History of quantum field theory
The third thread in the development of quantum field theory was the need to handle the statistics of many-particle systems consistently and with ease. In 1927, Pascual Jordan tried to extend the canonical quantization of fields to the many-body wave functions of identical particles using a formalism which is known as statistical transformation theory; this procedure is now sometimes called second quantization. In 1928, Jordan and Eugene Wigner found that the quantum field describing electrons, or other fermions, had to be expanded using anti-commuting creation and annihilation operators due to the Pauli exclusion principle (see Jordan–Wigner transformation). This thread of development was incorporated into many-body theory and strongly influenced condensed matter physics and nuclear physics.
Introduction to gauge theory
The earliest field theory having a gauge symmetry was Maxwell's formulation, in 1864–65, of electrodynamics ("A Dynamical Theory of the Electromagnetic Field"). The importance of this symmetry remained unnoticed in the earliest formulations. Similarly unnoticed, Hilbert had derived Einstein's equations of general relativity by postulating a symmetry under any change of coordinates. Later Hermann Weyl, inspired by success in Einstein's general relativity, conjectured (incorrectly, as it turned out) in year 1919 that invariance under the change of scale or "gauge" (a term inspired by the various track gauges of railroads) might also be a local symmetry of electromagnetism. Although Weyl's choice of the gauge was incorrect, the name "gauge" stuck to the approach. After the development of quantum mechanics, Weyl, Fock and London modified their gauge choice by replacing the scale factor with a change of wave phase, and applying it successfully to electromagnetism. Gauge symmetry was generalized mathematically in 1954 by Chen Ning Yang and Robert Mills in an attempt to describe the strong nuclear forces. This idea, dubbed Yang–Mills theory, later found application in the quantum field theory of the weak force, and its unification with electromagnetism in the electroweak theory.