Archive for January, 2010

St Thomas

Posted in Jazz with tags , , , , on January 25, 2010 by telescoper

Walking past a Jazz club during my recent trip to Copenhagen – sadly, I didn’t have time to go in – I remembered the many times I’d heard the great Danish bass player Niels-Henning Ørsted Pedersen (known universally to Jazz fans as NHØP) playing there in the past. He died suddenly of a heart attack in 2005, at the age of 58, bringing to a close a career that had started when he was only 17. He was an incredible virtuoso, playing his unwieldy instrument in an astonishingly nimble fashion. As a result he was number one choice as accompanist whenever leading jazz artists toured his native Denmark where he remained most of his life, despite frequent invitations to join big name bands abroad. Although he appeared quite frequently on TV in the United Kingdom with Oscar Peterson in the 1970s, he never really became as widely known as he should have been given what a great musician he was.

I looked around on youtube to find an appropriate example of his playing, and found this superlative performance which I’d never seen before and which also offers a fine helping  of the great Sonny Rollins on tenor saxophone.  He will be 80 later this year and is still playing with the immense drive and imagination that he has shown since he began his career at the age of 11. He also wrote the tune, St Thomas, which has a strong caribbean feel to it, and which is based on a song from the Virgin Islands that his mother sang to him when he was a child. I’ve seen him play a number of times live, including at Ronnie Scott’s club in London and at the Royal Festival Hall, and wherever it was he always set the place on fire.

I hope the lilting calypso beat,  infectiously happy tune and, most of all, superb playing by every member of the band here will give you as warm a feeling as it did me when I first heard it. The other members of the quartet alongside Sonny Rollins are Kenny Drew on piano and Albert “Tootie” Heath on drums, but listen out for NHØP’s fantastic bass solo, starting around 4:41. Brilliant.


The Management

Posted in Finance, Science Politics with tags , , , , , on January 24, 2010 by telescoper

After my little trip to Denmark last week, it’s now time to settle into the routine of academic life. Teaching starts tomorrow, and I’m actually quite looking forward to it. I find teaching very rewarding, in a way that’s quite different from research, to the extent that I would hate to see further separation between the two in British universities. Call me old-fashioned.

Inevitably, though, it’s been research that’s been occupying my mind for the past few days. I’ve posted a couple of times recently about the ongoing review of the way astronomy and particle physics research are funded here in the United Kingdom (see here and here). The Science Minister, Lord Drayson, seems keen to find a way to stop research grants  being massacred by overruns elsewhere in the Science and Technology Facilities Council (STFC). His aim appears to be come up with a plan before the end of February to find a way of preventing the situation from getting any worse for science. No doubt the idea of a dedicated British Space Agency will also be thrown into pot, so that the bit of STFC’s current portfolio that deals with space things will probably be hived off elsewhere.

The major question that is occupying the minds of scientists – but perhaps not those of the bureaucrats – is whether the research grants currently dispensed by STFC will continue to be held by whatever STFC morphs into or whether they should go elsewhere, probably to EPSRC.  I sense a predisposition towards the former possibility among many of my colleagues. I recognize that the EPSRC route is not without its problems, but I fear that if we remain with STFC then not only is there a very strong probability that recent history will repeat itself but that the damage done by the current STFC structure will be irreparable.

Behind all this is the issue of why STFC is in such a mess in the first place. When it came into being in 2007, it was immediately saddled with an £80 million operating deficit. Why? There are two theories. One is that it was a mistake, resulting from inept STFC management. The other is that the creation of STFC presented various grey eminences that inhabit the superstructure of British science politics represented by RCUK  with an opportunity to slash expenditure on “useless” science (i.e. particle physics and astronomy) without having to go through the tedious rigmarole of public consultation. I don’t know which of these is the truth but, given the choice, I’d put my money on the latter.

