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A review of the construction of a Weyl-invariant spinning-membrane action that is $polynomial$ in the fields, without a cosmological constant term, comprised of quadratic and quartic-derivative terms, and where supersymmetry is linearly realized, is presented. The action is invariant under a $modified$ supersymmetry transformation law which is derived from a new $ Q + K + S $ sum-rule based on the 3D-superconformal algebra ., Comment: 7 pages, Latex file

Recently we have presented a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in phase spaces . The maximal proper-acceleration bound is $a = c^2/ \Lambda$ where $ \Lambda$ is the Planck scale. The group transformation laws of this Maximal-acceleration Relativity theory under velocity and acceleration boosts are analyzed in full detail. For pure acceleration boosts it is shown that the minimal Planck-areas (maximal tension) are universal invariant quantities in any frame of reference. The implications of this minimal Planck-area (maximal tension) principle in future developments of string theory, $ W$-geometry and Quantum Gravity are briefly outlined., Comment: 21 pages, Latex file

We presented a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in the spacetime tangent bundle and in phase spaces (cotangent bundle). The maximal proper-acceleration bound is a = c^2/ \Lambda in full agreement with the old predictions of Caianiello, the Finslerian geometry point of view of Brandt and more recent results in the literature. Inspired by the maximal-acceleration corrections to the Lamb shifts of one-electron atoms by Lambiase, Papini and Scarpetta, we derive the exact integral equation that governs the Renormalization-Group-like scaling dependence of the fractional change of the fine structure constant as a function of the cosmological redshift factor and a cutoff scale L_c, where the maximal acceleration relativistic effects are dominant. A particular physical model exists dominated entirely by the vacuum energy, when the cutoff scale is the Planck scale, with \Omega_\Lambda \sim 1 . The implications of this extreme case scenario are studied., Comment: Latex file, 24 pages, corrected version

A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective eigenfunctions \psi_s (t), parameterized by continuous real indexes k and l. Orthogonality of the eigenfunctions is connected to the zeros of the Riemann zeta-function. Due to the fundamental Gauss-Jacobi relation and the Riemann fundamental relation Z (s') = Z (1-s'), one can show that there is a direct concatenation among the following symmetries, t goes to 1/t, s goes to \beta - s (\beta a real), and s' goes to 1 - s', which establishes a one-to-one correspondence between the label s of one orthogonal state to a unique vacuum state, and a zero s' of the \zeta. It is shown that the RH is a direct consequence of these symmetries, by arguing in particular that an exclusion of a continuum of the zeros of the Riemann zeta function results in the discrete set of the zeros located at the points s_n = 1/2 + i \lambda_n in the complex plane., Comment: Latex file, 18 pages, revised text, stronger and improved arguments with a figure are added, submitted to Annals of Mathematics

We present a new physical model that links the maximum speed of light with the minimal Planck scale into a maximal-acceleration Relativity principle in the spacetime tangent bundle and in phase spaces (cotangent bundle). Crucial in order to establish this link is the use of Clifford algebras in phase spaces. The maximal proper-acceleration bound is a = c^2/ \Lambda in full agreement with the old predictions of Caianiello, the Finslerian geometry point of view of Brandt and more recent results in the literature. We present the reasons why an Extended Scale Relativity based on Clifford spaces is physically more appealing than those based on kappa-deformed Poincare algebras and the inhomogeneous quantum groups operating in quantum Minkowski spacetimes. The main reason being that the Planck scale should not be taken as a deformation parameter to construct quantum algebras but should exist already as the minimum scale in Clifford spaces., Comment: 15 pages, revised Latex file

Based on the old results of Cho, Soh, Park and Yoon, it is shown how higher m + n dimensional pure gravitational actions restricted to AdS_m times S^n backgrounds admit a holographic reduction to a lower m-dimensional Yang-Mills-like gauge theory of diffeomorphisms of S^n interacting with a charged non-linear sigma model plus boundary terms by a simple tuning of the AdS_m-throat and S^n-radius sizes. After performing a harmonic expansion of the fields with respect to the internal coordinates and a subsequent integration one obtains an m-dimensional effective action involving an infinite-component field theory. The supersymmetrization program can be carried out in a straighforward fashion., Comment: 4 pages, plain Tex

