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.

R.I.P Roberta Flack (1937-2025)

Posted in Music, R.I.P. with tags , on February 24, 2025 by telescoper

I just heard the sad news of the death at the age of 88 of the wonderful singer Roberta Flack. Many years ago I posted my favourite song of hers (performed live on Top of the Pops way back in 1972) but it seems the link I used is now broken so I’ll post the original version. Roberta Flack’s The First Time Ever I Saw Your Face first appeared on her album First Take back in 1969 but it didn’t become a hit until it featured in Clint Eastwood’s (1971) film Play Misty For Me. Incidentally, the bass player on this track is the great Ron Carter.

It’s probably the most beautiful love song I’ve ever heard and to my mind it hasn’t dated at all in 50 years. I suppose if you’ve ever felt about someone the way it describes you never forget…

Rest in peace, Roberta Flack (1937-2025). To the dark, and the endless skies.

Particle Physics Masterclass at Maynooth

Posted in Education, Maynooth, The Universe and Stuff with tags , , on February 23, 2025 by telescoper

Last week’s announcement about Ireland joining CERN reminded me that I should have advertised the annual Particle Physics Masterclass at Maynooth University long before now, not least because I’m actually teaching particle physics this year. My only excuse is that I’m old and forgetful. Anyway, better late than never; there’s still almost a week until the registration closes.

Since 2012 the Department of Theoretical Physics hosted the International Particle Physics Masterclasses for secondary school students each spring (except for 2020 when it was cancelled due to Covid-19 restrictions).  Now the Department of Theoretical Physics is no more, having been incorporated last year into the Department of Physics, but the Particle Physics Masterclasses continue; the next event will be on Tuesday 18 March 2025.

These Masterclasses give secondary school students the opportunity to discover the world of quarks and leptons for themselves, by performing measurements on real data from CERN, meeting active particle physics researchers and linking up with like-minded students from other countries.  We will join thousands of other secondary school students at more than 100 universities and laboratories around Europe and worldwide in a programme stretching over four weeks.

Physics at the most fundamental level – the smallest and most basic building blocks of matter – is an exotic world.  But a few introductory talks and working with data from CERN will give the students insight into the fundamental particles of matter and the forces between them, as well as what went on during the Big Bang.

In the morning the students are introduced to particle physics, experiments and detectors in lectures given by active particle physics researchers.  After an early lunch, they work on their own with data from the ALICE detector at CERN. Afterwards they participate in a video conference with students from other countries and moderators at CERN, where they discuss and compare their results.  For more information on the masterclasses, see the International Masterclasses web site.

You can find more information about the event here and you can register here. Hurry up though as the deadline for registration is the end of this month, i.e. this Friday, February 28th!

Das Lied von der Erde

Posted in Biographical, Music with tags , , , , , , , on February 22, 2025 by telescoper

And it came to pass that yesterday evening I travelled into Dublin for another concert at the National Concert Hall. The main item on the menu was Das Lied von der Erde (“The Song of the Earth”), an orchestral work for two voices and orchestra by Gustav Mahler. Sometimes described as a song cycle this piece is a symphony in all but name (and number). Mahler was suspicious about counting this work as his 9th Symphony because of the Curse of the Ninth. He did go on to composer another (numbered) Symphony but did not live to hear it performed. He didn’t live to hear Das Lied von der Erde performed either.

Das Lied von der Erde (“The Song of the Earth”) is a long work – performance time is just over an hour – and it is spread over six movements, thematically linked by translations of classical Chinese poems translated into German by Has Bethge. The final movement, by far the longest, incorporates two texts whereas the others include one each.

