The most brilliant students, over the years, notice that certain topics in the teaching of physics are not well founded, quite the contrary. Especially in engineering courses, we tend to use certain mathematical techniques without asking too many questions, as long as they work.
But people who for years have trained the intellect with the rigor of mathematics and the meticulousness of philosophy, cannot let certain anomalies pass under the radar.
Here are some of them:
- The unification of physics is beautiful, speaking about contents, but from the point of view of languages and mathematical techniques, why are we faced with such a varied panorama?
- algebraic structures (groups, rings, fields) are part of the scientific high school syllabus. When you start studying physics, the language of vectors is presented as the key to master the physical world, yet you realize that there is no single multiplication operation between vectors and there is no division between vectors!
Instead, two products are presented … half-capables and not even reversible.
Scalar product issues
Vector product issues
- in high school physics we meet physical quantities treated as vectors, but they are not! They are called axial vectors because this problem is evident, but it’s never clarified that they are quantities of a completely different nature.
- complex numbers still maintain an aura of mystery, as Leibniz said, they seem “suspended between being and non-being”. In technical books we still read sentences of this tone today: Although complex numbers are fundamentally disconnected from our reality, they can be used to solve science and engineering problems (The Scientist and Engineer’s Guide to Digital Signal Processing, S.W. Smith, California Technical Publishing).
Yet in physics they play an essential role in the equations of quantum mechanics and are a fundamental tool in electrical engineering applications. What is discouraging is that they are treated with the usual approach “[…] we solve the equations and we keep only the real part of the solution, discarding the imaginary one”.
The description of physics with vector analysis that you start learning in high school is not – as you might think – the only possible way. In fact, this turns out to be a historical path that established itself at the beginning of the last century. Gibbs and Heaviside treated successfully electromagnetism with vectors but, to be honest, at the time there was already another way: the quaternions and in general the formalism of geometric algebra, developed by Clifford, Hamilton and Grassmann. Yet, the Gibbs and Heaviside formalism found widespread diffusion because the alternative had even more serious problems.
The modern synthesis, which we present here, is a decisive step forward in the comprehensibility of the mathematics that describes the physical world. Nothing changes regarding the results of the calculations, but their geometric meaning is finally clarified.
Had we to express the advantages of Geometric Algebra (which we will often indicate with GA) in the form of a manifesto, here are some.