Four years ago the Extended Scale Relativity (ESR) theory in C-spaces (Clifford manifolds) was proposed as the plausible physical foundations of string theory. In such theory the speed of light and the minimum Planck scale are the two universal invariants. All the dimensions of a C-space can be treated on equal footing by implementing the holographic principle associated with a nested family of p-loops of various dimensionalities. This is achieved by using polyvector valued coordinates in C-spaces that encode in one stroke points, lines, areas, volumes,.....We review the derivation of the minimal length/time string/brane uncertainty relations and the maximum Planck temperature thermodynamical uncertainty relation. The Weyl-Heisenberg algebra in C-spaces is constructed which $induces$ a Noncommutative Geometric structure in the $ X^A $ coordinates. Hence quantization in C-spaces involves in a natural fashion a Noncommutative Quantum Mechanics and Field Theory, rather than being postulated ad-hoc. A QFT in C-spaces may very likely involve (Braided Hopf) Quantum Clifford algebras and generalized Moyal-like star products associated with multisymplectic geometry., Comment: 16 pages, Revised Latex file. Important Corrections in Section 2 are made

An outline is presented of the Extended Scale Relativity (ESR) in C-spaces (Clifford manifolds), where the speed of light and the minimum Planck scale are the two universal invariants. This represents in a sense an extension of the theory developed by L. Nottale long ago. It is shown how all the dimensions of a C-space can be treated on equal footing by implementing the holographic principle associated with a nested family of p-loops of various dimensionalities. This is achieved by using poly-vector valued coordinates in C-spaces that encode in one stroke points, lines, areas, volumes,... In addition, we review the derivation of the minimal-length string uncertainty relations; the logarithmic corrections (valid in any dimension) to the black hole area-entropy relation. We also show how the higher derivative gravity with torsion and the recent results of kappa-deformed Poincare theories of gravity follow naturally from the geometry of C-spaces. In conclusion some comments are made on the cosmological implications of this theory with respect to the cosmological constant problem, the two modes of time, the expansion of the universe, number four as the average dimension of our world and a variable fine structure constant., Comment: 19 pages, Latex file. Invited talk at the NATO advanced workshop on the nature of time, geometry, physics and perception. Slovakia, May 2002

A novel approach that does not rely on supersymmetry, nor in the AdS/CFT correspondence, to evaluate the Wilson loops asociated with a gauge theory of area-preserving diffeomorphisms in terms of pure string degrees of freedoms is presented. It is based on the Guendelman-Nissimov-Pacheva formulation of composite antisymmetric tensor field theories of volume-preserving diffeomorphisms. Such theories admit p-brane solutions. The quantum effects are discussed and we evaluate exactly the vev of the Wilson loops, in the large N limit of quenched-reduced SU(N) QCD, in terms of a path integral involving pure string degrees of freedom. It is consistent with the recent results based on the AdS/CFT correspondence and dual QCD models (dual Higgs model with dual Dirac strings). More general Loop wave equations in C-spaces (Clifford manifolds) are proposed in terms of holographic variables that contain the dynamics of an aggregate of closed branes (p-loops) of various dimensionalities., Comment: 8 pages, plain Tex

Beginning with the most general fractal strings/sprays construction recently expounded in the book by Lapidus and Frankenhuysen, it is shown how the complexified extension of El Naschie's Cantorian-Fractal spacetime model belongs to a very special class of families of fractal strings/sprays whose scaling ratios are given by suitable pinary (pinary, $p$ prime) powers of the Golden Mean. We then proceed to show why the logarithmic periodicity laws in Nature are direct physical consequences of the complex dimensions associated with these fractal strings/sprays. We proceed with a discussion on quasi-crystals with p-adic internal symmetries, von Neumann's Continuous Geometry, the role of wild topology in fractal strings/sprays, the Banach-Tarski paradox, tesselations of the hyperbolic plane, quark confinement and the Mersenne-prime hierarchy of bit-string physics in determining the fundamental physical constants in Nature., Comment: 14 pages, plain Tex, Revised. An appendix with additional Comments on arithmetic QFT and other topics has been added. References are added

