E8

(Chastened by the stern but just admonishments of our Canadian colleague, regarding the tinsel and pinchbeck allure of Web celebrity based merely upon citizens’ obsessions with CUTE HEDGEHOGS and PLAYFUL PENGUINS,

http://pyesetz.livejournal.com/

we return now to the straight and narrow of Cantorian realism, leaving aside all reference to our animal friends, not excluding the lowly hedgehog (who knows but One Big Thing) or the HUMBLE WOODCHUCK.   You will find no mention of the HUMBLE WOODCHUCK in anything that follows;  and he that were so foolish as to search on the string

!! => “humble woodchuck” <= !!

in hopes of googling-up this essay, would surely search in vain.)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

E8

Before we proceed further, so that we have some acquaintances in common, let us consider our new friend,  E8.  This way, when I have occasion to refer to him again in future, we shall all give a knowing nod.

http://en.wikipedia.org/wiki/E%E2%82%88

Now, E8 is, frankly, not such a big deal:  it’s just a single, exceptional simple Lie group (albeit not an exceptionally simple one), even if it does turn out to be the symmetry group of string theory, and thus of all the world.  For actually, all our major experiences of life – our loves, our sorrows, our proofs of the Riemann Hypothesis --  are carried out  quite independently of string theory (be that worthy enterprise  well-founded or no).  No need to get all misty and mystical about it, though journalists and publishers love to do so because it moves more newsprint and books.  (The “God Particle”, egad; it’s just a frigging Higgs boson.)  You could with as much reason wax mushtical over the mere numbers zero and one – the Nihil and the Ens, if you wish – who heroically alone shoulder all the burden of binary description, which encompasses so much.

            Nevertheless, in its own small way, E8 does have a certain fascination for us, the fascination of a small thing, perfected past admiration, as by a master craftsman, with all eternity wherein to work, and whose very existence seems a paradox, like a spinning top.  After all, word of this fait divers from the normally cleidoic mathematical world  did manage to make it into the print editions of both Le Monde and the New York Times,

http://www.nytimes.com/2007/03/20/science/20math.html?ref=science

thus elbowing out whatever might otherwise have occupied those column inches, be it an account of an auto accident, or a lost cat.  And this, despite the utter incapacity of the journalists to give us the least idea of what has been actually discovered.  It is as though they had sent a correspondent to cover a major speech, and then reported, “We could not make out a word he said.”  As the Times reporter put it (beneath the swooning headline  “The Scientific Promise of Perfect Symmetry”):

Eighteen mathematicians spent four years and 77 hours of supercomputer computation to describe this structure, with the results unveiled Monday at a talk at the Massachusetts Institute of Technology.

But it still is not easy to describe the description, at least not in words.

“It’s pretty abstract,” conceded Jeffrey D. Adams …

“You can’t really picture it,” Brian Conrey, executive director of the American Institute of Mathematics, said of E8--

and then offered his own endearingly goofy stab at a depiction: “It’s some sort of curvy, torus type of thing.”  (A torus, for those who are not aware of this, is a donut with a college education.)

Most previous symmetries have been simple enough in themselves -- like, translation (that is, just boogying along in a straight line), which is as simple as it gets -- but startling in physical consequences.  Thus:  symmetry in translation along the time axis -- and  voilà , Conservation of Energy!  Rotational symmetry -- conservation of angular momentum!  Likewise, to say that “the gauge symmetry of the electro-magnetic field is U(1)” is again to invoke the familiar wagon-wheel.  Even the PCT intricacies (Parity, Charge, Time) are based simply on the shuffling of easily visualisable bivalences (left vs. right, positive vs. negative, sooner vs. later);  the implications of these for physics, however, are beyond the reach of most of us.  With E8 we arrive at a symmetry group that is itself well-nigh incomprehensible -- certainly not surveyable without very extensive practice and training.  As for its ultimate physical implications -- anybody’s guess.   But whatever its ultimate fate in physics, the fact is, E8 is already there, in just the same way that the symmetry of a rotating wheel is there, and would still be there even if our cosmos happened not to contain anything physical that actually rotates.  We discovered E8;  we didn’t invent it.

[Update 4 Jan 2011:  The above provoked a playful and entertaining meditation by our Canadian Colleague:
http://pyesetz.livejournal.com/103055.html
Enjoy.]

[continued here]