Monday, 4 March 2013
Quantum Field Theory- A Matrix of Symmetries
It's a matrix of symmetries spun round a sum over histories. An extrapolation of the law of conservation of weirdness upon particles upon fields with infinite degrees of freedom. For like the weather, with its fields of temperature and wind and like the orchestra, with fields of soft passages and abrupt chords; it presents a grand region of quantised influence. Quantum field theory persists as the most matured form of quantum mechanics; a regime of particles, groups and symmetries; a field formulation that has a value at every point in space and solutions that are nothing short of baffling. Its appeal to symmetries as an 'invariance under a specified group of transformations' means that some operation can be executed on a quantum system leaving it indiscernible from its starting state. For instance the standard model; which can be geometrically quantised in terms of fiber bundles and lie groups, has individual fibres or figures attached at every point in space-time each relating to a different type of particle. The resulting symmetry groups are U(1), consisting of circles at every spacetime point for electromagnetism, SU(2) for the weak interaction, SU(3) for the strong force and potentially Spin(1,3) for gravity. Some basic formalisms are QED; the field theory of electromagnetic vacuum perturbation along with QCD; the field theory of the strong force mediated by gluons upon quarks. QCD allows quarks to share confinement, colour charge and encounter asymptotic freedom at high energies (a paradoxical phenomenon that causes weaker attaction between quarks as distance increases). QED is quite distinct given its single charge and a less energetic response of photons to electric charge compared to the trinity of colour charges for quarks and the vigorous response of gluons to each other, making the prospects for a photon lightsaber bleak! Moreover, the precise calculation of the Lamb shift (a tiny spliting in energy values between the 2s and 2p states of the hydrogen atom) and the anomaly of the electron's magnetic dipole moment (the value of a particle that determines the force that it may release upon electric currents and the torque that a magnetic field will exert on it) are among the most accurate of predictions in quantum field theory. If you picture an electron dipped in a magnetic field, it's intrinsic spin gives it a magnetic moment producing an energy of interaction which is dependant on the angle between the direction of the imposed magnetic field and the electron's own magnetic field. Aligning the two fields will produce a low energy, opposing them will produce a high energy and at intermediary angles the energy level will differ between these values. So accurate these calculations are that in fact some common sense notions breakdown such as 'nothing'. Nothing becomes the lowest energy state of a physical theory, Maxwell's equations may describe the electric field in the classical sense but if we reduce its total energy to 0, it disappears and a vacuum is all that remains. The quantum field theory description however reveals a sea of fluctuations due to the uncertainty principle, a quantum 'vacuum' where virtual and anti-particles transiently pop in and out of existence from nothing from the instability of empty space. Such a redefinition provides the basis for explaining quantum phase transitions such as superconductivity expounded by the BCS theory. The idea that Cooper pairs or twins of electrons act distinctly from 'singlets' governed by the exclusion principle, they act in a way that resembles bosons, condensing into the same energy level leading to the inhibition of the collision reactions that cause resistance; thus causing a infinite flow of current. And above all, the magic of spontaneous symmetry breaking is an phenomenon worth mentioning whereby small, infinitesimal changes influencing a quantum system cause a sort of exhange between order and chaos resulting in a compromise. Basically like a pencil balanced on its tip which has symmetry in that it looks indistinguishable from its starting state if rotated along a vertical axis (an overall symmetry of the gravtitational field); but its instability causes the breaking of the symmetry and the pencil to collapse into an asymmetric horizontal state. Or like a Mexican sombrero?