A new nonlinear Schroedinger equation is obtained explicitly from the fractal Brownian motion of a massive particle with a complex-valued diffusion constant. Real-valued energy (momentum) plane wave and soliton solutions are found in the free particle case. The hydro-dynamical model analog yields another (new) nonlinear QM wave equation with physically meaningful soliton solutions. One remarkable feature of this nonlinear Schroedinger equation based on a fractal Brownian motion model, over all the other nonlinear QM models, is that the quantum-mechanical energy functional coincides with the field theory one., Comment: Latex file, 16 pages

It is shown that an action inspired from a BF and Chern-Simons model, based on the $AdS_4$ isometry group SO(3, 2), with the inclusion of a Higgs potential term, furnishes the MacDowell-Mansouri-Chamseddine-West action for gravity, with a Gauss-Bonnet and cosmological constant term. The $AdS_4$ space is a natural vacuum of the theory. Using Vasiliev's procedure to construct higher spin massless fields in AdS spaces and a suitable star product, we discuss the preliminary steps to construct the corresponding higher-spin action in $AdS_4$ space representing the higher spin extension of this model. Brief remarks on Noncommutative Gravity are made., Comment: 6 pages, plain Tex, Revised. References are are added

We present, using spectral analysis, a possible way to prove the Riemann's hypothesis (RH) that the only zeroes of the Riemann zeta-function are of the form s=1/2+i\lambda_n. A supersymmetric quantum mechanical model is proposed as an alternative way to prove the Riemann's conjecture, inspired in the Hilbert-Polya proposal; it uses an inverse eigenvalue approach associated with a system of p-adic harmonic oscillators. An interpretation of the Riemann's fundamental relation Z(s) = Z(1-s) as a duality relation, from one fractal string L to another dual fractal string L' is proposed., Comment: 18 pages, 3 figures, Latex, corrected typos, two references added, minor changes

It is shown that the large $N$ limit of SU(N) YM in $curved$ $m$-dim backgrounds can be subsumed by a higher $m+n$ dimensional gravitational theory which can be identified to an $m$-dim generally invariant gauge theory of diffs $N$, where $N$ is an $n$-dim internal space (Cho, Sho, Park, Yoon). Based on these findings, a very plausible geometrical interpretation of the $AdS/CFT$ correspondence could be given. Conformally invariant sigma models in $D=2n$ dimensions with target non-compact SO(2n,1) groups are reviewed. Despite the non-compact nature of the SO(2n,1), 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. The relation between Self Dual branes and Chern-Simons branes, High Dimensional Knots, follows. A detailed discussion on $W_\infty $ symmetry is given and we outline the Vasiliev procedure to construct an action involving higher spin massless fields in $AdS_4$. This $AdS_4$ spacetime higher spin theory should have a one-to-one correspondence to noncritical $W_\infty$ strings propagating on $AdS_4 \times S^7$., Comment: 43 pages, Tex file

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 $\zeta (0) = - 1/2$ plays a fundamental role. Final remarks on the recent developments in the proof of the Riemann conjecture are made., Comment: Revised Latex file. 22 pages. Remarks on the recent developments in the proof of the Riemann conjecture are made. Plots were included

This paper has been withdrawn, Comment: v3: The incorrect affiliation of M. S. El Naschie was removed. v4: This paper has been withdrawn

A straightforward explanation of the Young's two-slit experiment of a quantum particle is obtained within the framework of the Noncommutative Geometric associated with El Naschie's Cantorian-Fractal transfinite Spacetime continuum., Comment: Tex file 4 pages

It has been recently argued by some authors that is impossible to construct a Weyl invariant spinning membrane action, where the $S$-supersymmetry associated with the 3D superconformal algebra, is relinquished without gauge fixing. Contrary to those assertions, we show why it is possible to construct a Weyl-invariant spinning polynomial membrane action, without curvature terms,where $both$ the conformal boost symmetry and $S$-supersymmetry are explicitly broken by the action. It is shown that the gauge algebra $closes$ despite that the two latter symmetries are broken . For this to happen, a modifed $Q$-supersymmetry transformation, a sort of new $Q+K+S$ ``sum `` rule, is required that generates the compensating terms to cancel the spurious contributions fromthe $S$ and conformal boost anomalous transformations. A substantial discussion of the quantization of the spinning membrane and anomalies is given. We review briefly the role that this spinning membrane action may have in the theory of $D$-branes, Skyrmions and BPS monopoles in the large $N$-limit of SU(N) Yang-Mills ., Comment: 35 page, revised Tex file. Correction of minor typos and an one reference is added

