On a microscopic representation of space-time

Advances in Applied Clifford Algebras

Using the Dirac (Clifford) algebra γ µ as initial stage of our discussion, we summarize previous work with respect to the isomorphic 15dimensional Lie algebra su*(4) as complex embedding of sl(2,H), the relation to the compact group SU(4) as well as subgroups and group chains. The main subject, however, is to relate these technical procedures to the geometrical (and physical) background which we see in projective and especially in line geometry of R 3. This line geometrical description, however, leads to applications and identifications of line Complexe and the discussion of technicalities versus identifications of classical line geometrical concepts, Dirac's 'square root of p 2 ', the discussion of dynamics and the association of physical concepts like electromagnetism and relativity. We outline a generalizable framework and concept, and we close with a short summary and outlook.

1995

Abstract. A classification of hadrons and their interactions at low energies according to SU(4) allows to identify combinations of the fifteen mesons π, ω and ρ within the spin-isospin decomposition of the regular representation 15. Chirally symmetric SU(2)×SU(2) hadron interactions are then associated with transformations of a subgroup of SU(4). Nucleon and Delta resonance states are represented by a symmetric third rank tensor 20 whose spin-isospin decomposition leads to 4 ⊕ 16 ‘tower states ’ also known from the large-Nc limit of QCD. Towards a relativistic hadron theory, we consider possible generalizations of the stereographic projection S2 → C and the related complex spinorial calculus on the basis of the division algebras with unit element. Such a geometrical framework leads directly to transformations in a quaternionic projective ‘plane ’ and the related symmetry group SL(2,H). In exploiting the Lie algebra isomorphism sl(2,H) ∼ = su∗(4) ∼ = so(5,1), we focus on the Lie alge...

Journal of Physics A: Mathematical and General, 2006

The space-time symmetry group of a model of a relativistic spin 1/2 elementary particle, which satisfies Dirac's equation when quantized, is analyzed. It is shown that this group, larger than the Poincaré group, also contains space-time dilations and local rotations. It has two Casimir operators, one is the spin and the other is the spin projection on the body frame. Its similarities with the standard model are discussed. If we consider this last spin observable as describing isospin, then, this Dirac particle represents a massive system of spin 1/2 and isospin 1/2. There are two possible irreducible representations of this kind of particles, a colourless or a coloured one, where the colour observable is also another spin contribution related to the zitterbewegung. It is the spin, with its twofold structure, the only intrinsic property of this Dirac elementary particle.

2001

A quaternionic projective theory based on the symmetry group Sl(2,H) allows one to identify various hadron models and many well-known particle transformation laws in its subgroup chains. Identifying the 16-dimensional Dirac algebra {␥ } with Sl(2,H), we use a well-established group-theoretic framework as well as the framework of projective geometry to classify elementary particles and describe their interactions at low energies. It is straightforward to derive Chiral Dynamics and explain the spinorial ('quark') structure of hadrons. Spontaneous symmetry breaking occurs naturally by coset reductions, whereas 'classical' physics is obtained via well-defined limits in terms of a group contraction. The Dirac equation can be identified within a Riemannian globally symmetric space and thus allows one to investigate the fermionic mass as a well-defined parameter. In addition, we suggest an identification of the second quantization scheme and an approach to sum up the perturbation series.

Journal of Physics: Conference Series

In order to extend our approach based on SU * (4), we were led to (real) projective and (line) Complex geometry. So here we start from quadratic Complexe which yield naturally the 'light cone' x 2 1 + x 2 2 + x 2 3 − x 2 0 = 0 when being related to (homogeneous) point coordinates x 2 α and infinitesimal dynamics by tetrahedral Complexe (or line elements). This introduces naturally projective transformations by preserving anharmonic ratios. Referring to old work of Plücker relating quadratic Complexe to optics, we discuss (linear) symplectic symmetry and line coordinates, the main purpose and thread within this paper, however, is the identification and discussion of special relativity as direct invariance properties of line/Complex coordinates as well as their relation to 'quantum field theory' by complexification of point coordinates or Complexe. This can be established by the Lie mapping which relates lines/Complexe to sphere geometry so that SU(2), SU(2)×U(1), SU(2)×SU(2) and the Dirac spinor description emerge without additional assumptions. We give a short outlook in that quadratic Complexe are related to dynamics e.g. power expressions in terms of six-vector products of Complexe, and action principles may be applied. (Quadratic) products like F µν F µν or F a µν F a µν , 1 ≤ a ≤ 3 are natural quadratic Complex expressions ('invariants') which may be extended by line constraints λk • ǫ = 0 with respect to an 'action principle' so that we identify 'quantum field theory' with projective or line/Complex geometry having applied the Lie mapping.

