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
Astroparticle physics Astroparticle physics Astroparticle physics, also called particle astrophysics, is a branch of particle physics that studies elementary particles of astronomical origin and their relation to astrophysics and cosmology. It is a relatively new field of research emerging at the intersection of particle physics, astronomy, astrophysics, detector physics, relativity, solid state physics, and cosmology. Partly motivated by the discovery of neutrino oscillation, the field has undergone rapid development, both theoretically and experimentally, since the early 2000s. The field of astroparticle physics is evolved out of optical astronomy. With the growth of detector technology came the more mature astrophysics, which involved multiple