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+\phi^3 $ (where $\phi$ 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+$\phi^3$) 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 $\phi^3$., Comment: Revised (tex,latex) file. Number of pages = 39 Number of figures = 2. The incorrect affiliation of M. S. El Naschie was removed

A straightforward two-line derivation of the Bekenstein-Hawking Area-Entropy relation for Black-Holes in {\bf any} dimension is shown based on Shannon's information theory and Clifford algebras required by the New Relativity Principle., Comment: Tex file, 3 pages

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., Comment: Revised Tex file, 8 pages. A remarkably simple resolution of the cosmological constant is proposed

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., Comment: Revised Tex file. 5 pages. Important change of notation for the polymomenta coordinates has been made instead of using misleading wedge products. The Cosmological constant term is included

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 $\Phi$. 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., Comment: 16 pages Revised Tex File. Remarks about Fractals, Nocommutative Geometry, the Monster and other topics have been added

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., Comment: 8 pages, revised Tex file

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., Comment: Revised Tex file, 21 pages. An addition of the recent work on Chern-Simons Hadronic Bags (Ansoldi, Castro, Spallucci : hep-th/0011013) and p-branes from Moyal Deformation of Generalized Yang Mills (hep-th/9908115) has been added

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 $\Psi$., Comment: 8 pages, revised Tex file

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., Comment: 42 pages, revised latex file. Minor corrections are made and some additions are included

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., Comment: 15 pages, revised Tex file. A discussion of the chiral approach to Self Dual Gravity is added to the previous Yang Mills one

We present in detail a class of solutions to the $4D SU(\infty)$ Moyal Anti Self Dual Yang Mills equations that are related to $reductions$ of the generalized Moyal Nahm quations using the Ivanova-Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitary gauge groups. A further dimensional reduction yields solutions to the Moyal Anti Self Dual Gravitational equations. The Self Dual Yang Mills /Self Dual Gravity case requires a separate study. SU(2) and $SU(\infty)$ (continuous) Moyal Toda equations are derived and solutions to the latter equations in $implicit$ form are proposed via the Lax-Brockett double commutator formalism . An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction) into the SU(2) Moyal Toda field is found. Finally, the generalized Moyal Nahm equations are conjectured that contain the continuous $SU(\infty)$ Moyal Toda equation after a suitable reduction. Three different embeddings of the three different types of Moyal Toda equations into the Moyal Nahm equations are discussed., Comment: Revised TEX file. 31 pages. The Legendre transform between the Moyal heavenly form and the Moyal Toda field is solved

Nottale's special scale-relativity principle was proposed earlier by the author as a plausible geometrical origin to string theory and extended objects. Scale Relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale under dilatations in Nature is taken to be the Planck scale, which is not the same as the string scale. Starting with ordinary actions for strings and other extended objects, we show that gauge theories of volume-resolutions scale-relativistic symmetries, of the world volume measure associated with the extended ``fuzzy'' objects, are a natural and viable way to formulate the geometrical principle underlying the theory of all extended objects. Gauge invariance can only be implemented if the extendon actions in $D$ target dimensions are embedded in $D+1$ dimensions with an extra temporal variable corresponding to the scaling dimension of the original string coordinates. This is achieved upon viewing the extendon coordinates, from the fuzzy worldvolume point of view, as noncommuting matrices valued in the Lie algebra of Lorentz-scale relativistic transformations. Preliminary steps are taken to merge motion relativity with scale relativity by introducing the gauge field that gauges the Lorentz-scale symmetries in the same vain that the spin connection gauges ordinary Lorentz transformations and, in this fashion, one may go beyond string theory to construct the sought-after General Theory of Scale-Motion Relativity. Such theory requires the introduction of the scale-graviton (in addition to the ordinary graviton) which is the field that gauges the symmetry which converts motion dynamics into scaling-resolutions dynamics and vice versa (the analog of the gravitino that gauges supersymmetry). To go beyond the quantum string geometry most probably, Comment: 18 pages, revised Tex file

