IN-DEPTH 2022: GROUP THEORY, BIWEEKLY POST #6

How Will I Present?

I plan on presenting with a tall whiteboard, most likely from the TALONS class. The meat of the presentation will be me explaining to wandering visitors what group theory is, why it’s useful, and how you do it. Given the small amount of time each person will probably be at my exhibit, I’ll center the discussion around Cayley graphs of easy groups, as they are very visual and tactile. To help, I might even bring polyhedral blocks to motivate dihedral groups, or small coloured pieces of paper for symmetric groups. Under the assumption that I will have some downtime where nobody is at my exhibit, I might prepare some easy problems for me to do on the whiteboard as an accurate showcase of what group theory looks like in practice.

How Will I Prepare?

I will need make sure my understanding of the basics are crystal clear. Writing a Log post on group mappings might help ensure that. I may also revisit earlier lectures, check alternative sources I didn’t use for those early concepts, or run by my explanations with friends and family. I will also need to curate a small list of problems. Macauley has a list of homework pdf’s on his website, but some problems are unrealistic to my timeframe given, or are rather dull. I will thus need to skim through a couple of those to find ones that are the most interesting.

General Progress

My result is pretty much finished at this point. I didn’t end up creating a more general version as I had hoped. What ended up happening was that I finished the lectures on Galois theory, and it finally reached the part about group solvability. To my dismay, the information they provided was no more complete than my own, though it was from a slightly different perspective, and it was interesting to see how my own results could be translated into theirs’.

 Frustratingly, the lectures seemed to have left out the most important part of Galois theory: the connection between polynomials and groups. At the end of the last lecture, they outline all the steps they’ve explained and proved about Galois’ work, except for the fact that polynomials are solvable by radicals if and only if their Galois group is solvable, which is the lynchpin of the whole theory. I am currently reading Galois Theory by Tom Leinster to fill this gap in knowledge, but the proofs are dense, and I haven’t read the other 90% of the book.

Inexplicable omission

I would also like to finish off this blog post by relaying an interesting quote I found while reading The 1986 Dirac Memorial Lectures by Feynman and Weinberg. In the latter half of the lecture, Weinberg comes in to try and explain the most basic, fundamental physics they knew at the time, principally the fundamental particles and the laws that govern their behaviour. He had this to say:

Increasingly, many of us have come to think that the missing element that has to be added to quantum mechanics is a principle, or several principles, of symmetry. A symmetry principle is a statement that there are various ways you can change the way you look at nature, which actually change the direction the state vector is pointing, but which do not change the rules that govern how the state vector rotates with time. The set of all these changes in point of view is called the symmetry group of nature. It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today. I would like to suggest something here that that I am really not certain about but which is at least a possibility: that specifying the symmetry group of nature may be all we need to say about the physical world, beyond the laws of quantum mechanics.

There is another reason for believing that symmetries are fundamental, and possibly all that one needs to learn about the physical world beyond quantum mechanics itself. Consider how you describe an elementary particle. How do you tell one elementary particle from another? Well You have to give its energy and its momentum, and you have to give its electric charge, its spin and a few other numbers we know about. Now if you give those numbers, that is all you can say about an elementary particle; an electron with values of energy, momentum, etc., is identical to every other electron with these values. Now these numbers, the energy, momentum, and so on, are simply descriptions of the way that the particles behave when you perform various symmetry transformations. For instance, I have already said that when you rotate your laboratory, the state vector of an electron’s spin rotates by half the angle; that property is described by saying the electron is a spin one-half particle. In a similar way, though it may not be obvious to all of you, the energy of the particle just tells you how the sate vector of the particle changes when you change the way you set your clock, the momentum of the particle tells you how you change the state vector when you change the position of your laboratory, and so on. From this point of view, at the deepest level, all we find are symmetries and responses to symmetries. Matter itself dissolves, and the universe itself is revealed as some large reducible representation of the symmetry group of nature.”

Steven Weinberg, The 1986 Dirac Memorial Lectures

Keeping in mind that group theory is the tool that one uses to study symmetry, what could be a more fundamental thing to study?