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INDEX

Introduction2 What are Quaternions?3 Unifying Two Views of Events4 A Brief History of Quaternions

Mathematics6 Multiplying Quaternions the Easy Way7 Scalars, Vectors, Tensors and All That

11 Inner and Outer Products of Quaternions13 Quaternion Analysis23 Topological Properties of Quaternions28 Quaternion Algebra Tool Set

Classical Mechanics32 Newtons Second Law35 Oscillators and Waves37 Four Tests of for a Conservative Force

Special Relativity40 Rotations and Dilations Create the Lorentz Group43 An Alternative Algebra for Lorentz Boosts

Electromagnetism48 Classical Electrodynamics51 Electromagnetic Field Gauges53 The Maxwell Equations in the Light Gauge: QED?56 The Lorentz Force58 The Stress Tensor of the Electromagnetic Field

Quantum Mechanics62 A Complete Inner Product Space with Diracs Bracket Notation67

Multiplying quaternions in Polar Coordinate Form69 Commutators and the Uncertainty Principle74 Unifying the Representations of Integral and Half-Integral Spin79 Deriving A Quaternion Analog to the Schrdinger Equation83 Introduction to Relativistic Quantum Mechanics86 Time Reversal Transformations for Intervals

Gravity89 Unified Field Theory by Analogy101 Einsteins vision I: Classical unified field equations for gravity and electromagnetism

using Riemannian quaternions115 Einsteins vision II: A unified force equation with constant velocity profile solutions123 Strings and Quantum Gravity127

Answering Prima Facie Questions in Quantum Gravity Using Quaternions134 Length in Curved Spacetime136 A New Idea for Metrics138 The Gravitational Redshift

140 A Brief Summary of Important Laws in Physics Written as Quaternions

155 Conclusions

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What Are Quaternions?

Quaternions are numbers like the real numbers: they can be added, subtracted, multiplied, anddivided. There is something odd about them, hinted at by the Latin. Quaternions are composed offour numbers that work together as one. They were discovered by several people back in theeighteen hundreds. Some enthusiasts thought quaternions would be able to express everything thatcould happen in our three dimensions of space and one for time because quaternions naturally hadthat form too. Math accidents do not happen - they revel deep things about how Nature works.

Unfortunately, the big fans of quaternion mathematics claimed far more than they would deliver.Useful ideas born from initial quaternion work - for example the notion of scalars, vectors, div,grad, and curl - were stripped out of their initial context, and made more "general". I use thequotes because from my viewpoint, there is nothing more general than a number that can be added,subtracted, multiplied, and divided. I am trying to continue the project of applying four-dimen-sional quaternions to the four-dimensional spacetime we live in.