Archive for Dirac equation

Beautiful Equations

Posted in Biographical, mathematics, The Universe and Stuff with tags , , , , on February 25, 2025 by telescoper

I did a lecture today about the Dirac Equation (which is almost 100 years old, having been first presented in 1928). You might think this is a difficult topic to lecture on, but it’s really a piece of cake:

This reminds me that a a while ago I posted about an interesting article on the BBC website that discussed the way mathematicians’ brains appear to perceive “beauty”. A (slightly) more technical version of the story can be found here. According to functional magnetic resonance imaging studies, it seems that beautiful equations excite the same sort of brain activity as beautiful music or art.

The question of why we think equations are beautiful is one that has come up a number of times on this blog. I suspect the answer is a slightly different one for theoretical physicists compared with pure mathematicians. Anyway, I thought it might be fun to invite people offer suggestions through the comments box as to the most beautiful equation along with a brief description of why.

I should set the ball rolling myself, and I will do so with the Dirac Equation:

dirac_equation

This equation is certainly the most beautiful thing I’ve ever come across in theoretical physics, though I don’t find it easy to articulate precisely why. I think it’s partly because it is such a wonderfully compact fusion of two historic achievements in physics – special relativity and quantum mechanics – but also partly because of the great leaps of the imagination that were needed along the journey to derive it and my consequent admiration for the intellectual struggle involved. I feel it is therefore as much an emotional response to the achievement of another human being – such as one feels when hearing great music or looking at great art – as it is a rational response to the mathematical structure involved. But it’s not just that, of course. The Dirac Equation paved the way to many further developments in particle physics. It seems to encapsulate so much about the behaviour of elementary particles in so few symbols. Some of its beauty derives from its compactness- it uses up less chalk in a mathematical physics lecture.

Anyway, feel free to suggest formulae or equations, preferably with a brief explanation of why you think they’re so beautiful.

P.S. Paul Dirac was my (academic) great-grandfather.

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Back to Teaching

Posted in Biographical, Education, Maynooth, The Universe and Stuff with tags , , , on February 4, 2025 by telescoper

After yesterday’s holiday it was back to teaching full-time this morning with the first lecture of my module on Particle Physics. I just about managed to get everything ready in time for the teaching session at 1pm which, because it was an introductory lecture with lots of pictures, I decided to do via powerpoint rather than my usual chalk-and-talk. That didn’t get off to a very good start because the podium PC in my room had decided to do a Windows update just before I started and I had to wait for that to finish before I could show my slides. I suppose that happened because this was the first day of teaching after a lengthy break so nobody had used the room recently.

Most of the lecture was devoted to introducing natural units, which I intend to use throughout the module, like I have on previous occasions I have taught this sort of material for reasons I explained here. The last time I taught particle physics was some 15 years ago, so I had to update some things, especially the picture of the components of the standard model to include the Higgs. After extensive research (by which I mean looking at wikipedia) I found the above; the Higgs is on the right. Unfortunately the particle masses – which reveal themselves if you click on the image above – are not given in natural units, but have pesky factors of c-squared in them. You can’t have everything.

The bit I’m looking forward to most is doing the Dirac Equation which, years ago when I was at Sussex, was once the subject of a cake:

That particular cake was a lemon drizzle cake which unfortunately is not one of the flavours represented in the standard model.

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The Effect of Gravity on the Muon Magnetic Moment

Posted in The Universe and Stuff with tags , , , , , on February 3, 2018 by telescoper

Only time for a short post today, but I think this may turn out to be an important result. There’s a paper by Morishima et al. on the arXiv with the rather dry title Post-Newtonian effects of Dirac particle in curved spacetime – III : the muon g-2 in the Earth’s gravity, which suggests that the anomalous magnetic dipole moment of the muon.

Here is the abstract of the paper. You can click on it to make it bigger.

