be real is applicable to both, relativity theory and quantum mechanics. Consequently, one may expect that a solution of Einstein’s equation could be interpreted more smoothly at the quantum resolution using the Cantorian ^(∞) theory.

In the present paper we will attempt to implement the above strategy to study the Eguchi–Hanson gravitational instanton solution and its interpretation by ‘t Hooft in the context of quantum gravity Hilbert space as an event and a possible solitonic “extended” particle. Subsequently we do not only reproduce the result of ‘t Hooft but also find the mass of a fundamental “exotic” symplictic-transfinite particle m1.8 MeV as well as the mass M_x and M (Planck) which are believed to determine the GUT and the total unification of all fundamental interactions respectively. This may be seen as a further confirmation to an argument which we put forward in various previous publications in favour of an alternative mass acquisition mechanism based on unification and duality considerations. Thus even in case that we never find the Higgs particle experimentally, the standard model would remain substantially intact as we can appeal to tunnelling and unification arguments to explain the mass. In fact a minority opinion at present is that finding the Higgs particle is not a final conclusive argument since one could ask further how the Higgs particle came to its mass which necessitates a second Higgs field. By contrast the present argument could be viewed as an ultimate theory based on the existence of a “super” force, beyond which nothing else exists.     

Quarks confinement via Kaluza–Klein theory as a topological property of quantum classical spacetime phase transition

By introducing a curled fifth dimension, Kaluza–Klein theory predicted for the first time a connection between gravity and electromagnetism. An exacting look at this result shows that for a radius R of the fifth dimension equal to the Planck length, the coupling is exactly unity. The result is utilized to show that by introducing correction terms to the one loop renormalization equation of unification it can be made exact and subsequently quark confinement can be proven non-perturbatively as a property of the topology of quantum spacetime at the classical-quantum interface and the Planck phase transition.     

Functional tetrahedron equation

We describe a method for constructing classical integrable models in a (2+1)-dimensional discrete spacetime based on the functional tetrahedron equation, an equation that makes the symmetries of a model obvious in a local form. We construct a very general “block-matrix model,” find its algebraic-geometric solutions, and study its various particular cases. We also present a remarkably simple quantization scheme for one of those cases. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 370–384, December, 1998.     

A Remark on the Cosmic Microwave Background Radiation and the Hausdorff Dimension of Spacetime

We point out a possibility for a fundamental connection between the COBE satellite measurement of the microwave background radiation and the inverse of the Hausdorff dimension of quantum spacetime. Finally we give a derivation of the fine structure constant in terms of an 11-dimensional manifold.     

On the equations for p-adic closed and open strings

A survey of several pure mathematical results concerning the boundary-value problems for nonlinear pseudo-differential equation for closed and open strings in d-dimensional flat spacetime is presented. We obtained some results on existence or nonexistence of solutions. In particular, the absence of almost-periodic solutions was shown. We consider also some numerical approaches to the problems. The text was submitted by the author in English.     

Reduction of the n-dimensional static vacuum Einstein equation and generalized Schwarzschild solutions

We consider the static vacuum Einstein spacetime when the spatial factor is conformal to a n-dimensional pseudo-Euclidean space. The most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described. We obtain the entire set of solutions of the reduced system, where the classical Schwarzschild solution arises as a particular solution. In addition, we show that the Riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature.     

广义测不准关系与Reissner Nordstrom de Sitter黑洞熵

针对Reissner Nordstrom de Sitter时空背景，利用经广义测不准关系改进的薄层brick wall方法计算了黑洞熵。结果表明，由这种方法得到的黑洞熵上限与它的外视界和宇宙视界面积之和成正比，和人们预期的结果相符。从中揭示了黑洞 熵与视界面积之间的内在联系，也进一步表明了黑洞熵是视界面上量子态的熵，是一种量子效应。由广义测不准关系的引入看到，brick wall方法与引力场量子化可能存在着一些内在的联系。     

Gravitational radiation and the evolution of gravitational collapse in cylindrical symmetry

Using the Sparling form and a geometric construction adapted to spacetimes with a 2-dimensional isometry group, we analyse a quasi-local measure of gravitational energy. We then study the gravitational radiation through spacetime junctions in cylindrically symmetric models of gravitational collapse to singularities. The models result from the matching of collapsing dust fluids interiors with gravitational wave exteriors, given by the Einstein–Rosen type solutions. For a given choice of a frame adapted to the symmetry of the matching hypersurface, we are able to compute the total gravitational energy radiated during the collapse and state whether the gravitational radiation is incoming or outgoing, in each case. This also enables us to distinguish whether a gravitational collapse is being enhanced by the gravitational radiation.     