Note the behaviour of STFC’s Chief Executive after the yawning gap was discovered in his organization’s finances. If it was a result of management incompetence then he should have been fired. If he was stitched up by RCUK then the only honorable thing to do for someone with the best interests of science at heart was to resign in protest. Neither of these things happened. This leads me to the interpretation that Professor Mason was a willing participant in the game, a  point of view that is supported by his performance at the Town Meeting in December 2007 where the STFC’s delivery plan was presented to an audience of scientists. The document containing the delivery plan is notable for its upbeat and self-congratulatory tone containing no hints of the financial catastrophe engulfing the organization. It was clearly designed to say exactly what the Chief Executive’s political masters wanted it to say. The gross dishonesty of this publication was revealed by Professor Mason’s presentation, wherein he told us scientists something rather closer to the truth, that STFC was facing financial oblivion. It was an appaling performance.

After a botched and panicky initial attempt to cut science projects, and a public dressing down by the House of Commons select committee, it took another two years for its latest Programmatic Review to emerge. Once again, though, the management of STFC put an absurdly glowing light on the wreckage of UK astronomy, nuclear and particle physics; calling it “Investing in the Future” and making light of the devastating cull of research grants and projects that it is proposing. The message that I glean from all this is that STFC’s problems stem from deliberate policy at a high level, probably at the Treasury, and carried out enthusiastically by a hierarchy of yes-men who will do whatever they are told regardless of what it means for science. Some of these creatures may have started out as scientists, but they’ve definitely gone native when lured into the Whitehall jungle.

Of course the public purse is limited. We have to decide how much to spend on different bits of science. Astronomy or particle physics (or any other discipline, for that matter) has to make its case. Somehow a balance must be struck between all the competing demands for cash. Maybe Britain does have too many astronomers. Or too many particle physicists. Who knows?  My point is: who decides? This kind of thing is too important to be settled behind closed doors by  individuals who lap up whatever their masters feed them like mother’s milk.

The STFC debacle  is just one manifestation of the rampant managerialism that is strangling British civil society. Gone are the days when scientists knew best about science, doctors knew best about medicine and teachers knew best about education. Now we’re all subservient to managers who think they know best about everything. Things are no better at EPSRC, an organization notorious  for its top-down structure, mania for meaningless initiatives, and wholehearted endorsement of the ill-considered impact agenda. What I am saying is that the Haldane principle is dead and buried.

While I was in Copenhagen last week attending the inauguration of the Discovery Center I was struck by the differences between how research is funded in Denmark and in the United Kingdom. This new initiative in particle physics and cosmology is funded as a rolling programme by the Danish National Research Foundation (Danmarks Grundforskningsfond). Way back in 1991, Denmark part-privatised its pension system and a large chunk of the resulting cash was invested in scientific research. The organization funds programmes across an entire range of disciplines (including arts and humanities)  for periods of10 years (or, more precisely, 5 years with an extension to 10 after satisfactory performance; most get extended). The primary criterion for funding these programmes is scientific excellence and the vast bulk of the funds goes to funding PhD students and postdoctoral researchers at Danish universities.

A representative of the foundation (whose name I have regrettably forgotten) spoke at the official inauguration of the Discovery Center to describe the parent organization’s philosophy. In a nutshell his message was: “You’re the scientists. You know about science. We don’t. We’re here to help you hire the best people, then get out of your way. Excellence is what we want to fund, wherever it lies. That’s our only agenda.” As it happens, two out of the nine programmes funded in the last round, including the Discovery Center, were in particle physics.

Of course I was jealous. I was also struck by how similar this organization sounds to the suggestion I made in a blog post before christmas. Of course Denmark is a much smaller country than Britain and it has  a very different economic structure. I’m not saying we could simply copy what the Danes have done without any modification. But the  real reason why such an organization could never get set up in Britain, is that The Management would never allow it…

Astronomy (and Particle Physics) Look-alikes, No. 10

Posted in Astronomy Lookalikes, Opera, The Universe and Stuff with tags , , , , on January 23, 2010 by telescoper

I was struck by the similarity between the design of the  ATLAS detector, at the Large Hadron Collider in CERN, and that of a recent production of Les Troyens by Hector Berlioz  in Valencia, Spain. How’s that for cultural impact?

Pity it had to be this Opera though. I hate it. Somebody should do a similar thing with the Magic Flute, which is actually all about particle physics

Cardiff City 1 Bristol City 0

Posted in Football with tags , , on January 19, 2010 by telescoper

It seems like ages since my last football blog post, but tonight has provided me with another opportunity. My local team, Cardiff City, was drawn against Bristol City in the 3rd round of the FA Cup. This promised to be quite a tasty fixture because Bristol City are the nearest English club to Cardiff and there’s consequently something of a local rivalry between the two. A couple of weekends ago I settled down to watch the match (away at Bristol) on the television, as it was shown live on welsh channel S4C. Unfortunately, the snow descended, the pitch froze and the game was postponed. When eventually played the result was 1-1 which meant, under FA Cup rules, a replay at the team initially drawn as the away side. Hence tonight’s fixture.

Cup ties like this often generate their own special kind of atmosphere, but it would be an exaggeration to say that this happened this evening. The attendance was pathetically small (only 6,731), partly because it was a very cold night and partly because the game was again being shown live on S4C. I left in good time from my house in P0ntcanna and found that it took me only 15 minutes from my front door until I was taking my seat in the stadium at Ninian Park. I’m certainly well placed for sporting venues where I live: football, rugby and cricket all within a quarter of an hour’s walk from my house!

The game started unimpressively. Both teams seemed lethargic, they were reluctant to press the ball in midfield, the quality of passing was poor and there were no clear-cut chances at either end. Then, after about 30 minutes, Cardiff’s teenage midfielder Aaron Wildig (who has just broken into the first team) picked up an injury and was replaced by Michael Chopra. This made a big difference to Cardiff’s attacking play and from then on they looked the more likely to score. Nevertheless it was an undistinguished first half which ended predictably at 0-0.

I warmed myself with a pint of ice-cold beer during the interval, and the game started again with Cardiff playing at a better tempo than in the first half. Although it was still a rather scrappy game, it started to get a bit stretched and both teams managed to string a few passes together and find a bit of space. Cardiff probed and made half-chances, but still didn’t look particularly likely to score primarily because they don’t have a particularly clinical finisher in the team, Chopra’s sporadic contributions notwithstanding. Thir best chances came when the Bristol City goalkeeper tried one of his speciality fumbles or botched clearance kicks. His name, by the way, was Gerken which sounded like gherkin on the public address; on a cold and frustrating evening this provided much-needed amusement.

Because the 4th round ties are being played this coming weekend, this game had to reach a conclusion tonight. I was beginning to worry we would have to sit through extra time and penalties, but at least I knew the ball would have to hit the back of the net at some point. I girded my teeth and gritted my loins for the long haul, wishing I had worn my thermal underwear too.

However, my fears turned out to be ill-founded. About 75 minutes into the game, Whittingham’s excellent pass found Chopra in space wide on Bristol City’s left inside the penalty area. The angle looked very tight, but he did brilliantly well to fire in a low shot that sped across the face of the goal. The danger seemed to have passed when it cannoned off the far post but, fortunately for Cardiff, the ball bounced off the woodwork into the path of the Bristol City defender Bradley Orr. I don’t know whether he was trying to put the ball out of danger wide of the post for a corner or just trying to get out of its way, but the result was that he smashed it with considerable force into the back of his own net. It was quite a comical moment, but they all count.

From then on, it was all Cardiff. They had made a good decision to keep going forward rather than try to sit on their lead. They don’t look the best defensive team I’ve seen, by a long way. There were two chances after that for Cardiff to score again, including one really good one for Chopra which he missed when it looked easier to score. Chopra is an enigmatic player, sometimes brilliant, sometimes ordinary.

