Quantum Mechanics in the Geometry of Space-Time

This book continues the fundamental work of Arnold Sommerfeld and David Hestenes formulating theoretical physics in terms of Minkowski space-time geometry. We see how the standard matrix version of the Dirac equation can be reformulated in terms of a real space-time algebra, thus revealing a geometric meaning for the “number i” in quantum mechanics. Next, it is examined in some detail how electroweak theory can be integrated into the Dirac theory and this way interpreted in terms of space-time geometry. Finally, some implications for quantum electrodynamics are considered. The presentation of real quantum electromagnetism is expressed in an addendum. The book covers both the use of the complex and the real languages and allows the reader acquainted with the first language to make a step by step translation to the second one.

Table of Contents

1 Introduction

Part I The Real Geometrical Algebra or Space–Time Algebra. Comparison with the Language of the Complex Matrices and Spinors

2 The Clifford Algebra Associated with the Minkowski Space–Time M

3 Comparison Between the Real and the Complex Language

Part II The U(1) Gauge in Complex and Real Languages. Geometrical Properties and Relation with the Spin and the Energy of a Particle of Spin 1/2

4 Geometrical Properties of the U(1) Gauge

5 Relation Between the U(1) Gauge, the Spin and the Energy of a Particle of Spin 1/2

Part III Geometrical Properties of the Dirac Theory of the Electron

6 The Dirac Theory of the Electron in Real Language

7 The Invariant Form of the Dirac Equation and Invariant Properties of the Dirac Theory

Part IV The SU(2) Gauge and the Yang–Mills Theory in Complex and Real Languages

8 Geometrical Properties of the SU(2) Gauge and the Associated Momentum–Energy Tensor

Part V The SU(2) 3 U(1) Gauge in Complex and Real Languages

9 Geometrical Properties of the SU(2) 3 U(1) Gauge

Part VI The Glashow–Salam–Weinberg Electroweak Theory

10 The Electroweak Theory in STA: Global Presentation

11 The Electroweak Theory in STA: Local Presentation

Part VII On a Change of SU(3) into Three SU(2) 3 U(1)

12 On a Change of SU(3) into Three SU(2) 3 U(1)

Part VIII Addendum

13 A Real Quantum Electrodynamics

Part IX Appendices

14 Real Algebras Associated with an Euclidean Space

15 Relation Between the Dirac Spinor and the Hestenes Spinor

16 The Movement in Space–Time of a Local Orthonormal Frame

17 Incompatibilities in the Use of the Isospin Matrices

18 A Proof of the Tetrode Theorem

19 About the Quantum Fields Theory