Friday, November 30, 2007

In Our Time: Symmetry, Moonshine, String Theory and E8

It's good to stretch one's mind on the way in to the office. It makes modeling relationships between arcane procedural descriptions and CPT codes seem pleasantly relaxing.

Today's stretch is courtesy of an In Our Time podcast [1] on symmetry. The professors do discuss Galois, but so far they've missed Emily Noether. Excellent anyway.

I'm not quite done, but I've learned about Galois, a bit of history, and, for the first time, I have a vague understanding of what Group Theory is about and it's relationship to geometry and topology. Not bad for free.

Towards the end of the podcast they talk about mysterious relationships between one "monstrous" element in an "atlas" of "Groups", where a Group is a collection of mathematical objects with shared symmetric transformations. One of them, the "monster", is described as having a mysterious relationship with mathematical physics.

Cue Wikipedia:
Monstrous moonshine - Wikipedia, the free encyclopedia

Specifically, Conway and Norton, following an initial observation by John McKay, found that the Fourier expansion of j(τ) ... could be expressed in terms of linear combinations of the dimensions of the irreducible representations of M ...

... lying behind monstrous moonshine is a certain string theory having the Monster group as symmetries; the conjectures made by Conway and Norton were proven by Richard Ewen Borcherds in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac-Moody superalgebras.

... Igor Frenkel, James Lepowsky and Arne Meurman explicitly constructed this representation using vertex operators in conformal field theory describing bosonic string theory compactified on a 24-dimensional torus generated by the Leech lattice and orbifolded by a reflection. The resulting module is called the Monster module...

Well, ok, so I didn't exactly follow all of that. Impressive at a cocktail party no doubt, but we don't do that sort of thing.

All quite exciting in a geeky sort of way, except bosonic string theory seems to have been a bit of a dead end. Again, from Wikipedia (emphases mine, I thought this was a lovely explanation btw_:
Bosonic String theory is the original version of string theory, developed in the late 1960s. Although it has many attractive features, it has a pair of features that render it unattractive as a physical model. Firstly it predicts only the existence of bosons whereas we know many physical particles are fermions. Secondly, it predicts the existence of a particle whose mass is imaginary implying that it travels faster than light. The existence of such a particle, commonly known as a tachyon, would conflict with much of what we know about physics, and such particles have never been observed.

Another feature of bosonic string theory is that in general the theory displays inconsistencies due to the conformal anomaly. In a spacetime of 26 dimensions, however, with 25 dimensions of space and one of time, the inconsistencies cancel. Another way to look at this is that in general bosonic string theory predicts unphysical particle states called 'ghosts'. In 26 dimensions the no-ghost theorem predicts that these ghost states have no interaction whatsoever with any other states and hence that they can be ignored leaving a consistent theory. So bosonic string theory predicts a 26 dimensional spacetime. This high dimensionality isn't a problem for bosonic string theory because it can be formulated in such a way that along the 22 excess dimensions, spacetime is folded up to form a small torus. This would leave only the familiar four dimensions of spacetime visible.

In the early 1970s, supersymmetry was discovered in the context of string theory, and a new version of string theory called superstring theory (supersymmetric string theory) became the real focus. Nevertheless, bosonic string theory remains a very useful "toy model" to understand many general features of perturbative string theory, and string theory textbooks usually start with the bosonic string...
So why does Time always get only one dimension? Space seems awfully greedy.

So the monster/physics connection didn't quite hold up, but one supposes Supersymmetry might have some familial connection to the Monster.

Or perhaps there's another step up?
... E8 is the symmetries of a geometric object that is 57-dimensional. E8 itself is 248-dimensional...

...Hermann Nicolai, Director of the Albert Einstein Institute in Potsdam, Germany. "While mathematicians have known for a long time about the beauty and the uniqueness of E8, we physicists have come to appreciate its exceptional role only more recently - yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner."
Or a step down?
... Since 1997, physicists have proposed countless variations on Maldacena's theme, all of which interpret a string as a swarm of particles living in a small number of dimensions. Perhaps the easiest case to visualize is when that number is two. In such a scenario, anything that takes place in your many-dimensional, stringy universe has a sort of shadow representation in terms of particles moving on that universe's 'sphere at infinity.' This esoteric-sounding concept is actually similar to the familiar celestial sphere of the night sky as seen from Earth: It's the two-dimensional surface spanning all possible directions one can point to infinitely far in space..
I probably need to stop now. Work beckons, and my furhter inquires on relationship between E8 and the Monster Group started running into "alternative physics" posts. I have enough trouble with "conventional" physics, thanks.

Ok, one last comment. In the nice, sane, quiet world of the humble Higgs (God) particle CV tells us:
... if you impose upon our relativistic, complex, quantum-mechanical wavefunctions the requirement that they be invariant under these U(1) transformations, then you get electromagnetism. Conservation of electric charge. A massless photon. QED - quantum electrodynamics, in all its 12-digit precision glory. Electromagnetism is a simple consequence of the U(1) symmetry of any wavefunction....
It's all relative (sorry). QED seems perfectly pedestrian now.

[1] From a prior post:

Melvyn Bragg's BBC show, In Our Time, has begun a new season. I'm a fan.

The bad news is that the BBC is sticking with its execrable latest-episode-only download policy. So if you want to listen to the superb Opium War episode on your MP3 player you need to either use Audio Hijack Pro to capture the RealAudio stream or (if you know me) ask me for a DVD with the entire series [1]. Incidentally, this is a good time to write a quick email to set IOT free.

The good news is there's a new page that makes it easy to subscribe to a feed. I used to subscribe via iTunes, but if I went a week without using iTunes I missed the show. Now I subscribe via iTunes and Bloglines; I use Bloglines at least daily so it's easy for me to save the MP3 and email it to myself.

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