El Naschie’s structures in the electrodynamics of polarizable media

Using the concept of ‘combined field’, an electrodynamics of polarizable media on a fractal space–time is constructed. In this context, using the scale relativity theory, the permanent electric moment, the induced electric moment, the vacuum fluctuations, the paraelectrics, the diaelectrics, the electric Zeeman-type effect, the electric Einstein–de Haas-type effect, the electric Aharonov–Bohm-type effect, the superconductors in the ‘combined field’, the double layers as coherent structures, the magnetic Aharonov–Casher-type effect, are analyzed. Correspondence with the ε^(∞) space–time is accomplished either by admitting an anomal electric Zeeman-type effect, or through a fractal string as in the case of a superconductor in ‘combined field’, or, by phase coherence of the electron–ion pairs from the electric double layers (El Naschie’s coherence). Moreover, the electric double layer or multiple layer may be considered as two-dimensional projections of the same El Naschie’s fractal strings (higher-dimensional strings in ε^(∞) space–time).     

El Naschie’s ε theory and effects of nanoparticle clustering on the heat transport in nanofluids

Effects of nanoparticle clustering on the heat transfer in nanofluids using the scale relativity theory in the topological dimension D_T = 3 are analyzed. In the one-dimensional differentiable case, the clustering morphogenesis process is achieved by cnoidal oscillation modes of the speed field. In such conjecture, a non-autonomous regime implies a relation between the radius and growth speed of the cluster while, a quasi-autonomous regime requires El Naschie’s ε^(∞) theory through the cluster–cluster coherence (El Naschie global coherence). Moreover, these two regimes are separated by the golden mean. In the one-dimensional non-differentiable case, the fractal kink spontaneously breaks the ‘vacuum symmetry’ of the fluid by tunneling and generates coherent structures. This mechanism is similar to the one of superconductivity. Thus, the fractal potential acts as an energy accumulator while, the fractal soliton, implies El Naschie’s ε^(∞) theory (El Naschie local coherence). Since all the properties of the speed field are transferred to the thermal one, for a certain conditions of an external load (e.g. for a certain value of thermal gradient) the soliton and fractal one breaks down (blows up) and release energy. As result, the thermal conductibility in nanofluids unexpectedly increases. Here, El Naschie’s ε^(∞) theory interferes through El Naschie global and local coherences.     

El Naschie’s ε space–time, hydrodynamic model of scale relativity theory and some applications

A generalization of the Nottale’s scale relativity theory is elaborated: the generalized Schrödinger equation results as an irrotational movement of Navier–Stokes type fluids having an imaginary viscosity coefficient. Then ψ simultaneously becomes wave-function and speed potential. In the hydrodynamic formulation of scale relativity theory, some implications in the gravitational morphogenesis of structures are analyzed: planetary motion quantizations, Saturn’s rings motion quantizations, redshift quantization in binary galaxies, global redshift quantization etc. The correspondence with El Naschie’s ε^(∞) space–time implies a special type of superconductivity (El Naschie’s superconductivity) and Cantorian-fractal sequences in the quantification of the Universe.     

El Naschie’s coherence on the subquantum medium

In the hydrodynamic formulation of the Scale Relativity theory one shows that a stable vortices distribution of bipolaron type induces superconducting pairs by means of the quantum potential. One builds the superconducting fractal by an iterated map and demonstrates that the superconducting pairs results as projections of this fractal. Thus, usual mechanisms (as example the exchange interaction used in the bipolaron theory) are reduced to the coherence on the subquantum medium in a ε^(∞) space (El Naschie’s coherence).     

The maximum number of elementary particles in a super symmetric extension of the standard model

In a series of papers over the last few years El Naschie addressed the question of the minimum and maximum number of elementary particles which a mathematically consistent and a physically meaning full extended standard model should contain. El Naschie’s minimum is 62 particles namely 60 believed to have been already discovered in addition to one Higgs boson and one graviton which are theoretically needed but are not jet experimentally conformed. By contrast the maximum number of 69 particles is although consistent with many quantum field theories based models as well as a classical result by Dyson may not be the only possibility. In the present work we show that a larger number of 72 or even 84 particles are easily shown to be consistent with super string theory and super symmetry. Our work consists of two parts. The first part is a reappraisal of El Naschie’s results and the second is a derivation of the proposed possibility of an upper bound of 72 or 84 elementary particles.     