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., Comment: 18 pages, revised Tex file. It is further discussed how the Skyrmion action contains the action of the large N limit of N coincident D0-branes

Since the lightcone self dual spherical membrane, moving in flat target backgrounds, has a direct correspondence with the SU(\infty) Nahm equations and the continuous Toda theory, we construct the quantum/Moyal deformations of the self dual membrane in terms of the q-Moyal star product . The q deformations of the SU(\infty) Nahm equations are studied and explicit solutions are given. The continuum limit of the q Toda chain equations are obtained furnishing q deformations of the self dual membrane. Finally, the continuum Moyal-Toda chain equation is embedded into the SU(\infty) Moyal-Nahm equations, rendering the relation with the Moyal deformations of the self dual membrane. W_{\infty} and q-W_{\infty} algebras arise as the symmetry algebras and the role of ( the recently developed ) quantum Lie algebras associated with quantized universal enveloping algebras is pointed out pertaining the formulation of a q Toda theory. We review as well the Weyl-Wigner-Moyal quantization of the 3D continuous Toda field equation, and its associated 2D continuous Toda molecule, based on Moyal deformations of rotational Killing symmetry reductions of Plebanski first heavenly equation., Comment: 25 pages. Revised tex file. More references have been included and minor corrections

Since the lightcone self dual spherical membrane, moving in flat target backgrounds, has a direct correspondence with the $SU(\infty)$ Nahm equations and the continuous Toda theory, we construct the Moyal deformations of the self dual membrane in terms of the Moyal deformations of the continuous Toda theory. This is performed by using the Weyl-Wigner-Moyal quantization technique of the 3D continuous Toda field equation, and its associated 2D continuous Toda molecule, based on Moyal deformations of rotational Killing symmetry reductions of Plebanski first heavenly equation associated with 4D Self Dual Gravity., Comment: 15 pages, revised Tex file

The exact quantum integrability aspects of a sector of the membrane is investigated. It is found that spherical membranes moving in flat target spacetime backgrounds admit a class of integrable solutions linked to SU(infty) SDYM equations (dimensionally reduced to one temporal dimension). After a suitable ansatz, the SDYM equations can be recast in the form of the continuous Toda molecule equations whose symmetry algebra is the dimensional reduction of the W (infty} plus {\bar W}(infty} algebra. The latter algebra is explicitly constructed. Highest weight representations are built directly from the infinite number of defining relations among the highest weight states of W(\infty) algebras and the quantum states of the Toda molecule. Discrete states are also constructed. The full (dimensionaly reduced) quantum SU(infty) YM theory remains to be explored., Comment: 46 pages, tex file. Extended updated version of hep-th/9605029 and hep-th/9612160

The exact quantum integrability aspects of a sector of the membrane is investigated. It is found that spherical membranes ( in the lightcone gauge) moving in flat target spacetime backgrounds admit a class of integrable solutions linked to $SU(\infty)$ SDYM equations ( dimensionally reduced to one temporal dimension) which, in turn, are related to Plebanski 4D SD Gravitational equations. A further rotational Killing-symmetry reduction yields the 3D continuous Toda theory. It is precisely the latter which bears a direct relationship to non critical $W_\infty$ string theory. The expected critical dimensions for the ( super) membrane , (D=11) and D=27, are easily obtained. This suggests that this particular sector of the membrane's spectrum (connected to the $SU(\infty)$ SDYM equations ) bears a direct connection to a critical $W_\infty$ string spectrum adjoined to a q=N+1 unitary minimal model of the W_N algebra in the $N\rightarrow \infty$ limit. Final comments are made about the connection to Jevicki's observation that the 4D quantum membrane is linked to dilatonic-self dual gravity plus matter . 2D dilatonic ( super) gravity was studied by Ikeda and its relation to nonlinear $W_\infty$ algebras from nonlinear integrable deformations of 4D self dual gravity was studied by the author.The full $SU(\infty)$ YM theory remains to be explored as well as the incipient role that noncritical nonlinear $W_\infty$ strings might have in the full quantization program., Comment: 15 pages, tex file. A review of work that appeared in the Journal of Chaos, Solitons and Fractals with some new results