This is one of my favourite works but I’ve only ever heard it on the radio or on a recording so I was delighted to see it was coming up at the National Concert Hall. I enjoyed last night’s performance enormously. The National Symphony Orchestra was conducted by Jennifer Cottis with soloists Samuel Sakker (tenor) and Karen Cargill (mezzo-soprano). The start of this piece is difficult for the tenor who has to come in at full volume. At first I thought he was going to struggle, but he hit his straps very quickly and delivered a strong performance. Karen Cargill was superb throughout, her voice very well matched to the demands of the music. I have heard her sing Mahler before, incidentally, in Cardiff, and she was great then too. The whole orchestra played beautifully, but I would pick out the woodwind section for special mention.

That wasn’t the entire concert. There was also the Irish premier of a new work work by Ailís Ní Ríain called The Land Grows Weary of its Own, which is a meditation on the effects on bird populations and migration thereof caused by Earth’s changing climate. It’s an interesting piece, with some fascinatingly complex passages, especially for the percussion. The composer was the audience, but unforunately the auditorium was only about half full for the performance. It only lasts about 20 minutes so the interval came quite quickly. Returning after my glass of wine I could see a much fuller Concert Hall so some people obviously skipped the first piece, which is a shame.

Yesterday was a very rainy and blustery day and on the way home I thought about the number of times I’ve walked from Connolly Station to the NCH yet never been rained on. Last night was no different.

Weekly Update from the Open Journal of Astrophysics – 22/02/2025

Posted in OJAp Papers, Open Access, The Universe and Stuff with tags , , , , , , , , , , , , , , , , , , on February 22, 2025 by telescoper

It’s Saturday morning again so it’s time for an update of papers published at the Open Journal of Astrophysics. Things have picked up a bit after a quiet couple of weeks. Since the last update we have published four new papers which brings the number in Volume 8 (2025) up to 18 and the total so far published by OJAp up to 253.

In chronological order of publication, the four papers published this week, with their overlays, are as follows. You can click on the images of the overlays to make them larger should you wish to do so.

The first paper to report is in fact our 250th paper:  “Untangling Magellanic Streams” by Dennis Zaritsky (Steward Observatory), Vedant Chandra (Harvard), Charlie Conroy (Harvard), Ana Bonaca (Carnegie Observatories), Phillip A. Cargile (Harvard), and Rohan P. Naidu (MIT), all based in the USA. This paper is in the folder marked Astrophysics of Galaxies and it reports on spectroscopic study aimed at teasing out the stellar populations of different strands of the Magellanic Stream. It was published on Tuesday 18th February 2025. Here is the overlay:

You can read the officially accepted version of this paper on arXiv here.

The second paper of the week  is “Compressed ‘CMB-lite’ Likelihoods Using Automatic Differentiation” by Lennart Balkenhol (Institut d’Astrophysique de Paris, France) which was one of two papers published on Wednesday 19th February. It appears in the folder Cosmology and Nongalactic Astrophysics and it describes an implementation of the CMB-lite framework relying on automatic differentiation to reduce the computational cost of the lite likelihood construction.  The overlay is here:

You can find the officially accepted version of this paper on arXiv here.

The next paper, also published on Wednesday 19th February in the folder Cosmology and Nongalactic Astrophysics is “Bayesian distances for quantifying tensions in cosmological inference and the surprise statistic” by Benedikt Schosser (Heidelberg, Germany), Pedro Riba Mello & Miguel Quartin (Rio de Janeiro, Brazil) and Bjoern Malte Schaefer (Heidelberg).  It presents a discussion of statistical divergences applied to posterior distributions following from different data sets and their use in quantifying discrepancies or tensions.

Here is the overlay:

The official published version can be found on the arXiv here.

Finally in this batch we have “Precise and Accurate Mass and Radius Measurements of Fifteen Galactic Red Giants in Detached Eclipsing Binaries” by Dominick M. Rowan,  Krzysztof Z. Stanek,  Christopher S. Kochanek & Todd A. Thompson (Ohio State University), Tharindu Jayasinghe (independent researcher),  Jacqueline Blaum (UC Berkeley),  Benjamin J. Fulton (NASA/Caltech),  Ilya Ilyin (AIP Potsdam, Germany),  Howard Isaacson, Natalie LeBaron  &  Jessica R. Lu (UC Berkeley), and  David V. Martin (Tufts University, USA).  This paper was published on Thursday 20th February 2025 in the folder Solar and Stellar Astrophysics and it presents a compilation of mass and readius measurements of red giant stars obtained using spectroscopic measurements together with light curves and the eclipsing binary models obtained using PHOEBE.