From experimental quantum optics to quantum gravity via a fuzzy Kähler manifold

Starting from the two-slit experiment we show that the so-called particle–wave duality could be resolved amicably by assuming space–time to be a fuzzy K₃ manifold akin to that of E-Infinity theory. Subsequently, we show how many of the fundamental constants of nature such as the electromagnetic fine structure as well as the quantum gravity coupling may be deduced from the topology and geometry of the space–time manifold.     

From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and superstrings to Cantorian space–time

In this work we present a historical overview of mathematical discoveries which lead to fundamental developments in super string theory, super gravity and finally to E-infinity Cantorian space–time theory. Cantorian space–time is a hierarchical fractal-like semi manifold with formally infinity many dimensions but a finite expectation number for these dimensions. The idea of hierarchy and self-similarity in science was first entertain by Right in the 18th century, later on the idea was repeated by Swedenborg and Charlier. Interestingly, the work of Mohamed El Naschie and his two contra parts Ord and Nottale was done independently without any knowledge of the above starting from non- linear dynamics and fractals.     

Quantum decoherence and El Naschie’s complex temporality

It is well-known that the Schrödinger equation reduces to a classical diffusion equation by means of Wick rotation (t → it), suggesting a correspondence between quantum and classical mechanics. Nonetheless, this result does not admit a clear conceptual interpretation. In the framework of his fractal space-time theory, El Naschie showed that great conceptual advantage could be achieved by extending the imaginary time, it, to a perfectly symmetric, complex conjugate time 0 ± it. In this note we show through a simple analysis, involving formal analytic continuation (t → 0 ± it), that El Naschie’s time complexification provides the basis for a physical interpretation of the correspondence between quantum and classical mechanics in terms of quantum decoherence. We find that decoherent states inevitably arise due to time symmetry breaking as we go from the micro Cantorian space-time, where the two symmetric times, 0 + it and 0 − it, coexist to our 4-dimensional smooth space-time, where t is the only time.     

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.     

Arithmetic of plane Cremona transformations and the dimensions of transfinite heterotic string space-time

It is shown that the two sequences of characteristic dimensions of transfinite heterotic string space-time found by El Naschie can be remarkably well accounted for in terms of the arithmetic of self-conjugate homaloidal nets of plane algebraic curves of orders 3–20. A firm algebraic geometrical justification is thus given not only for all the relevant dimensions of the classical theory, but also for the other two dimensions proposed by El Naschie, viz. the inverse of the quantum gravity coupling constant (≃42.36067977) and that of (one half of) the fine structure constant (≃68.54101967). A non-trivial coupling between the two El Naschie sequences is also revealed.     

Fermion bound states in the Aharonov-Bohm field in 2+1 dimensions

We find exact solutions of the Dirac equation that describe fermion bound states in the Aharonov-Bohm potential in 2+1 dimensions with the particle spin taken into account. For this, we construct self-adjoint extensions of the Hamiltonian of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions. The self-adjoint extensions depend on a single parameter. We select the range of this parameter in which quantum fermion states are bound. We demonstrate that the energy levels of particles and antiparticles intersect. Because solutions of the Dirac equation in the Aharonov-Bohm potential in 2+1 dimensions describe the behavior of relativistic fermions in the field of the cosmic string in 3+1 dimensions, our results can presumably be used to describe fermions in the cosmic string field.     

An equivalence of two constructions of permutation-twisted modules for lattice vertex operator algebras

The problem of constructing twisted modules for a vertex operator algebra and an automorphism has been solved in particular in two contexts. One of these two constructions is that initiated by the third author in the case of a lattice vertex operator algebra and an automorphism arising from an arbitrary lattice isometry. This construction, from a physical point of view, is related to the space–time geometry associated with the lattice in the sense of string theory. The other construction is due to the first author, jointly with C. Dong and G. Mason, in the case of a multifold tensor product of a given vertex operator algebra with itself and a permutation automorphism of the tensor factors. The latter construction is based on a certain change of variables in the worldsheet geometry in the sense of string theory. In the case of a lattice that is the orthogonal direct sum of copies of a given lattice, these two very different constructions can both be carried out, and must produce isomorphic twisted modules, by a theorem of the first author jointly with Dong and Mason. In this paper, we explicitly construct an isomorphism, thereby providing, from both mathematical and physical points of view, a direct link between space–time geometry and worldsheet geometry in this setting.     

