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 group of a locally compact abelian group "K", equipped with a Haar measure. Such a group has a Pontrjagin dual formula_101, consisting of all continuous formula_102-valued characters on "K", which is also a locally compact abelian group if endowed with the compact-open topology. The Heisenberg group associated with the locally compact abelian group "K" is the subgroup of the unitary group of formula_103 generated by translations from "K" and multiplications by elements of formula_101. In more detail, the Hilbert space formula_103 consists of square-integrable complex-valued functions formula_106 on "K". The translations in "K" form a unitary representation of "K" as