Natural Sciences
Life Sciences
Scientific Computing
Scientific Computing

Eckhard Hitzer, Department of Material Sciences, International Christian University, Mitaka, Japan

Institute for Theoretical Physics, Seminar Room, Philosophenweg 19

D.W. Heermann; M. Salmhofer; U. Schwarz, CellNetworks Member; M. Haverkort

This talk first reviews the integration of relativistic physics through the works of Hamilton, Grassmann, Maxwell, Clifford, Einstein, Hestenes and lately the Cambridge (UK) Geometric Algebra Research Group. We start with the geometric algebra (Clifford algebra) of spacetime (STA). We show how frames and trajectories are described and how Lorentz transformations acquire their fundamental rotor form. Spacetime dynamics deals with spacetime rotors, which have invariant and frame dependent splits. Spacetime rotor equations yield the proper acceleration (bivector) and the Fermi (vector) derivative. A first application is given with the relativistic STA formulation of the Lorentz force law, leading to the description of spin precession in magnetic fields and Thomas precession. Now the stage is ready for introducing the STA Maxwell equation, which combines all 4 equations in one single STA equation. STA has procedures to extract from the electromagnetic field strength bivector F, electric and magnetic fields (also for relative motion observers) and field invariants, field momentum and stress-energy tensor. The Liénard-Wiechert potential gives the retarded field of a point charge. In addition, we formulate the Dirac equation in STA, both massless and massive. From the Dirac equation we can derive STA expressions for Dirac observables. Plane wave states are described with the help of rotor decomposition. Secondly, we briefly explain how the geometric product of crystal cell vectors generates all point groups. In the conformal version of geometric algebra, rotors also generate translations by cell vectors, providing a unified versor description of all space group symmetry transformations. This description has proven ideal for creating an animated, interactive, explorative software visualization of all three-dimensional space groups.

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