Applied Geometric Algebra
László Tisza
László Tisza was Professor of Physics Emeritus at MIT, where he began teaching in 1941. This online publication is a reproduction the original lecture notes for the course "Applied Geometric Algebra" taught by Professor Tisza in the Spring of 1976.
Over the last 100 years, the mathematical tools employed by physicists
have expanded considerably, from differential calculus, vector algebra
and geometry, to advanced linear algebra, tensors, Hilbert space,
spinors, Group theory and many others. These sophisticated tools provide
powerful machinery for describing the physical world, however, their
physical interpretation is often not intuitive. These course notes represent Prof. Tisza's attempt at bringing conceptual clarity and unity to the application and interpretation of these advanced mathematical tools. In particular, there is an emphasis on the unifying role that Group theory plays in classical, relativistic, and quantum physics. Prof. Tisza revisits many elementary problems with an advanced treatment in order to help develop the geometrical intuition for the algebraic machinery that may carry over to more advanced problems.
The lecture notes came to MIT OpenCourseWare by way of Samuel Gasster, '77 (Course 18), who had taken the course and kept a copy of the lecture notes for his own reference. He dedicated dozens of hours of his own time to convert the typewritten notes into LaTeX files and then publication-ready PDFs. You can read about his motivation for wanting to see these notes published in his Preface below. Professor Tisza kindly gave his permission to make these notes available on MIT OpenCourseWare.
Contents
| Lecture Notes |
|---|
| Chapter 1: Introduction (PDF) |
| Chapter 2: Algebraic Preliminaries (PDF) |
| 2.1 Groups 2.2 The geometry of the three-dimensional rotation group. The Rodrigues-Hamilton theorem 2.3 The n-dimensional vector space V(n) 2.4 How to multiply vectors? Heuristic considerations 2.5 A short survey of linear groups 2.6 The unimodular group SL(n, R) and the invariance of volume 2.7 On “alias” and “alibi”. The Object Group |
| Chapter 3: The Lorentz Group and the Pauli Algebra (PDF) |
| 3.1 Introduction 3.2 The corpuscular aspects of light 3.3 On circular and hyperbolic rotations 3.4 The Pauli Algebra |
| Chapter 4: Pauli Algebra and Electrodynamics (PDF) |
| 4.1 Lorentz transformation and Lorentz force 4.2 The Free Maxwell Field |
| Chapter 5: Spinor Calculus (PDF) |
| 5.1 From triads and Euler angles to spinors. A heuristic introduction 5.2 Rigid Body Rotation 5.3 Polarized light 5.4 Relativistic triads and spinors. A preliminary discussion 5.5 Review of SU(2) and preview of quantization |
| Supplementary Material on the Pauli Algebra (PDF) |
| A.1 Useful formulas A.2 Lorentz invariance and bilateral multiplication A.3 Typical Examples A.4 On the us of Involutions A.5 On Parameterization and Integration |
| Homework Assignments (PDF) |
| References (PDF) |
Other Publications
Tisza, László. Integration of Classical and Quantum Physics (PDF - 1.8MB). Physical Review A. (1989) Volume 40, Issue 12: 6781-6790. (Abstract)
Physics as Natural Philosophy: Essays in Honor of László Tisza.
Edited by Abner Shimony and Herman Feshbach. Cambridge, Mass. The
MIT Press, (1982). ISBN: 9780262693080.
The Thermodynamics of Phase Equilibrium. Annals of Physics 13, 1-92 (1961).
With P. M. Quay, Statistical Thermodynamics of Equilibrium. Ibid. 25, 49-90 (1963).
Generalized Thermodynamics, Cambridge, MA, MIT Press (1966). ISBN: 9780262200103.
The Reasonable Effectiveness of Mathematics in the Natural science, in Experimental Metaphysics. R. S. Cohen et al. (eds.), Kluwer Academic Publishers (1997).
Resolve the Paradoxes of the Beginning, Part I, Fall 1999; Part II, December 1999. SPS Newsletter, Society of Physics Students.