Bristol tried to get back into it, but didn’t really threaten and betrayed their feelings a bit with a string of wild tackles, many of which went unpunished by an extremely indulgent referee. After a very long 4 minutes of extra time, the referee finally decided to blow his whistle and send Cardiff City into the next round. They will play Leicester City in the 4th Round on saturday. I might go, if I get back from my travels in time…

OK, so it wasn’t the greatest game I’ve ever seen but I still think it’s a shame there were so few supporters there. Along with many football clubs, Cardiff City has quite a few financial problems at the moment but I think if they showed a bit of imagination they could get the fans a bit more on their side. These cup games are not covered by season tickets, so why not cut the price to get more people in? Surely it’s better for the team to have a bigger crowd paying smaller prices, than one that’s depressingly empty?

You can read the BBC report on the match here.

There’s a certain slant of light

Posted in Poetry with tags , on January 18, 2010 by telescoper

Once again I haven’t had time to put together anything of much significance for the old blog today so, as usual when this happens, I’ll cheat by posting a poem. I picked this one for its wintry and appropriately melancholic theme; it is by the great American poet Emily Dickinson. I bought a collection of her poems in a very cheap edition in a bookshop in the States many years ago, but have never really managed to figure many of them out. I gather she features much more regularly in Eng. Lit. classes on the other side of the Atlantic than over here in Britain, and its probably my foreigner status that makes me find her poems so difficult.

This a very famous example of her work. At one level it expresses the unsettling effect that changes in light can have on the human psyche, but that’s just the start. The deeper meanings elude me, except that it is probably to do with the poet’s uncomfortable relationship with organized religion.  At least when she was young,  Emily Dickinson was a devotee of the Transcendentalist movement, which saw human experience, Nature and God as aspects of a transcendent unity. The  view expressed in this poem is certainly nothing like that. This is a fractured, lonely world where Nature and God are both alien and oppressive influences.

The strange use of punctuation and capitalization is also very typical.

There’s a certain Slant of light,
Winter Afternoons —
That oppresses, like the Heft
Of Cathedral Tunes —

Heavenly Hurt, it gives us —
We can find no scar,
But internal difference,
Where the Meanings, are —

None may teach it — Any —
‘Tis the Seal Despair —
An imperial affliction
Sent us of the Air —

When it comes, the Landscape listens —
Shadows — hold their breath —
When it goes, ’tis like the Distance
On the look of Death —

Herschel News

Posted in The Universe and Stuff with tags , , , , on January 17, 2010 by telescoper

I’ve been a bit slow to mention recent news about the European Space Agency‘s Herschel mission so this is by way of a quick update.

The first thing is to remind you that there was a big meeting of Herschel scientists in Madrid just before Christmas, which was attended by quite a number of Cardiff astronomers. It also happened to coincide with  less happy events. The purpose of this meeting was to share the preliminary results from the Science Demonstration Phase of Herschel’s operations. I did a quick post about some of the results, but didn’t have time to cover everything, which I still don’t. However, the complete set of presentations is now available online and I’d encourage you to sample some of the amazing results. Matt Griffin gave a nice overview of the key results at the RAS Ordinary Meeting just over a week ago.

You may recall that the Herschel telescope is fitted with three instruments:

  • The Photodetector Array Camera and Spectrometer (PACS)
  • The Spectral and Photometric Imaging REceiver (SPIRE)
  • The Heterodyne Instrument for the Far Infrared (HIFI)

The last of these instruments is basically a high-resolution spectrometer which, among other things will be great for detecting spectral lines from molecules, including good old H2O. In fact here’s a nice example of a water line seen in a comet

The problem is that HIFI has actually been switched off for quite a while – 160 days in fact – after a fault developed in its power supply. There is a backup power-supply, of course, but the engineers didn’t want to switch it over until they had figured out what had gone wrong, which took quite a while.  However, last Thursday, the HIFI instrument was switched back on and is now working fine. The full story can be found here. It was also covered quite a bit in the general media, including  the BBC.

