chickenfeet

Occasionally I have this urge to play around with mathematics again.  Usually it's nothing too hard as it's over 35 years since I did mathematics seriously.  One thing I've been fooling around with is the representation of the roots of x^n-1=0 on the unit circle in the complex plane.  This is is one of those lovely bits of mathematics which is both simple and very profound.

For example, consider the relationship between the solutions of x^12-1=0 and x^3-1=0.

If one takes the representations of the latter on the complex plane, one can get to the former by either of two methods

(a) Make three rotations of π/6 in either direction

(b) Make a reflection in x=0 and then make a single rotation by π/6

Pretty cool really.

I suspect this post won't be nearly as coherent as I would like but it's an attempt to get some of my thoughts about current developments in physics down in writing. It may also come off as more cynical than I intend.

The last hundred or so years has brought incredible advances in our understanding of how the universe works on both a very large and a very small scale. It's truly one of the great intellectual achievements of the human race. What I'm wondering is whether it isn't running out of steam and whether that "running out of steam" is in part an institutional phenomenon.

It's a generally accepted notion that scientific "paradigms" develop a life of their own which keeps them alive long after they have ceased to be useful and cause perfectly well intentioned and capable people to spend a great deal of time and effort developing ever more elaborate models to explain away anomalies. The ingeniousness of some of the models used to reconcile observation with the notion that the sun and planets circled the Earth is a well known example.

So is particle physics at that point now? General relativity and quantum mechanics are both relativity simple (if non intuitive) ideas but a "Standard Model" full of kluges of many orders of magnitude and an ever increasing array of particles isn't. It starts to look like a mess. Maybe the introduction of strings and branes can overcome that but when the theories seem to rely on incredibly arbitrary choices of all sorts of parameters one wonders. When reputable scientists like Hawking put forward solutions to scale problems that involve "bolting" together contradictory theories at (conveniently) the point where it becomes impossible to do experimental work because of the energy levels involved it becomes even fishier.

I also wonder how much of this is driven by the "academic industrial complex". The early breakthroughs in modern physics were often made by grad students or even people (like Einstein) who couldn't get academic posts. Now we have multi billion dollar accelerator complexes staffed by vast scientific bureaucracies. Why should a scientific bureaucracy be any less inclined to promote it's own survival and growth as a primary end than any other bureaucracy?

Look at what's happening with building large scale experimental facilities. The really tough theoretical issues turn on what's happening at the Planck scale (10^19 GeV) but the latest and greatest collider (the Large Scale Hadron Collider at CERN) is only 7 times more powerful than current facilities and gets all the way to 10^3 GeV. I'm sure it's going to allow for some neat experiments but it's pretty clear that a linear progression of bigger and better colliders isn't going to bridge a 16 order of magnitude energy gap.

I'm not nearly clever enough to see how one gets off an apparent tramline like this one but I reckon I'm pretty much as good at spotting a bureaucratic structure that's outlived its usefulness as the next guy.