First steps in incorporating Nottale's scale-relativity principle to string theory and extended objects are taken. Scale Relativity is to scales what motion Relativity is to velocities. The universal, absolute, impassible, invariant scale under dilatations, in Nature, is taken to be the Planck scale which is not the same as the string scale. Starting with Nambu-Goto actions for strings and other extended objects, we show that the principle of scale-relativity invariance of the world-volume measure associated with the extended objects ( Lorentzian-scalings transformations with respect to the resolutions of the world-volume coordinates) is compatible with the vanishing of the scale-relativity version of the $\beta$ functions : $\beta^G_{\mu\nu}=\beta^X=0$, of the target spacetime metric and coordinates, respectively. Preliminary steps are taken to merge motion relativity with scale relativity and, in this fashion, analogs of Weyl-Finsler geometries make their appearance. The quantum case remains to be studied., Comment: 16 pages, revised tex file, with minor changes

The exact quantum integrability problem of the membrane is investigated. It is found that the spherical membrane moving in flat target spacetime backgrounds is an exact quantum integrable system for a particular class of solutions of the light-cone gauge equations of motion : a dimensionally-reduced $SU(\infty)$ Yang-Mills theory to one temporal dimension. Crucial ingredients are the exact integrability property of the $3D~SU(\infty)$ continuous Toda theory and its associated dimensionally-reduced $SU(\infty)$ Toda $molecule$ equation whose symmetry algebra is the $U_\infty$ algebra obtained from a dimensional-reducion of the $W_\infty \oplus {\bar W}_\infty$ algebras that act naturally on the original $3D$ continuous Toda theory. The $U_\infty$ algebra is explicitly constructed in terms of exact quantum solutions of the quantized continuous Toda equation. Highest weight irreducible representations of the $W_\infty$ algebras are also studied in detail. Continuous and discrete energy levels are both found in the spectrum . Other relevant topics are discussed in the conclusion., Comment: 35 pages; Revised Tex file; some minor details have been added

$p'$-brane solutions to rank $p+1$ composite antisymmetric tensor field theories of the kind developed by Guendelman, Nissimov and Pacheva are found when the dimensionality of spacetime is $D=(p+1)+(p'+1)$. These field theories posses an infinite dimensional group of global Noether symmetries, that of volume-preserving diffeomorphisms of the target space of the scalar primitive field constituents. Crucial in the construction of $p'$ brane solutions are the duality transformations of the fields and the local gauge field theory formulation of extended objects given by Aurilia, Spallucci and Smailagic. Field equations are rotated into Bianchi identities after the duality transformation is performed and the Clebsch potentials associated with the Hamilton-Jacobi formulation of the $p'$ brane can be identified with the $duals$ of the original scalar primitive constituents. Different types of Kalb-Ramond actions are discussed and a particular covariant action is presented which bears a direct relation to the light-cone gauge $p$-brane action. A simple derivation of $S$ and $T$ duality is also given. \medskip, Comment: 17 pages, Tex file, revised version with the inclusion of a derivation of the analog of S and T duality symmetry

Extensions (modifications) of the Heisenberg Uncertainty principle are derived within the framework of the theory of Special Scale-Relativity proposed by Nottale. In particular, generalizations of the Stringy Uncertainty Principle are obtained where the size of the strings is bounded by the Planck scale and the size of the Universe. Based on the fractal structures inherent with two dimensional Quantum Gravity, which has attracted considerable interest recently, we conjecture that the underlying fundamental principle behind String theory should be based on an extension of the Scale Relativity principle where both dynamics as well as scales are incorporated in the same footing., Comment: 10 pages. Tex file. Revised version. Diagram not included Center for particle Theory, University of Texas at Austin and World Laboratory, Lausanne, Switzerland

An exact quantization of the spherical membrane moving in flat target spacetime backgrounds is performed. Crucial ingredients are the exact integrabilty of the $3D~SU(\infty)$ continuous Toda equation and the quasi-finite highest weight irreducible representations of $W_{\infty}$ algebras. Both continuous and discrete energy levels are found. The latter are found for periodic-like solutions. Membrane wavefunctionals solutions are found in terms of Bessel's functions and plausible relations to singleton field theory are outlined., Comment: 15 pages

