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.
Quantum spacetime
In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory.
As a result of this change some variables that are usually continuous may become discrete.
Often only such discrete variables are called "quantized"; usage varies.
The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko
as a way to eliminate infinities from quantum field theory.
The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field
can be regarded as moving in a quantum space-time, and to Robert Oppenheimer, who carried it
to Hartland Snyder,
who published the first concrete example.
Snyder's Lie algebra was made simple by C. N. Yang in the same year.