Some days, when I look at the world, I think Nature's purpose was for us to breed a lot of dogs, so that when we all disappear, fighting each other for meagre food and water sources, the dogs will inherit the Earth.
The gallant kneemo and I have been thinking about (and writing about) this very interesting work of Brown. As Louise has discovered, equation 7.31 on page 96 is indeed very reminiscent of the Veneziano amplitude. It might also be worth noting that the other integrals of this type for n points are precisely the n point functions. Who would have guessed? So much physics just falling out of a bunch of associahedra.
Of course this is what we expect for gluons. Modular operads and other such goodies can be used to deal with higher loops, and higher operads are brought in to investigate other Standard Model particles. It's still early days for M-theory, but that vacuum seems to have disappeared in a puff of smoke. Perhaps a few more people will become interested in it soon, because one begins to suspect that the String picture, built with manifolds upon manifolds, is not actually relevant to real physics at all. Oh, dear.
The Rios paper has aob = bob which is clearly a typo for aob=boa, but when you're an idiot like me this can leave you stumped for a minute.
ReplyDeleteaob = 0.5(ab+ba) is identical to the inner product definition in Geometric (Cliford) algebra. I wonder. But the underlying GA/CA satisfies (ab)c = a(bc) so the Jordan algebra is trivial. But if you add derivatives, then you can get this sort of thing? My work (so far) avoids derivatives.
When I see people deriving Hilbert space on the basis of Banach space, (or a Hilbert space built from observables), I see that the vacuum, and the splitting into bras and kets, is melting away. Which is exactly what I'm doing with operators.
Hi Carl
ReplyDeleteLet a,b,c be NxN Hermitian matrices over C.
Expanding a o (b o c) in terms of ordinary matrix multiplication gives:
1/4[A(BC)+A(CB)+(BC)A+(CB)A].
Expanding (a o b) o c yields:
1/4[(AB)C+(BA)C+C(AB)+C(BA)].
Given that NxN matrices over C are associative under ordinary matrix multiplication, we can assert that:
A(BC)=(AB)C and (CB)A=C(BA).
To show the equivalence of the remaining terms would require not only associativity, but commutivity under matrix multiplication -- which we don't have in general.
It's in this sense that a Jordan algebra is not quite trivial. For the exceptional Jordan algebra over the octonions (3x3), matrix multiplication fails even to be associative. This is why it is not a "special" Jordan algebra, which is a Jordan algebra isomorphic to a subalgebra of an associative algebra.
Carl
ReplyDeleteI forgot to mention that I've been reading your super-cool book dmaa.pdf. On page 89 you define a snuark as a "group of primitive idempotents that travel together" in the same direction. I think I missed that definition over a PF and have been stumped on snuarks till now.
On page 88 you calculate the square of the sum of two distinct primitive idempotents in the same complete set. You observe that the result is not a primitive idempotent, so is not an "elementary particle". As the result is still a rank-two idempotent you interpret the combination as two elementary particles at the same point in space.
Taking the projective space perspective, an NxN complex primitive idempotent corresponds to a point in CP^{n-1}, while a rank-two idempotent corresponds to a line in CP^{n-1}. To be more concrete, consider 4x4 idempotents over C, where primitive idempotents correspond to points of CP^3. A rank-two idempotent of the form (6.25) would correspond to a CP^1 projective line in CP^3. Similarly, a generalized version of (6.25) for three primitive idempotents would furnish a CP^2 in CP^3. Via this interpretation, it should be possible to describe snuarks in Witten's twistor string theory with target space CP^3.
p.s. In my last post it seems my little a,b,c's grew into big A,B,C's. Just sub in A o (B o C) and (A o B) o C for the analagous little letters. ;)
Gee, thanks kneemo. Hi Carl. I've just finished writing up a talk for next week. Can't wait to get out of Sydney.
ReplyDeleteThanks for the clarification, Keenmo. And thanks for the compliment. It's a lot easier to write things that get read.
ReplyDeleteTo improve the cool factor, I've put my cover art up on the web here: cover art
Hi Kea and Carl
ReplyDeleteBook cover, yes! How about adding some idempotents? Here's what I came up with: P^2=P
That's a great cover. Maybe too good. I'm still trying to work out the various consequences. Right now there are about three people who regularly download the book and that's already pushing on too many.
ReplyDeleteAnd what is an "anion" anyway?
Hi Carl
ReplyDeleteAn "anyon" as coined by Frank Wilczek is a particle with spin that can take "any" real value, rather than integer or half-integer. Check out this Anyon Primer for more info. Anyons play a central role in the Fractional Quantum Hall Effect, as described in Phys. Rev. Focus.
Kea is looking at the moduli spaces of anyons and tiling them with associahedra. By doing this one can study anyons with operads and spiffy category theory tools.
As far as this relates to your work, I think a snuark may be equivalent to a collection of anyons.
Before I go, a little note on the Jordan product: it's used to ensure that multiplication of two observables produces an observable. This comes in handy, say, when we (Jordan) multiply an arbitrary hermitian matrix and a primitive idempotent. We at least get a hermitian matrix back. In the cooler eigenmatrix case, we A o P = cP. So that our resulting hermitian matrix is a mere scalar multiple of a primitive idempotent.