The Dirac equation is most easily formulated in terms of functions from space-time to the Clifford algebra R3,1 and the gradient operator. In this setting we may construct wave packet solutions for all leptons by introducing a single operator which transforms a real valued function on the group Spin+(3,1) to a wave function by using integration on the group itself. Applying the operator to a certain space of real valued functions on the group, we produce Spin+(3,1)-invariant solution spaces which may then be classified. The results are one space each for electrons and positrons and two parameter families of spaces for neutrinos and antineutrinos. The electron and neutrino spaces display an asymptotic symmetry at high energies, as do the positron and antineutrino spaces. If we also insist that the solution spaces be translation invariant, then we get the familiar two-component neutrino theory and the asymptotic symmetry between leptons of the same handedness used in the theory of weak interactions. This symmetry is purely a result of the Dirac theory in the Clifford algebra setting.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics