We present the bundle (Aff(3)⊗ℂ⊗Λ)(ℝ3), with a geometric Dirac equation on it, as a three-dimensional geometric interpretation of the SM fermions. Each (ℂ⊗Λ)(ℝ3) describes an electroweak doublet. The Dirac equation has a doubler-free staggered spatial discretization on the lattice space (Aff(3)⊗ℂ)(ℤ3). This space allows a simple physical interpretation as a phase space of a lattice of cells.We find the SM SU(3) c ×SU(2) L ×U(1) Y action on (Aff(3)⊗ℂ⊗Λ)(ℝ3) to be a maximal anomaly-free gauge action preserving E(3) symmetry and symplectic (...) structure, which can be constructed using two simple types of gauge-like lattice fields: Wilson gauge fields and correction terms for lattice deformations.The lattice fermion fields we propose to quantize as low energy states of a canonical quantum theory with ℤ2-degenerated vacuum state. We construct anticommuting fermion operators for the resulting ℤ2-valued (spin) field theory.A metric theory of gravity compatible with this model is presented too. (shrink)

In this paper, we use the differential forms of three-dimensional Euclidean space to realize a Clifford algebra which is isomorphic to the algebra of the Pauli matrices or the complex quaternions. This is an associative vector-antisymmetric tensor algebra with division: We provide the algebraic inverse of an eight-component spinor field which is the sum of a scalar + vector + pseudovector + pseudoscalar. A surface of singularities is defined naturally by the inverse of an eight-component spinor and corresponds to a (...) generalized Minkowski “double” light cone in the parameter space. A general description of finite spatial rotations, which utilizes the Baker-Campbell-Hausdorff formula, generalizes the usual infinitesimal treatments of the rotation group. We derive an explicit expression for the angle corresponding to two successive finite rotations in any direction. We also discuss Lorentz transformations and duality rotations of the electromagnetic field and exhibit a relationship between the algebraic inverse and a duality rotated field. Using a combined transformation, one can always transform an arbitrary electromagnetic field (E≠0) into a pure electric field, but never into a pure magnetic field. (shrink)

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz (...) potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q. (shrink)

Elementary particles are considered as local oscillators under the influence of zeropoint fields. Such oscillatory behavior of the particles leads to the deviations in their path of motion. The oscillations of the particle in general may be considered as complex rotations in complex vector space. The local particle harmonic oscillator is analyzed in the complex vector formalism considering the algebra of complex vectors. The particle spin is viewed as zeropoint angular momentum represented by a bivector. It has been shown that (...) the particle spin plays an important role in the kinematical intrinsic or local motion of the particle. From the complex vector formalism of harmonic oscillator, for the first time, a relation between mass $m$ and bivector spin $S$ has been derived in the form $\varvec{\sigma }_3 mc^2{\mathcal {J}}_{\pm } =\lambda \Omega _{\mathbf{s}} \cdot \mathrm{{S}} {\mathcal {J}}_{\pm }$ . Where, $\Omega _{s}$ is the angular velocity bivector of complex rotations, $c$ is the velocity of light. The unit vector $\varvec{\sigma }_3$ acts as an operator on the idempotents ${\mathcal {J}}_{+}$ and ${\mathcal {J}}_{-}$ to give the eigen values $\lambda =\pm 1.$ The constant $\lambda $ represents two fold nature of the equation corresponding to particle and antiparticle states. Further the above relation shows that the mass of the particle may be interpreted as a local spatial complex rotation in the rest frame. This gives an insight into the nature of fundamental particles. When a particle is observed from an arbitrary frame of reference, it has been shown that the spatial complex rotation dictates the relativistic particle motion. The mathematical structure of complex vectors in space and spacetime is developed. (shrink)

This article reviews Hestenes' work on the Dirac theory, where his main achievement is a real formulation of the theory within thereal Clifford algebra Cl 1,3 ≃ M2 (H). Hestenes invented first in 1966 hisideal spinors $\phi \in Cl_{1,3 _2}^1 (1 - \gamma _{03} )$ and later 1967/75 he recognized the importance of hisoperator spinors ψ ∈ Cl 1,3 + ≃ M2 (C).This article starts from the conventional Dirac equation as presented with matrices by Bjorken-Drell. Explicit mappings are given for (...) a passage between Hestenes' operator spinors and Dirac's column spinors. Hestenes' operator spinors are seen to be multiples of even parts of real parts of Dirac spinors (real part in the decompositionC ⊗ Cl 1,3 andnot inC ⊗ M4 (R)=M4 (C)). It will become apparent that the standard matrix formulation contains superfluous parts, which ought to be cut out by Occam's razor.Fierz identities of bilinear covariants are known to be sufficient to study the non-null case but are seen to be insufficient for the null case ψ†γ0ψ=0, ψ†γ0γ0123ψ=0. The null case is thoroughly scrutinized for the first time with a new concept calledboomerang. This permits a new intrinsically geometric classification of spinors. This in turn reveals a new class of spinors which has not been discussed before. This class supplements the spinors of Dirac, Weyl, and Majorana; it describes neither the electron nor the neutron; it is awaiting a physical interpretation and a possible observation.Projection operators P±, Σ± are resettled among their new relatives in End(Cl 1,3 ). Finally, a new mapping, calledtilt, is introduced to enable a transition from Cl 1,3 to the (graded) opposite algebra Cl 3,1 without resorting to complex numbers, that is, not using a replacement γμ →iγμ. (shrink)

We explore the possibility thatzitterbewegung is the key to a complete understanding of the Dirac theory of electrons. We note that a literal interpretation of thezitterbewegung implies that the electron is the seat of an oscillating bound electromagnetic field similar to de Broglie's pilot wave. This opens up new possibilities for explaining two major features of quantum mechanics as consequences of an underlying physical mechanism. On this basis, qualitative explanations are given for electron diffraction, the existence of quantized radiationless states, (...) the Pauli principle, and other features of quantum mechanics. (shrink)