On a Microscopic Representation of Space-Time III

Physics of Atomic Nuclei, 2012

We start from a noncompact Lie algebra isomorphic to the Dirac algebra and relate this Lie algebra in a brief review to low energy hadron physics described by the compact group SU(4). This step permits an overall physical identification of the operator actions. Then we discuss the geometrical origin of this noncompact Lie algebra and 'reduce' the geometry in order to introduce in each of these steps coordinate definitions which can be related to an algebraic representation in terms of the spontaneous symmetry breakdown along the Lie algebra chain su*(4) -→ usp(4) -→ su(2)×u(1). Standard techniques of Lie algebra decomposition(s) as well as the (physical) operator identification give rise to interesting physical aspects and lead to a rank-1 Riemannian space which provides an analytic representation and leads to a 5-dimensional hyperbolic space H 5 with SO(5,1) isometries. The action of the (compact) symplectic group decomposes this (globally) hyperbolic space into H 2 ⊕H 3 with SO(2,1) and SO(3,1) isometries, respectively, which we relate to electromagnetic (dynamically broken SU(2) isospin) and Lorentz transformations. Last not least, we attribute this symmetry pattern to the algebraic representation of a projective geometry over the division algebra H and subsequent coordinate restrictions.

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.

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.

European Journal of Physics, 2016

The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term by in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.

This article is intended as an addition to the book , since, in the first edition, I was double minded whether to introduce the Dirac theory for young students. Now I am quite sure that it should be introduced, and for several reasons. First, the Cl3 formulation of the Dirac theory is simple and the derivation of the Dirac's formula is straightforward. Second, it is relatively easy to show that gamma matrices are not the only possibility in linearizing the Klein-Gordon equation (we even do not need it in Cl3). Finally, the fact that it is possible to use the same mathematical (3D) formalism for classical mechanics, the special (and general) theory of relativity (without Minkowski space), electromagnetism, and both non-relativistic and relativistic quantum mechanics (without the imaginary unit) is remarkable. Not to mention the geometric clarity and possibilities of unifications, as well as generalizations. Moreover, all this without coordinates, matrices, tensors… In addition, we should appreciate the new concept of oriented numbers and simple fact that Cl3 contains complex, hypercomplex, and dual numbers, quaternions, spinors, etc. Geometric algebra of 3D Euclidean vector space (Cl3) is truly rich in structure and the question remains as to how physics would have developed had the ideas of Grassmann and Clifford been accepted in the late nineteenth and early twentieth centuries.

We recall some basic aspects of line and line Complex representations, of symplectic symmetry emerging in bilinear point transformations as well as of Lie transfer of lines to spheres. Here, we identify SU(2) spin in terms of (classical) projective geometry and obtain spinorial representations from lines, i.e. we find a natural non-local geometrical description associated to spin. We discuss the construction of a Lagrangean in terms of line/Complex invariants. We discuss the edges of the fundamental tetrahedron which allows to associate the most real form SU(4) with its various related real forms covering SO(n,m), n + m = 6.

arXiv: History and Philosophy of Physics, 2023

In order to ask for future concepts of relativity, one has to build upon the original concepts instead of the nowadays common formalism only, and as such recall and reconsider some of its roots in geometry. So in order to discuss 3-space and dynamics, we recall briefly Minkowski's approach in 1910 implementing the nowadays commonly used 4-vector calculus and related tensorial representations as well as Klein's 1910 paper on the geometry of the Lorentz group. To include microscopic representations, we discuss few aspects of Wigner's and Weinberg's 'boost' approach to describe 'any spin' with respect to its reductive Lie algebra and coset theory, and we relate the physical identification to objects in $P^{5}$ based on the case $(1,0)\oplus(0,1)$ of the electromagnetic field. So instead of following this -- in some aspects -- special and misleading 'old' representation theory, based on 4-vector calculus and tensors, we provide and use an alternat...

Advances in Applied Clifford Algebras, 2008

In a recent publication [1] it was shown how the geometric algebra G4,1 , the algebra of 5-dimensional space-time, can generate relativistic dynamics from the simple principle that only null geodesics should be allowed. The same paper showed also that Dirac equation could be derived from the condition that a function should be monogenic in that algebra; this construction of the Dirac equation allows a choice for the imaginary unit and it was suggested that different imaginary units could be assigned to the various elementary particles. An earlier paper [2] had already shown the presence of standard model gauge group symmetry in complexified G1,3 , an algebra isomorphic to G4,1. In this presentation I explore the possible choices for the imaginary unit in the Dirac equation to show that SU (3) and SU (2) symmetries arise naturally from such choices. The quantum numbers derived from the imaginary unit are unusual but a simple conversion allows the derivation of electric charge and isospin, quantum numbers for two families of particles. This association to elementary particles is not final because further understanding of the role played by the imaginary unit is needed.

2004

International Journal of Quantum Chemistry, 2012

The algebra of physical space (APS) is a name for the Clifford or geometric algebra, which can be closely associated with the geometry of special relativity and relativistic spacetime. For example, the Dirac Hamiltonian can be presented as the scalar product of the electron's four-momentum and Dirac's fourvector of gamma matrices, ðc 0 ;cÞ, the latter of which is a Clifford algebra. We show here that a geometric spacetime or four-space solution of Dirac's equation conforms to the principles of APS, an early example of which is Schroedinger's solution of Dirac's equation for a free electron, which exhibits Zitterbewegung. In a four-space solution the spacetime coordinates,r and the scaled time ct, are treated on an equal footing as physical observables to avoid any suggestion of a preferred frame of reference. The geometric spacetime theory is studied here for the Coulomb problem. The positive-energy spectrum of states is found to be identical within numerical error to that of standard Dirac's theory, but the wave function exhibits Zitterbewegung. It is shown analytically how the geometric spacetime solution can be reduced to the standard solution of Dirac's equation, in which Zitterbewegung is absent. The rigor of APS and of its conforming geometric spacetime solution provide strong support for further investigation into the physical interpretation of the geometric spacetime Dirac's wave function and Zitterbewegung.