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.
History of quantum field theory
The first approach that bore fruit is known as the "interaction representation" (see the article Interaction picture), a Lorentz-covariant and gauge-invariant generalization of time-dependent perturbation theory used in ordinary quantum mechanics, and developed by Tomonaga and Schwinger, generalizing earlier efforts of Dirac, Fock and Podolsky. Tomonaga and Schwinger invented a relativistically covariant scheme for representing field commutators and field operators intermediate between the two main representations of a quantum system, the Schrödinger and the Heisenberg representations. Within this scheme, field commutators at separated points can be evaluated in terms of "bare" field creation and annihilation operators. This allows for keeping track of the time-evolution of both the "bare" and "renormalized", or perturbed, values of the Hamiltonian and expresses everything in terms of the coupled, gauge invariant "bare" field-equations. Schwinger gave the most elegant formulation of this approach. The next and most famous development is due to Feynman, who, with his brilliant rules for assigning a "graph"/"diagram" to the terms in the scattering matrix (see S-matrix and Feynman diagrams). These directly corresponded (through the Schwinger-Dyson equation) to the measurable physical processes (cross sections, probability amplitudes, decay widths and lifetimes of excited states) one needs to be able to calculate. This revolutionized how quantum field theory calculations are carried-out in practice.