Heisenberg group by 90°. The other reflections are "xyx" and "yxy", and rotation by 180° is "xyxy" (="yxyx"). The Lie algebra formula_16 of the Heisenberg group formula_1 (over the real numbers) is the space of formula_18 matrices of the form with formula_20. The following three elements form a basis for formula_16: The basis elements satisfy the commutation relations: The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics: where formula_25 is the position operator, formula_26 is the momentum operator, and formula_27 is Planck's constant. The Heisenberg group formula_1 has
Werner Heisenberg in physics—Heisenberg had participated in the doctoral examination of one of them at the Universität Leipzig. The most influential of the three was Johannes Juilfs. During their investigation, they became supporters of Heisenberg as well as his position against the ideological policies of the "Deutsche Physik" movement in theoretical physics and academia. On 29 June 1936, a Nazi Party newspaper published a column attacking Heisenberg. On 15 July 1937, he was attacked in a journal of the SS. In mid-1936, Heisenberg presented his theory of cosmic-ray showers in two papers. Four more papers appeared in the next two years. In