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Carlos Castro

Publications and source records attributed to Carlos Castro.

At least 55 records · Page 3Linked to original sources
arXiv2001

Comments on the Riemann conjecture and index theory on Cantorian fractal space-time

An heuristic proof of the Riemman conjecture is proposed. It is based on the old idea of Polya-Hilbert. A discrete/fractal derivative self adjoint operator whose spectrum may contain the nontrivial zeroes of the zeta function is presented. To substantiate this heuristic proposal we show using generalized index-theory arguments, corresponding to the (fractal) spectral dimensions of fractal branes living in Cantorian-fractal space-time, how the required $negative$ traces associated with those derivative operators naturally agree with the zeta function evaluated at the spectral dimensions. The $ζ(0) = - 1/2$ plays a fundamental role. Final remarks on the recent developments in the proof of the Riemann conjecture are made.

hep-th↗
arXiv2001

On p-adic Stochastic Dynamics, Supersymmetry and the Riemann Conjecture

We construct (assuming the quantum inverse scattering problem has a solution ) the operator that yields the zeroes of the Riemman zeta function by defining explicitly the supersymmetric quantum mechanical model (SUSY QM) associated with the p-adic stochastic dynamics of a particle undergoing a Brownian random walk . The zig-zagging occurs after collisions with an infinite array of scattering centers that fluctuate randomly. Arguments are given to show that this physical system can be modeled as the scattering of the particle about the infinite locations of the prime numbers positions. We are able then to reformulate such p-adic stochastic process, that has an underlying hidden Parisi-Sourlas supersymmetry, as the effective motion of a particle in a potential which can be expanded in terms of an infinite collection of p-adic harmonic oscillators with fundamental (Wick-rotated imaginary) frequencies $ω_p = i log~p$ (p is a prime) and whose harmonics are $ω_{p, n} = i log ~ p^n$. The p-adic harmonic oscillator potential allow us to determine a one-to-one correspondence between the amplitudes of oscillations $a_n$ (and phases) with the imaginary parts of the zeroes of zeta $λ_n$, after solving the inverse scattering problem.

physics.gen-ph↗
arXiv2000

The status and programs of the New Relativity Theory

A review of the most recent results of the New Relativity Theory is presented. These include a straightforward derivation of the Black Hole Entropy-Area relation and its $logarithmic$ corrections; the derivation of the string uncertainty relations and generalizations ; ; the relation between the four dimensional gravitational conformal anomaly and the fine structure constant; the role of Noncommutative Geometry, Negative Probabilities and Cantorian-Fractal spacetime in the Young's two-slit experiment. We then generalize the recent construction of the Quenched-Minisuperspace bosonic $p$-brane propagator in $D$ dimensions ($AACS$ [18]) to the full multidimensional case involving all $p$-branes : the construction of the Multidimensional-Particle propagator in Clifford spaces ($C$-spaces) associated with a nested family of $p$-loop histories living in a target $D$-dim background spacetime . We show how the effective $C$-space geometry is related to $extrinsic$ curvature of ordinary spacetime. The motion of rigid particles/branes is studied to explain the natural $emergence$ of classical spin. The relation among $C$-space geometry and ${\cal W}$, Finsler Geometry and (Braided) Quantum Groups is discussed. Some final remarks about the Riemannian long distance limit of $C$-space geometry are made.

physics.gen-ph↗
arXiv2000

Conformally Invariant Sigma Models on Anti de Sitter Spaces, Chern-Simons p-branes and W Geometry

Conformally invariant sigma models in $D=2n$ dimensions with target non-compact O(2n,1) groups are studied. It is shown that despite the non-compact nature of the O(2n,1) groups, the classical action and Hamiltonian are positive definite. Instanton field configurations are found to correspond geometrically to conformal ``stereographic'' mappings of $R^{2n}$ into the Euclidean signature $AdS_{2n}$ spaces. Zaikov's relationship between Self Dual $p$-branes and Chern-Simons $p'$-branes, provided $p=p'+1$ and the embedding $D=p+1$-dimensional manifold is Euclidean, is elaborated further. Branes actions can be obtained also from a Moyal deformation quantization of Generalized Yang Mills Theories. Using this procedure, we show how four dimensional SU(N) YM theories contain Chern-Simons membranes and hadronic bags in the large $N$ limit. Since Chern-Simons $p'$-branes have an underlying infinite dimensional algebra containing $W_{1+\infty}$, as shown by Zaikov, we discuss the importance that $W$ geometry should have in the final formulation of $M$ theory.

hep-th↗
arXiv2000

On the four dimensional Conformal Anomaly, Fractal Spacetime and the Fine Structure Constant

