Recently, the Dirac and Einstein equations were unified in a tetrad formulation of a Kaluza–Klein model with gauge group SL(2,R)×U(1). In this model, the self-adjoint modes of the tetrad describe gravity, whereas the isometric modes of the tetrad together with a scalar field describe fermions. This model gives precisely the usual Dirac–Einstein Lagrangian. In this paper we generalize the tensor Dirac theory to the larger gauge group SL(2,C)×U(1) acting on bispinors. We show that each SL(2,R)×U(1) subgroup of SL(2,C)×U(1) corresponds to a different factorization of the second-order Klein–Gordon equation into a first-order Dirac equation. Since the Noether currents are different for each factorization, the solutions describe different flavors of fermions. We show that electric charge, lepton number, and baryon number are conserved in this generalization of the Dirac theory.

A recent article characterized all classical tensor systems which admit Fermi quantization as those having unitary Lie–Poisson brackets. Examples include Euler’s tensor equation for a rigid body and Dirac’s bispinor equation in tensor form. It was further shown that the tensor form of Dirac’s bispinor Lagrangian can be derived from a tetrad formulation of a Kaluza–Klein model, which unifies the Dirac and Einstein Lagrangians. Thus, fermions, like bosons, are represented as gauge fields in the tensor formulation of the Dirac theory. In this article boson gauge fields are added to the unified Dirac–Einstein Lagrangian by defining the gauge group of the Kaluza–Klein model to be a semi-direct product. It is shown that the semi-direct product structure uniquely prescribes the usual ‘‘minimal coupling’’ between bosons and fermions.

Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using

In a recent article all classical tensor systems which admit Fermi quantization are characterized as those having unitary Lie–Poisson brackets. Examples include Euler’s tensor equation for a rigid body and Dirac’s equation in tensor form. In this article it is shown that the tensor form of the Dirac Lagrangian can be derived from a tetrad formulation of a Kaluza–Klein model, which unifies the Dirac and Einstein Lagrangians. In this formulation, the isometric modes of the tetrad propagate as fermions, whereas the self-adjoint modes propagate as gravitons. An analogy is made with the rigid and elastic modes of a deformable body, where the rigid modes are Fermi quantized and the elastic modes are Bose quantized. However, unlike a deformable body for which the Euclidean metric is positive definite, fermion and graviton modes are not generally separable, unless the gravitational fluctuations are limited by a certain bound. It is shown that this bound applies whenever bispinors are defined in a general relativistic framework. That is, the bound does not depend on whether bispinors or tensors are used to describe the fermion modes.

In this paper we characterize classical tensor systems which admit Fermi quantization as those having unitary Lie Poisson brackets. Examples include Euler’s tensor equation for a rigid body and Dirac’s equation in tensor form, for which we give a new derivation which is simple and direct.