Feuilletages des espaces temps globalement hyperboliques par des hypersurfaces à courbure moyenne constante

We announce the following result: every maximal globally hyperbolic 3-dimensional spacetime with compact Cauchy surface, and with nonpositive constant curvature admits a unique time function whose fibers are constant mean curvature surfaces. We discuss the extension of this result in higher dimensions. To cite this article: T. Barbot et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Fractal space,cosmic strings and spontaneous symmetry breaking

The present paper is conceived within the framework of El Naschie's fractal-Cantorian program and proposes to develop a model of the fractal properties of spacetime. We show that, starting from the most fundamental level of elementary particles and rising up to the largest scale structure of the Universe, the fractal signature of spacetime is imprinted onto matter and fields via the common concept for all scales emanating from the physical spacetime vacuum fluctuations. The fractal structure of matter, field and spacetime (i.e. the nature and the Universe) possesses a universal character and can encompass also the well-known geometric structures of spacetime as Riemannian curvature and torsion and includes also, deviations from Newtonian or Einsteinian gravity (e.g. the Rössler conjecture). The leitmotiv of the paper is generated by cosmic strings as a fractal evidence of cosmic structures which are directly related to physical properties of a vacuum state of matter (VSM). We present also some physical aspects of a spontaneous breaking of symmetry and the Higgs mechanism in their relation with cosmic string phenomenology. Superconducting cosmic strings and the presence of cosmic inhomogeneities can induce to cosmic Josephson junctions (weak links) along a cosmic string or in connection with a cosmic string (self) interactions and thus some intermittency routes to a cosmic chaos can be explored. The key aspect of fractals in physics and of fractal geometry is to understand why nature gives rise to fractal structures. Our present answer is: because a fractal structure is a manifestation of the universality of self-organisation processes, as a result of a sequence of spontaneous symmetry breaking (SSB). Our conclusion is that it is very difficult to prescribe a certain type of fractal within an empty spacetime. Possibly, a random fractal (like a Brownian motion) characterises the structure of free space. The presence of matter will decide the concrete form of fractalisation. But, what does it mean the presence of matter? Can there exist a spacetime without matter or matter without spacetime? Possibly not, but consider on the other hand a space far removed from usual matter, or a space containing isolated small particles in which a very low density matter can exist. Very low density matter might be influenced by a fractal structure of space, for example in the sense that it is subject also to fluctuations structured by random fractals. Diffraction and diffusion experiments in an empty space and very low density matter could provide evidence of a fractal structure of space. However, at very high (Planck) densities, and a spacetime in which fluctuations represent also the source of matter and fields (which is very resonable within the context of a quantum gravity), we can assert that Einstein's dream of geometrising physics and El Naschie's hope to prove the fractalisation (or Cantorisation) of spacetime are fully realised.     

A geometric approach to error estimates for conservation laws posed on a spacetime

We consider a hyperbolic conservation law posed on an (N+1)-dimensional spacetime, whose flux is a field of differential forms of degree N. Generalizing the classical Kuznetsov’s method, we derive an L¹ error estimate which applies to a large class of approximate solutions. In particular, we apply our main theorem and deal with two entropy solutions associated with distinct flux fields, as well as with an entropy solution and an approximate solution. Our framework encompasses, for instance, equations posed on a globally hyperbolic Lorentzian manifold.     

n-Dimensional dual complex numbers

It is well-known that the quadratic algebrasQ _a,b = {z|z =x +qy,q ² =a +qb ,a, b, x, y ε ℝ,q ∉ ℝ }, also expressible as ℝ[x]/(x ² -bx -a), are, up to isomorphism, equivalent to just three algebras, corresponding to elliptic, parabolic and hyperbolic. These three types are usually represented byQ _−1,0,Q _0,0,Q _1,0 and called complex numbers, dual complex numbers and hyperbolic complex numbers, respectively. Each in turn describes a Euclidian, Galilean and Minkowskian plane. The hyperbolic complex numbers thus provide a 2-dimensional spacetime for special relativity physics (see e.g. [6]) and the dual complex numbers a 2-dimensional spacetime for Newtonian physics (see e.g. [17]). The present authors considered extensions of the hyperbolic complex numbers ton dimensions in [8], and here, in somewhat parallel fashion, some elements of algebra (in Section 1) and analysis (in Section 2) will be presented forn-dimensional dual complex numbers.      

Global Null Controllability of the 1-Dimensional Nonlinear Slow Diffusion Equation

The authors prove the global null controllability for the 1-dimensional nonlinear slow diffusion equation by using both a boundary and an internal control. They assume that the internal control is only time dependent. The proof relies on the return method in combination with some local controllability results for nondegenerate equations and rescaling techniques.