Antoniadis, Mazur and Mottola (AMM) two years ago computed the intrinsic Hausdorff dimension of spacetime at the infrared fixed point of the quantum conformal factor in 4D Gravity. The fractal dimension was determined by the coefficient of the Gauss-Bonnet topological term associated with the conformal gravitational anomaly and was found to be greater than 4. It is explicitly shown how one can relate the value of the Hausdorff dimension computed by AMM to the universal dimensional fluctuation of spacetime $ε$ given by $ϕ^3/2$, where $ϕ$ is the Golden Mean $0.618..$. Based on the infrared scaling limit of the theory and using recent Renormalization Group arguments by El Naschie, we conjecture that the unknown coefficient $Q^2$, associated with the four dimensional gravitational conformal anomaly, could be precisely equal to the inverse fine structure constant values ranging between $137.036 $ and 137.081. Our results generate decimal digits up to any $arbitrary$ number.

physics.gen-ph↗
arXiv2000

Why we live in 3 Dimensions

A Cantorian fractal spacetime, a family member of von Neumann's noncommutative geometry is introduced as a geometry underlying a new relativity theory which is similar to the relation between general relativity and Riemannian geometry. Based on this model and the new relativity theory an ensemble distribution of all the dimensions of quantum spacetime is derived with the help of Fermat grand theorem. The calculated average dimension is very close to the value of $4+ϕ^3 $ (where $ϕ$ is the golden mean) obtained by El Naschie on the basis of a different approach. It is shown that within the framework of the new relativity the cosmological constant problem is nonexistent, since the Universe self-organizes and self-tunes according to the renormalization group (RG) flow with respect to a local scaling microscopic arrow of time. This implies that the world emerged as a result of a non-equilibrium process of self-organized critical phenomena launched by vacuum fluctuations in Cantorian fractal spacetime $\cal E^{\infty}$. It is shown that we are living in a metastable vacuum and are moving towards a fixed point ($ D$ = 4+$ϕ^3$) of the RG. After reaching this point, a new phase transition will drive the universe to a quasi-crystal phase of the lower average dimension of $ϕ^3$.

hep-th↗
arXiv2000

On M Theory, Quantum Paradoxes and the New Relativity

Recently a New Relativity Principle has been proposed by one of the authors as the underlying physical and geometrical foundations of String and {\bf M} Theory. It is explicitly shown that within the framework of the New Relativity Theory, some Quantum Mechanical Paradoxes like the Einstein-Rosen Podolsky and the Black Hole Information Loss, are easily resolved. Such New Relativity Theory requires the introduction of an Infinite Dimensional Quantum Spacetime as has been shown recently by one of us. This can be viewed as just another way of looking at Feynman's path integral formulation of Quantum Mechanics. Instead of having an infinite dimensional funcional integral over $all$ paths, smooth, forwards and backwards in time, random and fractal, in a finite-dimensional spacetime, one has a finite number of paths in an Infinite Dimensional Quantum Spacetime. We present a few-lines proof why there is no such a thing as an {\bf EPR} Paradox in this New Relativity theory. The reason is {\bf not} due to a superluminal information speed but to a {\bf divergent} information charge density. In the infinite dimensional limit, due to the properties of gamma functions, the hypervolume enclosed by a $D$-dim hypersphere, of finite nonzero radius, shrinks to zero : to a {\bf hyperpoint}, the infinite-dimensional analog af a point. For this reason, Information flows through the infinite-dimensional hypersurface of nonzero radius, but zero size, the hyperpoint, in an instant. In this fashion we imbue an abstract mathematical "point" with a true physical meaning : it is an entity in infinite dimensions that has zero hypervolume at nonzero radius . A plausible resolution of the Information Loss Paradox in Black Holes is proposed.

physics.gen-ph↗
arXiv2000

Is Quantum Spacetime Infinite Dimensional ?

The Stringy Uncertainty relations, and corrections thereof, were explicitly derived recently from the New Relativity Principle that treats all dimensions and signatures on the same footing and which is based on the postulate that the Planck scale is the minimal length in Nature in the same vein that the speed of light was taken as the maximum velocity in Einstein's theory of Special Relativity. A simple numerical argument is presented which suggests that Quantum Spacetime may very well be $infinite$ dimensional. A discussion of the repercusions of this new paradigm in Physics is given. A truly remarkably simple and plausible solution of the cosmological constant problem results from the New Relativity Principle : The cosmological constant is $not$ constant, in the same vein that Energy in Einstein's Special Relativity is observer dependent. Finally, following El Naschie, we argue why the observed D=4 world might just be an $average$ dimension over the infinite possible values of the Quantum Spacetime and why the compactification mechanisms from higher to four dimensions in String theory may not be actually the right way to look at the world at Planck scales.

hep-th↗
arXiv2000

The String Uncertainty Relations follow from the New Relativity Principle

The String Uncertainty Relations have been known for some time as the stringy corrections to the original Heisenberg's Uncertainty principle. In this letter the Stringy Uncertainty relations, and corrections thereof, are explicitly derived from the New Relativity Principle that treats all dimensions and signatures on the same footing and which is based on the postulate that the Planck scale is the minimal length in Nature in the same vein that the speed of light was taken as the maximum velocity in Einstein's theory of Special Relativity. The Regge behaviour of the string's spectrum is also a natural consequence of this New Relativity Principle.