While HIFI was offline, the calibration and verification of PACS and SPIRE went ahead at a good speed and now HIFI will have to catch up which has meant a bit of juggling around with schedules but, other than that, it’s all systems go…

Finally, I’ll just point out in case you didn’t know or have forgotten, that the Herschel Mission has its own wordpress blog, which is regularly updated  and is well worth checking out.

A Little Bit of Quantum

Posted in The Universe and Stuff with tags , , , , , , , , , , , on January 16, 2010 by telescoper

I’m trying to avoid getting too depressed by writing about the ongoing funding crisis for physics in the United Kingdom, so by way of a distraction I thought I’d post something about physics itself rather than the way it is being torn apart by short-sighted bureaucrats. A number of Cardiff physics students are currently looking forward (?) to their Quantum Mechanics examinations next week, so I thought I’d try to remind them of what fascinating subject it really is…

The development of the kinetic theory of gases in the latter part of the 19th Century represented the culmination of a mechanistic approach to Natural Philosophy that had begun with Isaac Newton two centuries earlier. So successful had this programme been by the turn of the 20th century that it was a fairly common view among scientists of the time that there was virtually nothing important left to be “discovered” in the realm of natural philosophy. All that remained were a few bits and pieces to be tidied up, but nothing could possibly shake the foundations of Newtonian mechanics.

But shake they certainly did. In 1905 the young Albert Einstein – surely the greatest physicist of the 20th century, if not of all time – single-handedly overthrew the underlying basis of Newton’s world with the introduction of his special theory of relativity. Although it took some time before this theory was tested experimentally and gained widespread acceptance, it blew an enormous hole in the mechanistic conception of the Universe by drastically changing the conceptual underpinning of Newtonian physics. Out were the “commonsense” notions of absolute space and absolute time, and in was a more complex “space-time” whose measurable aspects depended on the frame of reference of the observer.

Relativity, however, was only half the story. Another, perhaps even more radical shake-up was also in train at the same time. Although Einstein played an important role in this advance too, it led to a theory he was never comfortable with: quantum mechanics. A hundred years on, the full implications of this view of nature are still far from understood, so maybe Einstein was correct to be uneasy.

The birth of quantum mechanics partly arose from the developments of kinetic theory and statistical mechanics that I discussed briefly in a previous post. Inspired by such luminaries as James Clerk Maxwell and Ludwig Boltzmann, physicists had inexorably increased the range of phenomena that could be brought within the descriptive framework furnished by Newtonian mechanics and the new modes of statistical analysis that they had founded. Maxwell had also been responsible for another major development in theoretical physics: the unification of electricity and magnetism into a single system known as electromagnetism. Out of this mathematical tour de force came the realisation that light was a form of electromagnetic wave, an oscillation of electric and magnetic fields through apparently empty space.  Optical light forms just part of the possible spectrum of electromagnetic radiation, which ranges from very long wavelength radio waves at one end to extremely short wave gamma rays at the other.

With Maxwell’s theory in hand, it became possible to think about how atoms and molecules might exchange energy and reach equilibrium states not just with each other, but with light. Everyday experience that hot things tend to give off radiation and a number of experiments – by Wilhelm Wien and others – had shown that there were well-defined rules that determined what type of radiation (i.e. what wavelength) and how much of it were given off by a body held at a certain temperature. In a nutshell, hotter bodies give off more radiation (proportional to the fourth power of their temperature), and the peak wavelength is shorter for hotter bodies. At room temperature, bodies give off infra-red radiation, stars have surface temperatures measured in thousands of degrees so they give off predominantly optical and ultraviolet light. Our Universe is suffused with microwave radiation corresponding to just a few degrees above absolute zero.

The name given to a body in thermal equilibrium with a bath of radiation is a “black body”, not because it is black – the Sun is quite a good example of a black body and it is not black at all – but because it is simultaneously a perfect absorber and perfect emitter of radiation. In other words, it is a body which is in perfect thermal contact with the light it emits. Surely it would be straightforward to apply classical Maxwell-style statistical reasoning to a black body at some temperature?

