This week marks a landmark in the class (Introduction to Quantum Field Theory – see also here) since it focuses on the seminal work of one my heroes, Paul Dirac, who quietly went about his business of puzzling over the issue of how to find an equation that 
describes the properties of electrons (particles of spin one-half), and in finding what is now called the Dirac equation (see snapshot from my notes on right), uncovered a hugely rich and bright cornerstone of fundamental physics. It is famously described as a sort of square root of the relativistic wave equation known at the time – the Klein-Gordon equation – and in that way of thinking you quickly arrive at the idea of anti-particles (as did Dirac), since taking a square root leads to two solutions (both +2 and -2 square to give you 4). One solution turned out to be the electron, and the other leads (by an appropriate path of reasoning) to its anti-particle, the positron*.
Besides leading inevitably to anti-particles, the equation (which Dirac pursued in large part using mathematical criteria -albeit physically rooted ones) also leads to a stronger understanding of spin than had been known up to then (the middle to late 1920s), forever leading us away from the idea of little spinning tops that were nevertheless structureless, and toward realizing things in terms of representations of the Lorentz group (those transformations of space and time coordinated that take you from one reference frame to another in Special Relativity). Spinor representations became firmly part of physics, and anticommuting variables and Clifford algebras (no relation) – some relatively new mathematics discovered half a century earlier by purely mathematical considerations – were seen to be intrinsically part of Nature’s scheme for the fundamental building blocks of the universe.
Fun stuff to teach about. We should be able to get to Quantum Electrodynamics (QED) by Thursday, and start computing with it next week.
-cvj
*There are cleaner ways to think about the positron than using the original route of messing around with worrying about negative energy and so forth, but that is the way he arrived at it first… also, sometimes that route is useful in constructing effective theories (“holes” in condensed matter spring to mind)…
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Again with the long, vaguely connected, quotations. And so many of them, on multiple threads. Why?
-cvj
Intuition and Logic in Mathematics by Henri Poincaré
“On the other hand, look at Professor Klein: he is studying one of the most abstract questions of the theory of functions to determine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.”
Felix Klein on intuition
“It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction ….
I am led to these remarks by the consciousness of growing danger in Germany of a separation between abstract mathematical science and its scientific and technical applications. Such separation can only be deplored, for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics …. “
Well, I suppose someone could say that, but I’m afraid I’ve not the slightest idea what it means…
-cvj
Clifford,
I remember your previous warnings about length/quotes so had curtail as much as possible. Thanks
So could yone say “Feynmans toy models” arose from the equations of Dirac?
Best,
Disconnect? I am not sure what you mean. I would say that the theoretical physicist of today is, at least in areas I know well like high energy physics, trained to use both geometric and algebraic methods quite a bit in their thinking. This has certainly been on the rise in the stirng theory community, given the nature to the research there, and some of those techniques spread far and wide to other fields, combining with algebraic and geometric practices there…
-cvj
Hi. Found it. It is restored. You often put links and so forth, and long quotes. This often looks like spam.
-cvj
Your spam filter keeps rejecting my comments. If this is what you want let me know and I will stop visiting and commenting.
Best,
Hi Clifford,
In reading Coxeter’s life story it seems that there was a disconnect toward geometry(visualization) once having the algebraic equations?
Have you found this to exist through your journey’s as a professor? I believe the visualization skills to be very important as to what can be encapsulated in the equations? I believe that you exhibit this very principle in those things you share as a blogger.
All the best,
Ah, such a great equation. I can’t believe it’s 12 or 13 years since I was studying this! It’s great to see you talking about the course, I’m sure the students are loving your class.
I still remember the “aha moment” when I was introduced to the idea of the Fermi sea. In that context, particle/anti-particle pairs started to make sense. A related image is that of the crest and trough of a wave which I’ve kept in mind if I ever have to explain something close to field theory to my relatives.