Exact instanton solutions to $D=11$ spherical supermembranes moving in flat target spacetime backgrounds are construted. Our starting point is Super Yang-Mills theories, based on the infinite dimensional $SU(\infty)$ group, dimensionally reduced to one time dimension. In this fashion the super-Toda molecule equation is recovered preserving only one supersymmetry out of the $N=16$ that one would have obtained otherwise. It is conjectured that the expected critical target spacetime dimensions for the (super) membrane, ($D=11$) $D=27$ is closely related to that of the $noncritical$ (super) $W_{\infty}$ strings. A BRST analysis of these symmetries should yield information about the quantum consistency of the ($D=11$) $D=27$ dimensional (super) membrane. Comments on the role that Skyrmions might play in the two types of Spinning- Membrane actions construted so far is presented at the conclusion. Finally, the importance that integrability on light-lines in complex superspaces has in other types of solutions is emphasized., Comment: 16 pages; Latex file

A proposal for constructing a universal nonlinear ${\hat W}_{\infty}$ algebra is made as the symmetry algebra of a rotational Killing-symmetry reduction of the nonlinear perturbations of Moyal-Integrable deformations of $D=4$ Self Dual Gravity (IDSDG). This is attained upon the construction of a nonlinear bracket based on nonlinear gauge theories associated with infinite dimensional Lie algebras. A Quantization and supersymmetrization program can also be carried out. The relevance to the Kadomtsev-Petviashvili hierarchy, $2D$ dilaton gravity, quantum gravity and black hole physics is discussed in the concluding remarks., Comment: 10, IAEC594 ( May, 1994. Revised, Sept. 94)

Starting from a self-dual $SU(\infty)$ Yang-Mills theory in $(2+2)$ dimensions, the Plebanski second heavenly equation is obtained after a suitable dimensional reduction. The self-dual gravitational background is the cotangent space of the internal two-dimensional Riemannian surface required in the formulation of $SU(\infty)$ Yang-Mills theory. A subsequent dimensional reduction leads to the KP equation in $(1+2)$ dimensions after the relationship from the Plebanski second heavenly function, $\Omega$, to the KP function, $u$, is obtained. Also a complexified KP equation is found when a different dimensional reduction scheme is performed . Such relationship between $\Omega$ and $u$ is based on the correspondence between the $SL(2,R)$ self-duality conditions in $(3+3)$ dimensions of Das, Khviengia, Sezgin (DKS) and the ones of $SU(\infty)$ in $(2+2)$ dimensions . The generalization to the Supersymmetric KP equation should be straightforward by extending the construction of the bosonic case to the previous Super-Plebanski equation, found by us in [1], yielding self-dual supergravity backgrounds in terms of the light-cone chiral superfield, $\Theta$, which is the supersymmetric analog of $\Omega$. The most important consequence of this Plebanski-KP correspondence is that $W$ gravity can be seen as the gauge theory of $\phi$-diffeomorphisms in the space of dimensionally-reduced $D=2+2,~SU^*(\infty)$ Yang-Mills instantons. These $\phi$ diffeomorphisms preserve a volume-three-form and are, precisely, the ones which provide the Plebanski-KP correspondence., Comment: 10 pages, IAEC 7-93R

A new local world volume supersymmetric Lagrangian for the bosonic membrane is presented. The starting Lagrangian is the one constructed by Dolan and Tchrakian with vanishing cosmological constant, with quadratic and quartic derivative terms. Our Lagrangian differs from the one constructed by Lindstrom and Rocek in the fact that it is polynomial in the fields facilitating the quantization process. It is argued, rigorously, that if one wishes to construct polynomial actions without a curvature scalar term and, where supersymmetry is linearly realized in the space of physical fields, after the elimination of auxiliary fields, one must relinquish $S$ supersymmetry, altogether, and concentrate solely on the $Q$ supersymmetry associated with the superconformal algebra in three dimensions. A full $''Q+S''$ supersymmetry cannot be implemented in a linearly realized way satisfying all of the above-mentioned requirements, unless a non-polynomial action is chosen. PACS:04.65.+e, 04.20.Fy., Comment: submitted to the Journal of Group Theory in Physics 14 pages