Book Review: Quantum fields and strings: A course for mathematicians, Volumes 1 and 2
Louis Hirsch Kauffman · Bulletin of the American Mathematical Society · 2001
Quantum theory began in an enigma about the structure of matter and radiation.This enigma pointed, in the hands of Planck and his successors, to a fundamental discreteness at the basis of physics.Such discrete behaviour would not be paradoxical in a fully discrete world, but it appeared at the time (and still does appear to many) that the world of our experience is well-approximated by a continuum.Certainly the classical phenomena of gravity, and electricity and magnetism appeared to be described by differential equations that modelled behaviour in a continuous world.Planck found that radiation needs to be quantized with a minimum allowed energy level, and the enigma was born.Einstein discovered that this same quantization hypothesis could explain the photoelectric effect, and atomic structure needed a new theory to allow the electrons to have stable orbits if they had orbits at all.One needed an explanation for the spectra of elements that would have the atoms emitting light at only certain characteristic frequencies.Bohr's theory of the atom gave the right numbers but was logically inconsistent.DeBroglie stepped into the picture and suggested that matter was accompanied by a wave and that this wave/particle duality of matter was the source of the elusive discrete.DeBroglie's suggestion explained the special orbits of electrons in atoms by a restriction due to the needed periodicity of his wave functions.Werner Heisenberg discovered an algebraic approach to the atom where one no longer tried to visualize the atomic orbits.He took an approach that gave the coordinates for position and momentum the attributes of operators in a noncommutative calculus.Concurrently, Erwin Schrödinger found a wave equation to go along with DeBroglie's waves.That wave equation was in accord with the quantum behaviour of the hydrogen atom and with many other systems.In the context of the wave equation it was natural to replace physical observables by differential operators with special commutation relations.Soon it became clear that Heisenberg's matrix mechanics (of infinite matrices) and Schrödinger's wave mechanics were formally identical.The unification of these approaches led to a version of quantum mechanics where the states of the quantum system are vectors in a complex Hilbert space, and the observables are self-adjoint linear operators on that space.In this formulation all the physics rests in the structure of the observables.The enigma remained, for it was still not clear how a particle could at the same time be a wave.One can see the dilemma clearly in the Schrödinger context, for there, without observation, the wave function evolves deterministically, but as soon as an observation is made the wave function is projected into an eigenstate.Observations are discontinuous interruptions of the smooth evolution of the wave 2000