Monday, August 16, 2010


Ahh summer – those long lazy days on holiday by the sea…with the peace… and quiet.. and… well boredom, which can be relieved by finding a good read in the local library. Now call me weird, but when I drop into the library in those sleepy far off towns I usually pass by the physics shelf in the science section for a bit of a squiz. The shelf’s contents are almost always desultory and mercifully I’m usually forced to pass those long afternoons with some pulp fiction.

In contrast, imagine my excitement to discover how absolutely well stocked with fabulous physics books is the DESY library. And those long shelves full of (actually not) dusty old journals like Soviet Physics JETP (journal of experimental and theoretical physics). Now to come across even such a journal title, as I did when I was an undergraduate in the still cold war world of the 80’s, was exciting enough. It seemed like a little bit of a communist 5th column in our bourgeois decadent science library. And that journal is jammed pack full of very intelligent theoretical work – no doubt drummed out of proletarian scientists as they were forced to think away in some Siberian institute that was bound to be a state secret.

I based a lot of my research on papers from that journal – in the 80’s it felt like I was the only person in the West who even knew of their existence! Not true of course. They had obviously been translated into english quite sometime before and I was being naieve – but its nice to daydream sometimes. In any case during the process of my research I ran into a mathematical problem (see my previous blog entry) and I found, tucked away in a little corner of one of those JETP papers, a delicious reference to a soviet maths book – that hadn’t been translated and in fact wasn’t in our library’s catalogue. Or indeed in any library catalogue in the country. Eventually my brave inter-library loans librarian established that it could be obtained from the Leningrad library itself!

Obtained? Obtained!?? Would such a thing be possible? Even if the Apparatchiks allowed it out of the country, could it make it past the Iron Curtain? Even then, wouldn’t some US blockade (like the one around Cuba) stop my precious Leningrad maths book in its tracks? Somehow my intrepid book found its way into my eager hands some satisfyingly long 6 weeks later. And it was satisfyingly jammed pack full of mathematical identities, none of which, sadly, helped me to solve my maths problem. So I sent my brave little book back from whence it came.

To its doom. Horrifyingly, just a few months later the Leningrad National Academy of Sciences library burnt to the ground – destroying some 400,000 precious books and damaging millions of others. As this was really before the whole OCR, electronic library thing, perhaps many were lost forever – a veritable chernobyl of the soviet library world. The only thin silver lining was that it wasn’t the gorgeous Kunstkammer – built by Peter the Great to house the science collection and the original home of the Science Library – that was destroyed, but a faceless brown cubish building, which cant have been as nice to work in in any case.

These days the internet has brought us the means to better preserve our collective science knowledge. Many physics text books are now available online and the academic papers likewise – though more work needs to be done to digitize the older papers in journals like the JETP. And its still the case that this information has to be made free. I mean it not now in an iron curtain sense but a financial one. Subscription rates to academic journals are prohibitively high meaning that you have to be an employee of a large university or laboratory to get access to it. The international science community however thrives on the free exchange of information and to that end set up the freely accessible preprint server ArXiv. Here you will find academic papers in electronic form before they get locked away in the journals. There are high hopes for ArXiv

“Its existence was one of the precipitating factors that led to the current revolution in scientific publishing, known as the open access movement, with the possibility of the eventual disappearance of traditional scientific journals.”

One advantage of the journals are that the work that appears in them is peer (or if you like) quality reviewed. However mechanisms exist for the endorsement of ArXiv material – in general the quality of the work is very high, and its a great resource.

To finish with an adage then, information wants to be free and Science thrives on openness, cooperativeness and the absence of a profit motive!

Physics and Mathematics

When I was an undergraduate I enjoyed both my mathematics and physics subjects. Pure maths was an exercise in precision, when in proving a simple enough looking theorem, you should be concerned about the minutest detail. Physics I enjoyed because it was mysterious, it was about the world and it involved maths. That seemed like a compelling combination – that a whole class of physical phenomena could be encapsulated in a single mathematical expression. However, there is always the question lurking in the background, what exactly is the relationship of physics to maths?

There are of course many different attitudes and indeed deeply philosophical attitudes. A mathematician might think something like this:
In mathematics, the pure notions of numbers and other structures do not need physics to exist or explain or even justify them. But the surprising thing is that often some newly discovered abstract formulation in mathematics turns out, years later, to describe physical phenomena which we hadn’t known about earlier. The only conclusion I can bring myself to is that mathematics is not just a tool of physics; it must be much, much more.
Conversely, mathematics alone is not enough to determine a physical system. For instance in studying magnetism historically, the English physicist Michael Faraday invisioned lines of forces in an invisible medium stretching between, say, your fridge magnet and your fridge as you bring the former towards the latter. Continental physicsts like Laplace and Poisson envisaged centres of force acting over a distance across empty space. James Maxwell showed that the two different visions were identical mathematically. However physically they were completely different systems giving rise to long debates and experiments about the existence of a universal aether which may transmit Faraday’s lines of force

One concept which occurs often in physics – and which gives rise to interesting mathematical expressions is that of symmetry. For instance the image above is caused by focusing light onto a circular hole and resulting in a centrally symmetric diffraction pattern. The mathematical function which describes how the brightness of the pattern varies is called after its creator – the Airy function.

These Airy functions (written Ai(z)) occur whenever we try to describe any physical system with the same type of symmetry. And indeed I have an ulterior purpose in making such a long-winded introduction – I study a particular physical system with such a symmetry – in fact an interaction between particles embedded in a strong centrally symmetric field. So naturally in my study I obtain Airy’s functions – and not just one or two, but an awkward combination of Airy’s and other functions.

To be frank this “awkward combination” has been driving me nuts for quite sometime – with me wishing that my undergraduate maths lectures hadn’t occurred so long ago. You didn’t think I was going to spare you the gory details did you?

There are two ways of simplifying this little bit of maths. Firstly, the squiggly, almost vertical line on the left – the integration – can be done analytically. That is we can perhaps find an algebraic expression exa
ctly equivalent to the above, but without the integration. This is usually the preferred result – to be able to see a physical system described in assimple mathematics as possible is not only asthetically pleasing, but leads to deep insights about the physical system in question. For instance the formula describing the entropy S (or amount of disorder) of a gas, developed by physicist Lugwig Boltzmann, was considered so important that it is engraved on his grave.

In the second method of simplification, my problem integration could be done numerically by using a computer to plot the functions to the right of the integration sign and then calculating the area under the plot. This is the “brute force” method and not very satisfactory if you expect a physical system to be written simply in the language of mathematics. On the other hand, since all simple functions like the Airy function are themselves written in terms of integrations over other functions, then it may be the case that I’m dealing with a new type of function that deserves to be “fundamental” in some sense.

This goes to a deeper question – what is more important, the abstract formulae that describe a physical system, or a the real numbers that arise from calculating such formulae and which are compared with real experiments on the system in question? I suspect the answer is that both are equally important and it is the interplay between the numbers and the formulae – the experiment and the theory – that leads to a deeper understanding.