Friday, 28 December 2012
Thursday, 27 December 2012
A butterfly flutters its wings in Guatemala and triggers a tornado in Guantanamo. While such stands as nothing short of a miracle, a perfect yet powerful expression of roll, pitch and yaw by a humble arachnid; it remains the metaphorical ideal for an entire dynamics bent on the eccentricities and capriciousness of nature. The bold and presumptuous notion that randomness is merely a mirage, that order and disorder are intimately linked and that small changes result in vast differences. The notion of chaos usually conjures the image of randomness and entropy whereas it is a manifestation of unpredictability and fluctuation played by the laws of nature. Everything from the emergent ethology of flocks of birds and the formation of the weather to the evolution of stock markets and economic crashes unveils a non-linear, irregular and unpredictable platform. One where the deterministic view collapses and makes room vacant for a chaotic framework of dynamically changing systems provoked by minute and seemingly trivial modifications. A double pendulum will assume a sporadic and capricious motion when released due to minuscule differences in the initial configuration, just as the weather varies due to fluctuations in atmospheric conditions such as convection as quantified by the Lorenz attractor. Chaos may be both observed and envisaged as a facet of nature; for mountains are not cones neither are clouds perfect spheres nor are trees flawless cylinders. It is the non-Euclidean geometry of the physical world that is addressed by the chaotic nature of fractals, essentially the byproducts of dynamically changing systems that convey a feedback loop of infinite regression, self-similarity and locally and globally oriented irregularity. Even simple contraptions such as a Poincare's three body astronomical system indicate that small errors in calculation result in chaotic subsequent outcomes, way out of proportion of the initial mathematical estimates and premises. Others such as the Belousov non-equilibrium thermodynamic reactions probe the chaos of chemical oscillation from a clear state to coloured to clear again; indicating a chemical basis of morphogenesis centred on the reaction-diffusion paradigm expounded by Turing. Chaos is kaleidoscopic; every small fluctuation, modification and variation constitute to the outcome and output. And however unpredictable, dubious and erratic that outcome may be, it is alway reducible to simplicity and lucidity regardless of whether its a Mandelbrot set or a Romanesco broccoli!
Sunday, 23 December 2012
Thursday, 13 December 2012
Moreover, because patterns of string vibration arise in pairs, varying by around half a unit of spin, it creates an internal supersymmetry (SUSY). SUSY allows one to merge bosons and fermions given that they return to their original states after 360 and 720 degree rotations respectively, if one can define such transformations as manifestations of a greater multidimensional geometry, each of the known particles can be related by heavier superpartners. The additional dimensions are for mathematical consistency and Kaluza was the first to realise that adding a fifth dimension to spacetime mirrored Maxwell's equations, the extra dimensions curled up like a cartesian plane passing via each point in space. But among the hallmarks of string theory included the ability to mimic spin-2 gravitons with closed strings, allowing the possilibity of describing gravity quantum mechanically but without invoking fields directly. The only free parameter in question is the tension of the string, which can be calibrated more precisely using the graviton to meet the strength of the strong force. As a result, the minimum energy of a string, the planck energy, can be multiplied many times over by the amount of wavelengths in each mode or oscillation to give a large resultant, such may be cancelled out by quantum fluctuations leading to a massless string. As for other particle properties, spin simply becomes the aspect of how a string vibrates, as these vibrations can extend over curled up dimensions it becomes relatively simply to imagine a fermion moving in a gyrating fashion in another dimension, thus gaining a 720 degree rotation in the 3D space to resume its original position. So if mass and spin can emerge from a multi-dimensional geometry, what about the other forces?
By the 80's, the 5 string theories which all paired their bosons and fermions differently were united by Witten into M-theory into a context of an 11 dimensional Calabi-Yau space-time (10+1). It incorporates one dimensional strings in conjuction with branes (membranes), allowing gravity and the non-gravitational forces to unite at a single energy. This wasn't previously possible, given that the gravitational coupling constant refused to match the quantum forces except until an energy density that exceeded the unification energy; in M-theory it matched neatly. But we are talking ridulous scales on the order of 10^-35m, around 10^-20 the diameter of a proton; at such tiny scales gravity competes with the other forces so we need to unite quantum field theory with general relativity. Such a stark incompatility between the two arises because of the Uncertainty principle; in general relativity, perfectly flat space arises in the absence of a significant mass (such as the quantum vacuum) and thus the value of the gravitational field should be exactly zero. However, the Uncertainty principle needs only a mean value of zero and so the value of the gravitational field can fluctuate in a random 'foam'; therefore we get infinities (or ultraviolet divergences). These infinites come about because in quantum field theory, you deal with points that act as sources of fields, but with one dimensional strings infinities never occur because the energy of a vibrating string is dependant on its frequency as well as the amplitude at which it oscillates (closed strings and loops rely on circumference) and more frenzied modes of vibration produce more energy than placid ones. Much like a Feynman diagram, which capture the history of a particle, a 'worldsheet' capture snippets of a bunch of strings and recording their history as a static network of connecting tubes in spacetime; when the loops interact, two loops may merge and split into a different figuration. Such allows gauge bosons to be expressed as aspects of vibrating strings rather than the conventional view of seperate fields for each charge and particles as point-like bundles of energy osillating in those fields.