Nonexistence of time‐periodic solutions of the Dirac equation in an axisymmetric black hole geometry

We prove that in the nonextreme Kerr‐Newman black hole geometry, the Dirac equation has no normalizable, time‐periodic solutions. A key tool is Chan‐drasekhar's separation of the Dirac equation in this geometry. A similar nonexistence theorem is established in a more general class of stationary, axisymmetric metrics in which the Dirac equation is known to be separable. These results indicate that, in contrast to the classical situation of massive particle orbits, a quantum mechanical Dirac particle must either disappear into the black hole or escape to infinity. © 2000 John Wiley & Sons, Inc.     

Statistical Theory of Equilibrium Fluctuations

An analysis of the BBGKY hierarchy shows that the fluctuation theory for hydrodynamic variables (i.e., the fluid density, velocity, and temperature) must be constructed based on the hydrodynamic equations. We show that fluctuations in the hydrodynamic variables uniquely determine the fluctuations in all other thermodynamic parameters of a fluid (such as the pressure, entropy, and intrinsic energy). We find the spectral structure of fluctuation waves of the hydrodynamic variables, calculate their amplitudes, and derive the dispersion equation establishing the relation between the phase velocity of the fluctuation wave propagation and its frequency. We explain the effects of the flicker noise generation and of the stability loss on the first-order phase equilibrium lines. We also analyze the fluctuation decay process and show that this process gives rise to the effect of the Mandelstam–Brillouin scattering. We compare the obtained results with the existing fluctuation theories.     

A new solution for the two-slit experiment

A completely new mathematical solution of the two-slit experiment is given. The corresponding physical model suggests, for the first time, that the wave-particle duality is simply an expression of the non-classical topology and geometry of quantum spacetime when projected into our 3 + 1 Euclidean space.

However if we disregard the topological cause of the projected behaviour and concentrate only on what is the case, then the observed wave-particle behaviour may be regarded as a physical realization of Gödel’s undecidability theorem.

The essence of the present solution may be summarized in the following statement: If we could perform the two-slit experiment in a laboratory of a size comparable to that of an elementary particle, the wave-particle duality would cease to exist. It is remarkable that we do not assume quantum mechanics but end up confirming its results.     

Chu duality theory and coalgebraic representation of quantum symmetries

Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general theory of categorical dualities on the basis of Chu space theory and closure conditions, which encompasses a variety of dualities for topological spaces, convex spaces, closure spaces, and measurable spaces (some of which are new duality results on their own). It works as a general method to generate analogues of categorical dualities between frames (locales) and topological spaces beyond topology, e.g., for measurable spaces, convex spaces, and closure spaces. After establishing the Chu duality theory, we apply the state-observable duality between quantum lattices and closure spaces to coalgebraic representations of quantum symmetries, showing that the quantum symmetry groupoid fully embeds into a purely coalgebraic category, i.e., the category of Born coalgebras, which refines, through the quantum duality that follows from Chu duality theory, Samson Abramsky's fibred coalgebraic representations of quantum symmetries (which, in turn, builds upon his Chu representations of symmetries).     

Geometry as seen by string theory

This is an introductory review of the topological string theory from physicist’s perspective. I start with the definition of the theory and describe its relation to the Gromov–Witten invariants. The BCOV holomorphic anomaly equations, which generalize the Quillen anomaly formula, can be used to compute higher genus partition functions of the theory. The open/closed string duality relates the closed topological string theory to the Chern–Simons gauge theory and the random matrix model. As an application of the topological string theory, I discuss the counting of bound states of D-branes.     

Stark Effect in Charge–Dyon System

We consider the linear Stark effect in the MIC–Kepler problem describing the interaction of a charged particle with a Dirac dyon and show that a constant homogeneous electric field completely removes the degeneracy of energy levels with respect to the azimuthal quantum number.     

Instability of cnoidal‐peak for the NLS‐δ equation

We study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in $H^1_{per}$ .     

Derivation of the energy–momentum and Klein–Gordon equations from El Naschie’s complex time

In a previous note, we have provided a formal derivation of the transverse Doppler shift of special relativity from the generalization of El Naschie’s complex time. Here, we show that the relativistic energy–momentum equation, and hence the Klein–Gordon equation, are also natural consequences of the complex time generalization.