Geometric Foundation of Spin and Isospin

Various theories of spinning particles are interpreted as realizing elements of an underlying geometric theory. Classical particles are described by trajectories on the Poincaré group. Upon quantization an eleven-dimensional Kaluza-Klein type theory is obtained which incorporates spin and isospin in a local SL(2, C) × U(1) × SU(2) theory with broken U(1)× SU(2) part. The theories that describe the spinning property of electrons and other particles developed completely different regarding the classical and the quantum description. The classical approach initiated by Frenkel [1] and Thomas [2] was continued with many efforts [3, 4, 5], but never settled to a generally accepted theory, as recent work shows [6, 7, 8, 9, 10]. One apparent reason was that its relation to the quantum theory, as developed by Dirac [11], never was fully clarified [5, 12]. On the other hand, Dirac’s theory was soon accepted, supported by experiment, further developed and immersed in the general framework of the representation theory of the Poincaré group [13, 14]; today it forms the basis of particle description in modern elementary particle physics. There are two characteristic problems common to most classical theories of spinning particles. The first is, that equations for momentum-like spin variables are defined so that total spin is conserved, but no configuration space is defined, on which corresponding position variables live. As a result, the equations of motion cannot be derived through variation of some action. The second problem is, that the constraint that is intended to reduce the number of independent spin variables from six to three, fails to be handled easily. In this short article we show that by taking the missing spin configuration space to be the Lorentz group, we obtain a variational principle and a full canonical formalism. We show that the unconstrained theory is in agreement with the main ideas of the standard model of electroweak interactions; the six independent variables describing spin as well as isospin. Classically spin is described in a Lorentz covariant theory by an antisymmetic tensor Sab; a, b = 0, 1, 2, 3 [1, 2, 4, 5]. The equations of motion for a particle with