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:
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.
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!
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 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 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.
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/(2k+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.
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.
Time for another quick update of papers published at the Open Journal of Astrophysics. Since the last update we have published two new papers, which brings the number in Volume 8 (2025) up to 14 and the total so far published by OJAp up to 249.
Here are quick descriptions of the two papers concerned; you can click on the images of the overlays to make them larger should you wish to do so.
First one up is “AI-assisted super-resolution cosmological simulations IV: An emulator for deterministic realizations” by Xiaowen Zhang & Patrick Lachance (Carnegie Mellon), Ankita Dasgupta (Penn State), Rupert A. C. Croft & Tiziana Di Matteo (Carnegie Mellon), Yueying Ni (Harvard), Simeon Bird (UC Riverside) and Yin Li (Shenzhen University, China). It presents a method of achieving super-resolution to rapidly enhance low-resolution runs with statistically correct fine details to generate accurate simulations and mock observations for large galaxy surveys and was published on Monday 10th February 2025 in the folder marked Cosmology and NonGalactic Astrophysics.
You can find the officially accepted version of this paper on arXiv here.
The strategic case for this Chair revolves around broader developments in the area of astrophysics and cosmology at Maynooth. Currently there are two groups active in research in these areas, one in the former Department of Experimental Physics (which is largely focussed on astronomical instrumentation) and the other, in the former Department of Theoretical Physics, which is theoretical and computational. We want to promote closer collaboration between these research strands. The idea with the new position is that the holder will nucleate and lead a research programme in the area between these existing groups as well as getting involved in outreach and public engagement.
It is intended that the position to appeal not only to people undertaking observational programmes using ground-based facilities (e.g. those provided by ESO, which Ireland recently joined), or those exploiting data from space-based experiments, such as Euclid, as well as people working on multi-messenger astrophysics, gravitational waves, and so on.
P. S. For those of you reading this from outside Ireland the job includes a proper public service pension, a defined benefit scheme way better than the UK’s USS.
What better way to start a cold February morning than with a lovely image from Euclid? The picture above on the left shows an image of the galaxy NGC 6505 and on the right a closer view of the central portion that reveals a near perfect Einstein Ring. This phenomenon is caused by gravitational lensing and is quite a rare occurrence because it requires a perfect alignment between a background source, a concentration of mass that acts as a lens, and the observer (in this case the Euclid telescope):
This find is all the more extraordinary because it was made using observations made during Euclid’s commissioning phase when the telescope was not yet fully focussed. The first release of (a small sample) of full-quality data from Euclid – the so-called Q1 release – will actually be announced next month.
The published paper by O’Riordan et al is available here, from which I have taken this image showing the two relationship between the two images above:
There has already been quite a lot of media coverage of this discovery (even in Ireland). Here is the Press Release from the European Space Agency explaining the background and some comments from people involved in the work:
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Euclid blasted off on its six-year mission to explore the dark Universe on 1 July 2023. Before the spacecraft could begin its survey, the team of scientists and engineers on Earth had to make sure everything was working properly. During this early testing phase, in September 2023, Euclid sent some images back to Earth. They were deliberately out of focus, but in one fuzzy image Euclid Archive Scientist Bruno Altieri saw a hint of a very special phenomenon and decided to take a closer look.
“I look at the data from Euclid as it comes in,” explains Bruno. “Even from that first observation, I could see it, but after Euclid made more observations of the area, we could see a perfect Einstein ring. For me, with a lifelong interest in gravitational lensing, that was amazing.”
The Einstein Ring, an extremely rare phenomenon, turned out to be hiding in plain sight in a galaxy not far away. The galaxy, called NGC 6505, is around 590 million light-years from Earth, a stone’s throw away in cosmic terms. But this is the first time that the ring of light surrounding its centre is detected, thanks to Euclid’s high-resolution instruments.
The ring around the foreground galaxy is made up of light from a farther out bright galaxy. This background galaxy is 4.42 billion light-years away, and its light has been distorted by gravity on its way to us. The far-away galaxy hasn’t been observed before and doesn’t yet have a name.
“An Einstein ring is an example of strong gravitational lensing,” explains Conor O’Riordan, of the Max Planck Institute for Astrophysics, Germany, and lead author of the first scientific paper analysing the ring. “All strong lenses are special, because they’re so rare, and they’re incredibly useful scientifically. This one is particularly special, because it’s so close to Earth and the alignment makes it very beautiful.”
Albert Einstein’s general theory of relativity predicts that light will bend around objects in space, so that they focus the light like a giant lens. This gravitational lensing effect is bigger for more massive objects – galaxies and clusters of galaxies. It means we can sometimes see the light from distant galaxies that would otherwise be hidden.
If the alignment is just right, the light from the distant source galaxy bends to form a spectacular ring around the foreground object. These Einstein rings are a rich laboratory for scientists. Studying their gravitational effects can help us learn about the expansion of the Universe, detect the effects of invisible dark matter and dark energy, and investigate the background source whose light is bent by dark matter in between us and the source.
“I find it very intriguing that this ring was observed within a well-known galaxy, which was first discovered in 1884,” says Valeria Pettorino, ESA Euclid Project Scientist. “The galaxy has been known to astronomers for a very long time. And yet this ring was never observed before. This demonstrates how powerful Euclid is, finding new things even in places we thought we knew well. This discovery is very encouraging for the future of the Euclid mission and demonstrates its fantastic capabilities.
By exploring how the Universe has expanded and formed over its cosmic history, Euclid will reveal more about the role of gravity and the nature of dark energy and dark matter. The space telescope will map more than a third of the sky, observing billions of galaxies out to 10 billion light-years. It is expected to find around 100 000 strong lenses, but to find one that’s so spectacular – and so close to home – is astonishing. Until now, less than 1000 strong lenses were known, and even fewer were imaged at high resolution.
“Euclid is going to revolutionise the field, with all this data we’ve never had before,” adds Conor.
Although this Einstein ring is stunning, Euclid’s main job is searching for the more subtle effects of weak gravitational lensing, where background galaxies appear only mildly stretched or displaced. To detect this effect, scientists will need to analyse billions of galaxies. Euclid began its detailed survey of the sky on 14 February 2024 and is gradually creating the most extensive 3D map of the Universe yet. Such an amazing find, so early in its mission, means Euclid is on course to uncover many more hidden secrets.
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In the Dark 