The overlay is here:

You can find the “final” version on arXiv here.

That’s all for this week. I’ll do another update next Saturday.

Ireland Joining CERN

Posted in Science Politics, The Universe and Stuff with tags , , , , , , on February 20, 2025 by telescoper

The big news in Irish physics this week was the announcement that Ireland’s application to join the European Organisation for Nuclear Research (CERN) has been accepted in principle, and the country is expected to become an associate member in 2026. The formal process to join began in late 2023, as described here. Maynooth University responded to the news in positive fashion here, including the statement that

This important decision represents a transformative step for Irish science, research, and innovation, unlocking unparalleled opportunities for students, researchers, and industry.

I think this is a very good move for Irish physics, and indeed for Ireland generally. I will, however, repeat a worry that I have expressed previously. There is an important point about CERN membership, however, which I hope is not sidelined. The case for joining CERN made at political levels is largely about the return in terms of the potential in contracts to technology companies based in Ireland from instrumentation and other infrastructure investments. This was also the case for Ireland’s membership of the European Southern Observatory, which Ireland joined almost 7 years ago. The same thing is true for involvement in the European Space Agency, which Ireland joined in 1975. These benefits are of course real and valuable and it is entirely right that arguments should involve them.

Looking at CERN membership from a purely scientific point of view, however, the return to Ireland will be negligible unless there is a funding to support scientific exploitation of the facility. That would include funding for academic staff time, and for postgraduate and postdoctoral researchers to build up an active community as well as, e.g., computing facilities. This need not be expensive even relative to the modest cost of associate membership (approximately  €1.9M). I would estimate a figure of around half that would be needed to support CERN-based science.

The problem is that research funding for fundamental science (such as particle physics) in Ireland has been so limited as to be virtually non-existent by a matter of policy at Science Foundation Ireland, which basically only funded applied research. Even if it were decided to target funding for CERN exploitation, unless there is extra funding that would just lead to the jam being spread even more thinly elsewhere.

As I have mentioned before, Ireland’s membership of ESO provides a cautionary tale. The Irish astronomical community was very happy about the decision to join ESO, but that decision was not accompanied by significant funding to exploit the telescopes. Few astronomers have therefore been able to benefit from ESO membership. While there are other benefits of course, the return to science has been extremely limited. The phrase “to spoil a ship for a ha’porth of tar” springs to mind.

Although Ireland joined ESA almost fifty years ago, the same issue applies there. ESA member countries pay into a mandatory science programme which includes, for example, Euclid. However, did not put any resources on the table to allow full participation in the Euclid Consortium. There is Irish involvement in other ESA projects (such as JWST) but this is somewhat piecemeal. There is no funding programme in Ireland dedicated to the scientific exploitation of ESA projects.

Under current arrangements the best bet in Ireland for funding for ESA, ESO or CERN exploitation is via the European Research Council, but to get a grant from that one has to compete with much better developed communities in those areas.

The recent merger of Science Foundation Ireland and the Irish Research Council to form a single entity called Research Ireland perhaps provides an opportunity to correct this shortfall. If I had any say in the new structure I would set up a pot of money specifically for the purposes I’ve described above. Funding applications would have to be competitive, of course, and I would argue for a panel with significant international representation to make the decisions. But for this to work the overall level of public sector research funding will have to increase dramatically from its current level, well below the OECD average. Ireland is currently running a huge Government surplus which is projected to continue growing until at least 2026. Only a small fraction of that surplus would be needed to build viable research communities not only in fundamental science but also across a much wider range of disciplines. Failure to invest now would be a wasted opportunity. There is currently no evidence of the required uplift in research spending despite the better-than-healthy state of Government finances.