It did indeed turn out to be straightforward, but the result was a catastrophe. One can see the nature of the disaster very straightforwardly by taking a simple idea from classical kinetic theory. In many circumstances there is a “rule of thumb” that applies to systems in thermal equilibrium. Roughly speaking, the idea is that energy becomes divided equally between every possible “degree of freedom” the system possesses. For example, if a box of gas consists of particles that can move in three dimensions then, on average, each component of the velocity of a particle will carry the same amount of kinetic energy. Molecules are able to rotate and vibrate as well as move about inside the box, and the equipartition rule can apply to these modes too.

Maxwell had shown that light was essentially a kind of vibration, so it appeared obvious that what one had to do was to assign the same amount of energy to each possible vibrational degree of freedom of the ambient electromagnetic field. Lord Rayleigh and Sir James Jeans did this calculation and found that the amount of energy radiated by a black body as a function of wavelength should vary proportionally to the temperature T and to inversely as the fourth power of the wavelength λ, as shown in the diagram for an example temperature of 5000K:

Even without doing any detailed experiments it is clear that this result just has to be nonsense. The Rayleigh-Jeans law predicts that even very cold bodies should produce infinite amounts of radiation at infinitely short wavelengths, i.e. in the ultraviolet. It also predicts that the total amount of radiation – the area under the curve in the above figure – is infinite. Even a very cold body should emit infinitely intense electromagnetic radiation. Infinity is bad.

Experiments show that the Rayleigh-Jeans law does work at very long wavelengths but in reality the radiation reaches a maximum (at a wavelength that depends on the temperature) and then declines at short wavelengths, as shown also in the above Figure. Clearly something is very badly wrong with the reasoning here, although it works so well for atoms and molecules.

It wouldn’t be accurate to say that physicists all stopped in their tracks because of this difficulty. It is amazing the extent to which people are able to carry on despite the presence of obvious flaws in their theory. It takes a great mind to realise when everyone else is on the wrong track, and a considerable time for revolutionary changes to become accepted. In the meantime, the run-of-the-mill scientist tends to carry on regardless.

The resolution of this particular fundamental conundrum is accredited to Karl Ernst Ludwig “Max” Planck (right), who was born in 1858. He was the son of a law professor, and himself went to university at Berlin and Munich, receiving his doctorate in 1880. He became professor at Kiel in 1885, and moved to Berlin in 1888. In 1930 he became president of the Kaiser Wilhelm Institute, but resigned in 1937 in protest at the behaviour of the Nazis towards Jewish scientists. His life was blighted by family tragedies: his second son died in the First World War; both daughters died in childbirth; and his first son was executed in 1944 for his part in a plot to assassinate Adolf Hitler. After the Second World War the institute was named the Max Planck Institute, and Planck was reappointed director. He died in 1947; by then such a famous scientist that his likeness appeared on the two Deutschmark coin issued in 1958.

Planck had taken some ideas from Boltzmann’s work but applied them in a radically new way. The essence of his reasoning was that the ultraviolet catastrophe basically arises because Maxwell’s electromagnetic field is a continuous thing and, as such, appears to have an infinite variety of ways in which it can absorb energy. When you are allowed to store energy in whatever way you like in all these modes, and add them all together you get an infinite power output. But what if there was some fundamental limitation in the way that an atom could exchange energy with the radiation field? If such a transfer can only occur in discrete lumps or quanta – rather like “atoms” of radiation – then one could eliminate the ultraviolet catastrophe at a stroke. Planck’s genius was to realize this, and the formula he proposed contains a constant that still bears his name. The energy of a light quantum E is related to its frequency ν via E=hν, where h is Planck’s constant, one of the fundamental constants that occur throughout theoretical physics.

Boltzmann had shown that if a system possesses a  discrete energy state labelled by j separated by energy Ej then at a given temperature the likely relative occupation of the two states is determined by a “Boltzmann factor” of the form:

n_{j} \propto \exp\left(-\frac{E_{j}}{k_BT}\right),

so that the higher energy state is exponentially less probable than the lower energy state if the energy difference is much larger than the typical thermal energy kB T ; the quantity kB is Boltzmann’s constant, another fundamental constant. On the other hand, if the states are very close in energy compared to the thermal level then they will be roughly equally populated in accordance with the “equipartition” idea I mentioned above.