Arxiv preprint arXiv:0901.4230, 2009

A toy model for the electroweak interactions(without chirality) is proposed in a six dimensional spacetime with 3 timelike and 3 spacelike coordinates. The spacetime interval ds 2 = dxµdx µ is left invariant under the symmetry group SO(3, 3). We obtain the six-dimensional version of the Dirac gamma matrices, Γµ, and write down a Dirac-like lagrangian density, L = iψΓ µ ∇µψ. The spinor ψ is decomposed into two Dirac spinors, ψ1 and ψ2, which we interpret as the electron and neutrino fields, respectively. In six-dimensional spacetime the electron and neutrino fields are then merged in a natural manner. The SO(3, 3) Lorentz symmetry group must be locally broken to the observable SO(1, 3) Lorentz group, with only one observable time component, tz. The tz-axis may not be the same at all points of the spacetime and the effect of breaking the SO(3, 3) spacetime symmetry group locally to an SO(1, 3) Lorentz group is perceived by the observers as the existence of the gauge fields. The origin of mass may be attributed to the remaining two hidden timelike dimensions. We interpret the origin of mass and gauge interactions as a consequence of extra time dimensions, without the need of the so-called Higgs mechanism for the generation of mass. Further, we are able to give a geometric meaning to the electromagnetic and non-abelian gauge symmetries.

1996

A classification of hadrons and their interactions at low energies according to SU(4) allows to identify combinations of the fifteen mesons $\pi$, $\omega$ and $\rho$ within the spin-isospin decomposition of the regular representation \rhdmulti{15}. Chirally symmetric SU(2)$\times$SU(2) hadron interactions are then associated with transformations of a subgroup of SU(4). Nucleon and Delta resonance states are represented by a symmetric third rank tensor \rhdmulti{20} whose spin-isospin decomposition leads to $4\oplus 16$ `tower states' also known from the large-N$_c$ limit of QCD. Towards a relativistic hadron theory, we consider possible generalizations of the stereographic projection {\bf S}$^{2}$ $\to$ {\bf C} and the related complex spinorial calculus {\it on the basis of the division algebras with unit element}. Such a geometrical framework leads directly to transformations in a quaternionic projective `plane' and the related symmetry group SL(2,{\bf H}). In exploiting t...

Based on previous work on the Dirac algebra and su * (4) Lie algebra generators, using Lie transfer we've associated spin to line and Complex reps. Here, we discuss the construction of a Lagrangean in terms of invariant theory using lines or linear Complex reps like F µν , its dual F αβ , or even quadratic terms like e.g. F µν F µν , or F a µν F a µν with respect to regular linear Complexe. In this context, we sketch briefly the more general framework of quadratic Complexe and show how special relativistic coordinate transformations can be obtained from (invariances with respect to) line transformations. This comprises the action of the Dirac algebra on 4 × 2 'spinors', real as well as complex. We discuss a classical picture to relate photons to linear line Complexe so that special relativity emerges naturally from a special line (or line Complex) invariance, and compare to Minkowski's fundamental paper on special relativity. Finally, we give a brief outlook on how to generalize this approach to general relativity using advanced projective and (line) Complex geometry related to P 5 and the Plücker-Klein quadric as well as transfer principles.

2018

There are several 3 + 1 parameter quantities in physics (like vector + scalar potentials, four-currents, space-time, four-momentum, …). In most cases (but space-time), the three-and the one-parameter characterised elements of these quantities differ in the field-sources (e.g., inertial and gravitational masses, Lorentz-and Coulomb-type electric charges, …) associated with them. The members of the field-source pairs appear in the vector-and the scalar potentials, respectively. Sections 1 and 2 of this paper present an algebra what demonstrates that the members of the fieldsource siblings are subjects of an invariance group that can transform them into each other. (This includes, e.g., the conservation of the isotopic field-charge spin (IFCS), proven in previous publications by the author.) The paper identifies the algebra of that transformation and characterises the group of the invariance; it discusses the properties of this group, shows how they can be classified in the known nomenclature, and why is this pseudo-unitary group isomorphic with the SU(2) group. This algebra is denoted by tau (). The invariance group generated by the tau algebra is called hypersymmetry (HySy). The group of hypersymmetry had not been described. The defined symmetry group is able to make correspondence between scalars and vector components that appear often coupled in the characterisation of physical states. In accordance with conclusions in previous papers, the second part (Sections 3 and 4) shows that the equations describing the individual fundamental physical interacions are invariant under the combined application of the Lorentztransformation and the here explored invariance group at high energy approximation (while they are left intact at lower energies). As illustration, the paper presents a simple form for an extended Dirac equation and a set of matrices to describe the combined transformation in QED. The paper includes a short reference illustration (in Section 2.2) to another applicability of this algebra in the mathematical description of regularities for genetic matrices.

Physical Review, 1965