hep-th↗
arXiv1999

Hints of a New Relativity Principle from $p$-brane Quantum Mechanics

This report is an extension of 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 {\bf C}-space, and also by Pezzaglia's Polydimensional Relativity, where all dimensions and signatures of spacetime should be included on the same footing.

hep-th↗
arXiv1999

Branes from Moyal Deformation Quantization of Generalized Yang Mills Theories

It is shown that a Moyal deformation quantization of the SO(4k) Generalized Yang-Mills (GYM) theory action in D=4k dimensions, for spacetime independent field configurations, in the $\hbar \to 0$ limit, yields the Dirac-Nambu-Goto p-brane actions (obtained from the conformally invariant Dolan-Tchrakian p-brane actions after elimination of the auxiliary world volume metrics), in the orthonormal gauge, for p+1=4k world volumes embedded in a D=4k target spacetime background. The gauge fields/target spacetime coordinates correspondence is required but no large N limit is necessary. The equivalence between Moyal SDYM and Self Dual p-branes is proposed without choosing the orthonormal gauge.

hep-th↗
arXiv1998

p-Brane Quantum Mechanical Wave Equations

Several quantum mechanical wave equations for $p$-branes are proposed based on the role that the volume-preserving diffeomorphisms group has on the physics of extended objects. The $p$-brane quantum mechanical wave equations determine the quantum dynamics involving the creation/destruction of $p$-branes in a $D$ dimensional spacetime background with a given world-volume measure configuration in a given quantum state $Ψ$.

hep-th↗
arXiv1998

The Search for the Origins of M Theory : Loop Quantum Mechanics, Loops/Strings and Bulk/Boundary Dualities

The construction of a $covariant$ Loop Wave functional equation in a 4D spacetime is attained by introducing a generalized $eleven$ dimensional categorical {\bf C}-space comprised of $8\times 8$ antisymmetric matrices. The latter matrices encode the generalized coordinates of the histories of points, loops and surfaces $combined$. Spacetime Topology change and the Holographic principle are natural consequences of imposing the principle of $covariance$ in {\bf C}-space. The Planck length is introduced as a necessary rescaling parameter to establish the correspondence limit with the physics of point-histories in ordinary Minkowski space, in the limit $l_P\to 0$. Spacetime quantization should appear in discrete units of Planck length, area, volume ,....All this seems to suggest that the generalized principle of covariance, representing invariance of proper $area$ intervals in {\bf C}-space, under matrix-coordinate transformations, could be relevant in discovering the underlying principle behind the origins of $M$ theory. We construct an ansatz for the $SU(\infty)$ Yang-Mills vacuum wavefunctional as a solution of the Schroedinger Loop Wave equation associated with the Loop Quantum Mechanical formulation of the Eguchi-Schild String . The Strings/Loops ($SU(\infty)$ gauge field) correspondence implements one form of the Bulk/Boundary duality conjecture in this case.

hep-th↗
arXiv1998

The Spinning Membrane and Skyrmions Revisited

A local world volume Q-supersymmetric Weyl invariant Lagrangian for the membrane is presented. An analysis is provided which solves the problems raised by some authors in the past concerning the algebraic elimination of the auxiliary fields belonging to the coupling function supermultiplet. The starting bosonic action is the one given by Dolan and Tchrakian with vanishing cosmological constant and with quadratic, quartic derivative terms. Our Lagrangian differs from the one of Lindstrom and Rocek in the fact that is polynomial in the fields facilitating the quantization process. It is argued, rigorously, that if one wishes to construct polynomial actions without curvature terms and where supersymmetry is linearly realized, after the elimination of auxiliary fields, one must relinquish S supersymmetry and concentrate solely on the Q-supersymmetry associated with the superconformal algebra in three dimensions. The role that this spinning membrane action may have in the theory of D-branes, Skyrmions and BPS monopoles is also pointed out.

hep-th↗
arXiv1998

W-Geometry from Fedosov's Deformation Quantization

A geometric derivation of $W_\infty$ Gravity based on Fedosov's deformation quantization of symplectic manifolds is presented. To lowest order in Planck's constant it agrees with Hull's geometric formulation of classical nonchiral $W_\infty$ Gravity. The fundamental object is a ${\cal W}$-valued connection one form belonging to the exterior algebra of the Weyl algebra bundle associated with the symplectic manifold. The ${\cal W} $-valued analogs of the Self Dual Yang Mills equations, obtained from a zero curvature condition, naturally lead to the Moyal Plebanski equations, furnishing Moyal deformations of self dual gravitational backgrounds associated with the complexified cotangent space of a two dimensional Riemann surface. Deformation quantization of $W_\infty$ Gravity is retrieved upon the inclusion of all the $\hbar$ terms appearing in the Moyal bracket. Brief comments on Non Commutative Geometry and M(atrix)theory are made.

hep-th↗