The trouble with the classical treatment of an electromagnetic field is that it makes it too easy for the field to store infinite energy in short wavelength oscillations: it can put  a little bit of energy in each of a lot of modes in an unlimited way. Planck realised that his idea would mean ultra-violet radiation could only be emitted in very energetic quanta, rather than in lots of little bits. Building on Boltzmann’s reasoning, he deduced the probability of exciting a quantum with very high energy is exponentially suppressed. This in turn leads to an exponential cut-off in the black-body curve at short wavelengths. Triumphantly, he was able to calculate the exact form of the black-body curve expected in his theory: it matches the Rayleigh-Jeans form at long wavelengths, but turns over and decreases at short wavelengths just as the measurements require. The theoretical Planck curve matches measurements perfectly over the entire range of wavelengths that experiments have been able to probe.

Curiously perhaps, Planck stopped short of the modern interpretation of this: that light (and other electromagnetic radiation) is composed of particles which we now call photons. He was still wedded to Maxwell’s description of light as a wave phenomenon, so he preferred to think of the exchange of energy as being quantised rather than the radiation itself. Einstein’s work on the photoelectric effect in 1905 further vindicated Planck, but also demonstrated that light travelled in packets. After Planck’s work, and the development of the quantum theory of the atom pioneered by Niels Bohr, quantum theory really began to take hold of the physics community and eventually it became acceptable to conceive of not just photons but all matter as being part particle and part wave. Photons are examples of a kind of particle known as a boson, and the atomic constituents such as electrons and protons are fermions. (This classification arises from their spin: bosons have spin which is an integer multiple of Planck’s constant, whereas fermions have half-integral spin.)

You might have expected that the radical step made by Planck would immediately have led to a drastic overhaul of the system of thermodynamics put in place in the preceding half-a-century, but you would be wrong. In many ways the realization that discrete energy levels were involved in the microscopic description of matter if anything made thermodynamics easier to understand and apply. Statistical reasoning is usually most difficult when the space of possibilities is complicated. In quantum theory one always deals fundamentally with a discrete space of possible outcomes. Counting discrete things is not always easy, but it’s usually easier than counting continuous things. Even when they’re infinite.

Much of modern physics research lies in the arena of condensed matter physics, which deals with the properties of solids and gases, often at the very low temperatures where quantum effects become important. The statistical thermodynamics of these systems is based on a very slight modification of Boltzmann’s result:

n_{j} \propto \left[\exp\left(\frac{E_{j}}{k_BT}\right)\pm 1\right]^{-1},

which gives the equilibrium occupation of states at an energy level Ej; the difference between bosons and fermions manifests itself as the sign in the denominator. Fermions take the upper “plus” sign, and the resulting statistical framework is based on the so-called Fermi-Dirac distribution; bosons have the minus sign and obey Bose-Einstein statistics. This modification of the classical theory of Maxwell and Boltzmann is simple, but leads to a range of fascinating phenomena, from neutron stars to superconductivity.

Moreover, the nature the ultraviolet catastrophe for black-body radiation at the start of the 20th Century perhaps also holds lessons for modern physics. One of the fundamental problems we have in theoretical cosmology is how to calculate the energy density of the vacuum using quantum field theory. This is a more complicated thing to do than working out the energy in an electromagnetic field, but the net result is a catastrophe of the same sort. All straightforward ways of computing this quantity produce a divergent answer unless a high-energy cut off is introduced. Although cosmological observations of the accelerating universe suggest that vacuum energy is there, its actual energy density is way too small for any plausible cutoff.

So there we are. A hundred years on, we have another nasty infinity. It’s a fundamental problem, but its answer will probably open up a new way of understanding the Universe.