Geometric and General Relativistic Techniques for Non-relativistic Quantum Systems

Posted in Maynooth, The Universe and Stuff with tags , , on February 19, 2025 by telescoper

The other day I mentioned the forthcoming graduation of a Maynooth PhD student. His name is Aonghus Hunter-McCabe and his main supervisor was Maynooth colleague Brian Dolan, and I just took over when Brian retired to see Aognhus through the latter stages. Anyway, asof yesterday, his thesis is available on arXiv (on hep-th) as well as on the Maynooth University Research Archive Library (MURA) here, so as it is all in the public domain I thought I would advertise it here, as I think it is very good indeed (though I would say that!) and also in case anyone out there is looking to employ a PDRA in a related area…

The abstract is:

This thesis explores the application of differential geometric and general relativistic techniques to deepen our understanding of quantum mechanical systems. We focus on three systems, employing these mathematical frameworks to uncover subtle features within each. First, we examine Unruh radiation in the context of an accelerated two-state atom, determining transition frequencies for a variety of accelerated trajectories via first-order perturbation theory. For harmonic motion of the atom in a vacuum, we derive transition rates with potential experimental realizations. Next, we investigate the quantum Hall effect in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field generated by a Wu-Yang monopole. The Atiyah-Singer index theorem constrains the degeneracy of the ground state, and the fractional quantum Hall effect is studied using the composite fermion model, where Dirac strings associated with the monopole field supply the statistical gauge field vortices. A unique, gapped ground state emerges, yielding fractions of the form ν=1/(2⁢k+1) for large particle numbers. Finally, we examine the AdS/CMT correspondence through a bulk fermionic field in an RN-AdS4 background (with a U(1) gauge field), dual to a boundary fermionic operator. Spherical and planar event horizon geometries are discussed, with the temperature of the RN black hole identified with that of the dual system on the boundary. By numerically solving for the spectral functions of the dual theory, for a spherical event horizon at zero temperature, we identify a shift in the Fermi surface from that which arises in the planar case. Preliminary evidence of a phase transition emerges upon examining these spectral functions, again for the spherical horizon, at non-zero temperature.

arXiv:2502.13009

The 250th Publication at the Open Journal of Astrophysics!

Posted in OJAp Papers, Open Access, The Universe and Stuff with tags , on February 18, 2025 by telescoper

It’s been a very busy day for various reasons so I’ll just mention that this morning I published the 250th paper at the Open Journal of Astrophysics. The lucky publication to garner this distinction is “Untangling Magellanic Streams” by Dennis Zaritsky (Steward Observatory), Vedant Chandra (Harvard), Charlie Conroy (Harvard), Ana Bonaca (Carnegie Observatories), Phillip A. Cargile (Harvard), and Rohan P. Naidu (MIT), all based in the USA. Here is the overlay

This will feature in the update on Saturday along with the other papers to be published this week, of which I expect several.

Two Weeks into Term…

Posted in Biographical, Education, Maynooth with tags , on February 17, 2025 by telescoper

I thought that this semester would have had an easier start than the last because I am teaching only one new module (and one familiar one) instead of the two new modules I had last semester. Unfortunately I wasn’t very well before the start of this one and have been struggling to recover so this term hasn’t been easy at all and I’ve had a lot to catch up. At least I’ve managed to stay on top of the lectures and lab sessions and everything so far has gone relatively smoothly. I have to admit though that only two weeks in, I already feel extremely tired. I went to bed at 9pm last night and slept straight through to my alarm at 7am. I’m getting old.

Last week we had our Examination Board meeting for Semester 1 in the Department of Physics. I also attended the Examination Board for the Department of Engineering because I was teaching an Engineering Mathematics module last semester. The students in both Departments should receive their provisional examination results this week, although there have been some gremlins in the campus systems which may lead to a delay in the marks being released.

