Werner Heisenberg the first to reinterpret the Dirac equation as a "classical" field equation for any point particle of spin ħ/2, itself subject to quantization conditions involving anti-commutators. Thus reinterpreting it as a (quantum) field equation accurately describing electrons, Heisenberg put matter on the same footing as electromagnetism: as being described by relativistic quantum field equations which allowed the possibility of particle creation and destruction. (Hermann Weyl had already described this in a 1929 letter to Albert Einstein.) Heisenberg's paper establishing quantum mechanics has puzzled physicists and historians. His methods assume that the reader is familiar with Kramers-Heisenberg transition probability calculations. 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