On the Photon Vertex and Half-Integer Spin

Abstract

When rewriting the photon vertex of quantum electrodynamics in terms of geometrical quantities, various elements can be mapped directly to objects and properties known from classical projective geometry (PG). Elements of P 5 when mapped to line reps in P 3 exhibit their intrinsic Lorentz invariance associated to automorphisms of the Plücker-Klein quadric , and line reps when expressed by point or plane coordinates introduce (one-parameter) pencils, or formally gl(2,ℝ), or gl(1,ℍ) which covers su(2)⊕u(1). This introduces binary forms and, using a potential approach of central forces, Schrödinger or Laplace equations and the respective special functions, as well as the projective generation of quadrics like in Dirac’s approach which legitimates Clifford algebra elements as linear factors in invariant theory and the quadratic algebra to represent geometry. Physically, this identification allows for the classical concept of moments in terms of tetrahedrons which on the one hand relates to previous work on SU(4) and SU*(4) in quantum representations. On the other hand, it relates to the classical physical definitions, however, exhibiting a factor 2 between contemporary (euclidean) moments and the tetrahedral construction used in the vertex. Finally, we discuss the equilibrium conditions with respect to gauge and Yang-Mills theories in general as well as the related objects and their transformation theory.