Semester 2 is a bit more complex than Semester 1 because of a number of interruptions for holidays, etc. The first Monday of this term was actually a holiday, but I don’t actually have lectures on Mondays this term so that didn’t matter, nor will Monday 5th May which is also a holiday. I do, however, have a Particle Physics lecture on Fridays so will miss one on because of the national holiday on April 18th for Good Friday and will have to plan accordingly. The following week (Mon-Fri, starting on 21st April) is the Easter Break, and the week from 17th to 21st March is Study Week (starting with the St Patrick’s Day national holiday on Monday 17th). Although the number of teaching weeks is the same as Semester 1, they are spread out over a longer period with two gaps instead of one. One break is coming up about a month from now, and then another a month after that. This schedule is rather kind to those of us Oldies who tire easily!

Although I’ve kept up with the teaching preparation reasonably well, I have let other things slip. In particular, I have a graduate student getting ready to submit their PhD thesis ahead of an imminent deadline. I promised to read it and supply suggestions/corrections which I have yet to. That’s right at the top of my list for this week.

Talking of PhD students, my first ever official PhD student at Maynooth has already already passed his viva voce examination – about a year ago actually – but owing to bureaucratic delays he won’t graduate until this year, at a conferring ceremony in the March study week mentioned above.

Introducing the Clartiverse™

Posted in Education, mathematics with tags , , , , on February 16, 2025 by telescoper

The recent decision by Maynooth University to appoint a Ranking Strategy and Insights Officer in an attempt to raise the University’s position in university league tables has inspired me to create a new spinout company to provide a service for higher education institutions who want to improve their standing in rankings while avoiding the expense and complication of actually improving the institution or indeed while continuing to pursue policies that drive performance in the opposite direction.

I have decided to name my new company CLARTIVERT™ and the extensive suite of services we will provide is called the Clartiverse™.

The idea of CLARTIVERT™ is to produce, in return for a modest payment equivalent to the salary cost of a Ranking Strategy and Insights Officer, a bespoke league table that guarantees a specified position for any given institution. This can be either your own institution whose position you would like to raise or some competitor institution that you wish to lower. We then promote the league table thus constructed in the world’s media (who seem to like this sort of thing).

The idea behind this company is that the existing purveyors of rankings deliberately manufacture artificial “churn” in the league tables by changing their weighting model every year. Why not take this process to its logical conclusion? Our not-at-all dodgy software works by including so many metrics that an appropriate combination can always be chosen to propel any institution to the top (or bottom). We then produce We achieve all this by deplying a highly sophisticated branch of mathematics called Linear Algebra which we dress up in the fancy terms “Machine Learning” and  “Artificial Intelligence” to impress potential buyers.

To begin we will concentrate on research assessment. This is, of course, covered by existing league tables but our approach is radically different. We will desploy a vastly expanded set of metrics, many of which are currently unused. For example, on top of the usual bibliometric indicators like citation counts and numbers of published papers, we add number of authors, number of authors whose names start with a given letter of the alphabet, letter frequencies occuring in published texts, etc. We adopt a similar approach to other indicators, such as number of academic staff, number of PhD students, number of research managers, initial letters of names of people in these different categories, distribution of salaries for each, and so on.

As well as these quantities themselves we calculate mathematical functions of them, including polynomials, exponentials, logarithms and trigonometricfunctions; sine and cosine have proved very useful in early testing. All these indicators are combined in various ways: not only added, but also subtracted, multiplied, and/or divided until a weighted combination can be found that places your institution ahead of all the others.

In future we will roll out additional elements of the Clartiverse™ to cover other aspects of higher education including not only teaching and student satisfaction but also more important things such as commercialisation and financial impropriety.

P.S. The name Clartiver is derived from the word clart and is not to be confused with that of any other companies providing similar but less impressive services.