Werner Heisenberg foundation of the matrix formulation of quantum mechanics through only the use of observables. In February 1943, Heisenberg was appointed to the Chair for Theoretical Physics at the "Friedrich-Wilhelms-Universität" (today, the Humboldt-Universität zu Berlin). In April, his election to the "Preußische Akademie der Wissenschaften" (Prussian Academy of Sciences) was approved. That same month, he moved his family to their retreat in as Allied bombing increased in Berlin. In the summer, he dispatched the first of his staff at the "Kaiser-Wilhelm Institut für Physik" to Hechingen and its neighboring town of Haigerloch, on the edge of the Black Forest, for the
Heisenberg group operators on formula_103: for formula_109. So too do the multiplications by characters: for formula_111. These operators do not commute, and instead satisfy multiplication by a fixed unit modulus complex number. So the Heisenberg group formula_113 associated with "K" is a type of central extension of formula_114, via an exact sequence of groups: More general Heisenberg groups are described by 2-cocyles in the cohomology group formula_116. The existence of a duality between formula_117 and formula_101 gives rise to a canonical cocycle, but there are generally others. The Heisenberg group acts irreducibly on formula_103. Indeed, the continuous characters separate points so any