There is redundancy in the mathematical description of physical reality. The redundancy has been increasing exponentially since the beginning of the 20’th century. We have different theories to describe the microscopic and macroscopic phenomena, for example. I am not talking about the redundancy in this sense. This kind of redundancy is due to the lack of a unified theory. The redundancy I am talking about is that there are multiple mathematical formulations of the same physical law. For example, you can take a look at my post titled “Concise Formulation of Maxwell Equations” to see that the Maxwell equations can be written in different mathematical notations. What is the purpose of notational redundancy?
- Mathematical elegance: conciseness in mathematical expression is a strong psychological driver among scientists
- Expressing the physical law in a different notation may lead to new insights
- It may be easier to solve an equation in a different notation.
- A certain notation may be more intuitive. For example, in the post “Mathematical Spirals” I used the polar coordinate notation. If I wrote the equations of those spirals in Cartesian coordinates they would look very ugly and not intuitive at all.
- Computer simulations of the physical phenomena and the requirement for speed in computations made it necessary to introduce new mathematical formulations or sometimes, as in the case of quaternions, bring back an old formulation to life. The point is that not all formulations (notations) are equal when it comes to computer simulations.
- The rise of the “duality” approach in physics. In my post titled “Meaning of Duality in Physics” I explained how dual theories help with the solutions of equations.
What is wrong with notational redundancy?
- Confusion! Physics students are confused! Physics teachers are confused! There is so much redundancy in the mathematical notation that teachers and students are confused both. There is a crisis in physics education.
- The best evidence for the crisis is in the treatment of the rotations in physics. There are multiple equivalent descriptions of rotations. You go through the system, get your Ph.D. in physics but you are still confused about the mathematics of rotations.
Notation that leads to new insights If I were a teacher I would teach the rotations in the notation that would most likely lead to new insights in physics. That would be the notation involving “Pauli spin matrices.” In physics the subject of rotations cover a wide range: it is not just rotations in 3D space. We deal with abstract rotations too. Properly speaking we should call those mathematical operations “transformations.” In this post I will mention 2 tutorials on the subject of abstract rotations (transformations) in physics. These tutorials are very exceptional! They exemplify perfection in teaching. I wish we had tutorials like this when I was in school. I examined all my notebooks from college and graduate school. With the objectivity gained in the last 20 years I have to say with great sadness that my professors did a very poor job of explaining this material. There are good books out there but they are not freely available unless you have access to a university library. We must have freely available quality tutorials on the web. This is very important. A motivated kid in a remote location should be able learn physics from the best teachers. I congratulate the authors of these tutorials with all my heart. They have done a great service.
Lie Groups in Physics: This lecture course was originally set up by M. Veltman, and subsequently modified and extended by Bernard de Wit and G. ‘t Hooft. Please save this pdf document. The reputation of the authors add to the value of this document but objectively speaking this is the best tutorial on the subject. I have always admired the clarity of thought in G. ‘t Hooft’s papers. G. ‘t Hooft and M. Veltman shared the Nobel Prize in Physics in 1999 for “for elucidating the quantum structure of electroweak interactions in physics.” This short description does not do justice to the totality of G. ‘t Hooft’s contributions to theoretical physics. If you are a beginning student you should also check out his “How to become a good theoretical physicist.” That site is also a good place to find links to other physics education resources. If you understand what is written in “Lie Groups in Physics” then you know more than a typical professional physicist. I bet that you can do research too.
Introduction to Spinors: Excellent review paper by Andrew Steane of Oxford University. His major works to date are on error correction in quantum information processing, including Steane codes. He was awarded the Maxwell Medal and Prize of the Institute of Physics in 2000.