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 "n"-dimensional systems, and most generally, to any symplectic vector space. In the three-dimensional case, the product of two Heisenberg matrices is given by: Since the multiplication is not commutative, the group is non-abelian. The neutral element of the Heisenberg group is the identity matrix, and inverses are given by The group is a subgroup of the 2-dimensional affine group Aff(2): formula_2 acting on formula_6 corresponds to the affine transform formula_7. There are several prominent examples of the three-dimensional case. If , are real numbers (in the ring R) then one has the continuous Heisenberg group H(R). It is a nilpotent