In a nutshell the anomaly is that according to basic relativistic quantum theory in the form of the Dirac equation, the muon (and any other charged spin-1/2 fermion) should have a magnetic dipole moment μ of magnitude (given in terms of its mass m and fundamental constants) by μ=geħ/4m with the g-factor g=2 for Dirac fermions. The anomaly is that this can be measured and it appears that g differs from zero by a small but significant amount, i.e. (g-2) is not zero. It has been widely suggested that this discrepancy suggests the existence of physics beyond the Standard Model of Partlce Physics. Well, gravity is not included in the Standard Model so I suppose this could still be right, but the it this calculation may well disappoint those who were hoping that (g-2) might provide evidence for, e.g., supersymmetry when it looks like it might be something rather more mundane, ie the Earth’s gravity!

UPDATE: It appears there is an error in the paper; see here. You may stand down.

UPDATE: Well, that was pretty fast. There’s now a paper on the arXiv by Matt Visser that gives a detailed refutation of the above claim. Here is the abstract:

In three very recent papers, (an initial paper by Morishima and Futamase, and two subsequent papers by Morishima, Futamase, and Shimizu), it has been argued that the observed experimental anomaly in the anomalous magnetic moment of the muon might be explained using general relativity. It is my melancholy duty to report that these articles are fundamentally flawed in that they fail to correctly implement the Einstein equivalence principle of general relativity. Insofar as one accepts the underlying logic behind these calculations (and so rejects general relativity) the claimed effect due to the Earth’s gravity will be swamped by the effect due to Sun (by a factor of fifteen), and by the effect due to the Galaxy (by a factor of two thousand). In contrast, insofar as one accepts general relativity, then the claimed effect will be suppressed by an extra factor of [(size of laboratory)/(radius of Earth)]^2. Either way, the claimed effect is not compatible with explaining the observed experimental anomaly in the anomalous magnetic moment of the muon.

That’s how science goes!

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The Cake Equation

Posted in The Universe and Stuff with tags , , , on May 31, 2014 by telescoper

Yesterday being the last Friday of the month of May it was time for another tea-and-cake event in the School of Mathematical and Physical Sciences. These provide an opportunity for staff to get together and chat while demolishing a specially-themed cake. The cakes themselves are organized by the inestimable Miss Lemon and I never know what the theme is before the goods arrive, so I have to ad lib a short introduction (for just a minute, without repetition, hesitation, deviation or repetition) before cutting the cake.

As you will observe, this time the (Lemon Drizzle) cake was decorated with the Dirac Equation (which I consider to be the most beautiful equation in physics)..

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The most beautiful equation?

Posted in The Universe and Stuff with tags , , , , on February 13, 2014 by telescoper

There’s an interesting article on the BBC website today that discusses the way mathematicians’ brains appear to perceive “beauty”. A (slightly) more technical version of the story can be found here. According to functional magnetic resonance imaging studies, it seems that beautiful equations excite the same sort of brain activity as beautiful music or art.

The question of why we think equations are beautiful is one that has come up a number of times on this blog. I suspect the answer is a slightly different one for theoretical physicists compared with pure mathematicians. Anyway, I thought it might be fun to invite people offer suggestions through the comments box as to the most beautiful equation along with a brief description of why.

I should set the ball rolling myself, and I will do so with this, the Dirac Equation:

dirac_equation

This equation is certainly the most beautiful thing I’ve ever come across in theoretical physics, though I don’t find it easy to articulate precisely why. I think it’s partly because it is such a wonderfully compact fusion of two historic achievements in physics – special relativity and quantum mechanics – but also partly because of the great leaps of the imagination that were needed along the journey to derive it and my consequent admiration for the intellectual struggle involved. I feel it is therefore as much an emotional response to the achievement of another human being – such as one feels when hearing great music or looking at great art – as it is a rational response to the mathematical structure involved. But it’s not just that, of course. The Dirac Equation paved the way to many further developments in particle physics. It seems to encapsulate so much about the behaviour of elementary particles in so few symbols. Some of its beauty also derives from its compactness.

Anyway, feel free to suggest formulae or equations through the comments box, preferably with a brief explanation of why you think they’re so beautiful.

19 Comments »