An Introduction to Geometric Algebra with some Preliminary Thoughts on the Geometric Meaning of Quantum Mechanics

16th Symmetries in Science

Horn, Martin Erik^1

Bbw University of Applied Sciences Berlin-Brandenburg, Leibnizstraße 11-13, Berlin

D-10625, Germany^1

关键词: Geometric Algebra;    Mathematical constructions;    Modern physics;    Non-commutative;    Pauli matrices;    Plane wave solutions;    Special relativity;    SU(3) symmetry;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/538/1/012010/pdf
DOI  :  10.1088/1742-6596/538/1/012010

来源: IOP

【 摘 要 】

It is still a great riddle to me why Wolfgang Pauli and P.A.M. Dirac had not fully grasped the meaning of their own mathematical constructions. They invented magnificent, fantastic and very important mathematical features of modern physics, but they only delivered half of the interpretations of their own inventions. Of course, Pauli matrices and Dirac matrices represent operators, which Pauli and Dirac discussed in length. But this is only part of the true meaning behind them, as the non-commutative ideas of Grassmann, Clifford, Hamilton and Cartan allow a second, very far reaching interpretation of Pauli and Dirac matrices. An introduction to this alternative interpretation will be discussed. Some applications of this view on Pauli and Dirac matrices are given, e.g. a geometric algebra picture of the plane wave solution of the Maxwell equation, a geometric algebra picture of special relativity, a toy model of SU(3) symmetry, and some only very preliminary thoughts about a possible geometric meaning of quantum mechanics.