Arcadian Functor

occasional meanderings in physics' brave new world

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Marni D. Sheppeard

Wednesday, July 11, 2007

GRG18 Day 3

Kip Thorne gave an excellent public lecture yesterday evening, focussing on impressive new work in numerical relativity (such as black hole collision simulations) and the search for gravitational waves. First up this morning was an informative talk by Schneider on probing cosmology with gravitational lensing techniques, covering mass determinations, direct estimates of the Hubble parameter $H_0$, substructure, the collisionless nature of Dark Matter from Bullet cluster observations and cosmic shear. Over 200 strong lensing multiple-image systems are now known. Fifteen lens systems were used to determine $H_0 = 72$ from the time delay dependent on mass distribution and background geometry, in agreement with the WMAP result and other estimates. It appears that lensing has firmly established the existence of dark halo objects for the Milky Way, expected from the visible undercount of only 20 such objects. Other examples of substructure systems included B1422+231, MG2016+112, for which one can visually see the small perturbing satellite galaxy, and the classic recent result from CL0024+17. Cosmic shear looks at the lensing effect of the 3D distribution of matter on large scales. This was detected by four groups in 2000. With more extensive surveys, it promises to further constrain cosmological parameters in the near future.

Renate Loll gave a clear introduction into basic problems with quantum gravity, and then outlined recent successes of the Causal Dynamical Triangulation approach, in particular the recovery of four dimensions at large scales for the Lorentzian path integral with pathological configurations removed and two dimensionality at small scales. Then Francis Everitt entertained us with his wry sense of humor, as he outlined the success of the Gravity Probe B mission. I caught a better glimpse of the unofficial frame-dragging result, and it looked possibly a little higher than the GR prediction by about 10%, but this will probably disappear in the final error analysis. He didn't really comment on this except to tentatively offer a December deadline for final results.

As expected, the poster session involved a notable lack of interest in Category Theory, but the sandwiches were yummy and the company pleasant.

3 Comments:

Blogger nige said...

"As expected, the poster session involved a notable lack of interest in Category Theory, but the sandwiches were yummy and the company pleasant." - Kea

The second paragraph of the Wiki entry on "Category theory" begins:

Category theory has several faces known, not just to specialists, but to other mathematicians. "Generalized abstract nonsense" refers, not entirely affectionately, to its high level of abstraction, compared to more classical branches of mathematics.

(Emphasis added.)

I don't think that this explanation is that helpful.

The most abstract mathematical tools are generally the most valuable once they are widely understood and applied to appropriate problems. It's still early days for category theory.

I also keep getting confused between functors and functions (because the symbols used are similar and I'm more used to thinking about functions) when I read about it, not to mention the different morphisms.

One thing is to keep working on it and to try to find successful ways of applying Category Theory to explaining preon or some related theory, about how the different sets of quantum numbers (including weak and strong charges, masses, etc.) of fundamental particles of physics are really related to one another by morphisms.

You don't necessarily have to have a physical mechanism explaining how the morphism physically occurs. It can just be a mathematical representation of what happens, and what the relationships between different fundamental particles really are.

I think the key thing here is the relationship of leptons to quarks. Quark properties are only known through composites of 2 or 3 quarks, because quarks can't be isolated. The fact of universality, e.g., similarities between lepton and quark decay processes

muon -> electron + electron antineutrino + muon neutrino

for leptons and

neutron -> proton + electron + electron antineutrino

for quarks, hints that quarks and leptons are surprisingly similar, when ignoring the strong force. (My preliminary investigations on the relationship are here,
here and here.)

The problem is how much time it takes to apply new maths to solving these physical problems.

I like the fact that although you are a mathematician, you are free to go to physics conferences and study that stuff, at least as far as your time allows. The standard mathematical tools of particle physics, like Lie and Clifford algebras, aren't focussed at modelling the morphisms between different fundamental particles (transformations between leptons and quarks obviously haven't been observed yet, but they probably are possible at very high energy in certain situations). That would appear to be an ideal area to try to apply Category Theory to, because you have a table of particle properties and just have to fund the correct morphisms between them. That's very important for trying to understand how unification can occur at high energy, and could lead to quantitative, falsifiable predictions (the old unification ideas like supersymmetry are not even wrong). I hope to learn a lot more about Category Theory.

July 12, 2007 7:27 AM  
Blogger Matti Pitkanen said...

I highly recommend "Quantum Quandaries" of John Baez if you are interested about how category theoretic thinking might be applied in physics. In topological QFT fundamental arrows are time developments by cobordisms connecting initial and final states represented as manifolds and functors would map these cobordisms to S-matrices. Generalize this by bringing in metric dynamics and you have done it;-)!

July 12, 2007 1:53 PM  
Blogger Kea said...

Matti, with all due respect, I do not need references to the naive categorical physics literature. Busy here!

July 12, 2007 2:52 PM  

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