M Theory Lesson 72
C. S. Peirce is probably best known today for his work on semiotics, based on his notion of triple for which the symbols were but one component. Let's just recall today his simple representation of Boolean complement, indicated by a circle marking the boundary between A and not A. In Boolean logic it follows that a double circle should act as an identity operation. This looks just like the holographic cylinder, which we know is the identity in a category of cobordisms! 
Sticking to disc operads gives a large class of algebras (and connections to many, many things) but it may be necessary to consider also intersecting discs. Each Jones diagram, shown on the left below, includes internal lines that may not intersect, but when we overlap these big discs we make a choice as to their vertical ordering, and a line from one disc may cross over a line from another, thereby introducing the possibility of braiding for strands from different discs. 
Note that the three dimensionality of braids is reduced here by viewing the time direction only in a countable number of slices. Not all knots can be ordered in this way, because a strand may weave both under and over another one, which would require an ill defined ordering of discs. However, the basic $B_3$ elements of Bilson-Thompson diagrams are examples of ordered braids.
Sticking to disc operads gives a large class of algebras (and connections to many, many things) but it may be necessary to consider also intersecting discs. Each Jones diagram, shown on the left below, includes internal lines that may not intersect, but when we overlap these big discs we make a choice as to their vertical ordering, and a line from one disc may cross over a line from another, thereby introducing the possibility of braiding for strands from different discs. 
Note that the three dimensionality of braids is reduced here by viewing the time direction only in a countable number of slices. Not all knots can be ordered in this way, because a strand may weave both under and over another one, which would require an ill defined ordering of discs. However, the basic $B_3$ elements of Bilson-Thompson diagrams are examples of ordered braids.
posted by Kea | 10:25 AM
7 Comments:
Interesting post. I'm interested in the ordering of braids to properly explain particle spins. String theory failed to account for the different standard model particles as being different vibrations of the same extra dimensional string.
So, how do the different particles come about? Some kind of Sundance Bilson-Thompson's braid model seems likely. Lubos once tried to ridicule the idea by calling it "octopusi swimming in the spin network", so you know it's worth investigating.
If you look at the results, differences in braiding account for differences in particle spin, while differences in electric charge account for the difference between a downquark and an electron.
This makes sense to me: compress 3 electrons in a small space (against the exclusion principle) so that the polarized vacuum of each (out to the Schwinger pair production cutoff of (m^2)*(c^3)/(e*h-bar) = 1.3*10^18 v/m, which occurs out to a radius of r = [e/(2m)]*[(h-bar)/(Pi*Permittivity*c^3)]^{1/2} = 3.2953 * 10^{-14} metre = 32.953 fm from the middle of an electron) overlaps substantially, and the shielding effect due to the vacuum polarized vacuum would be 3 times stronger, so the screened charge per electron sharing the vacuum shield will be increased 3 times, giving an observable charge at long distances of e/3 per electron, i.e., downquarks. The extra energy shielded by the vacuum when 3 leptons are compressed has to go somewhere: it goes into a new-short range force powered by the vacuum, mediated by colour charge. (There are also other complexities such as isospin charge for mesons, but this basic principle still holds good.)
The basic structure of a preon, or the unification particle behind leptons and quarks, has real spin if particles can be described by black holes consisting of trapped light velocity radiation. The principal magnetic moment of an electron, 1 Bohr magneton, is easily explained this way. I've an article in Electronics World, Apr. 2003 which shows that you get the spherically symmetric E-field, the dipole B-field, and time-dilation from the model of a fermion as radiation trapped into a loop by the black hole effect of gravity.
Theorem: for any effective gravitational mass M (which includes energy E/c^2 according to general relativity) there is a black hole event horizon radius of R = 2GM/c^2. If charged electromagnetic radiation has a wavelength of that scale, it's trapped into a tiny loop by the curvature of spacetime and becomes a fermion. It doesn't slow down; the motion is just circular as a small loop (i.e. electron "spin").
The connection to the Bilson-Thompson diagrams is that the Poynting vector of such trapped charged radiation can in some cases rotate while it goes round in the loop, producing an effect like a Mobius strip. This is one way that different types of particles can occur.
There are other mechanisms as well. Obviously, the main difference between the particles in the three different "generations" of the standard model is mass. Thus, muons are effectively heavy electrons. So to understand the different generations, you need to look closely at the model which describes the masses of particles (Higgs field bosons). It's possible to do that.
By the way, a brilliant Nov 64 Feynman lecture, "The relation of mathematics to physics", is now on google video!
Hi, Nigel. Interesting comments. Yes, I agree that the Bilson-Thompson diagrams should arise naturally from the right notion of observable, which I am taking to be in terms of categorical cohomology.
The problem with topological explanations of standard model symmetries is how to get out the continuous symmetry groups from structures involving only discrete symmetries primarily. Something like Mc-Kay correspondence might perhaps give hopes in this respect.
An alternative conjecture is that they bring in some genuinely new physics. Number theoretical braids are in key role in the formulation of quantum TGD where standard model quantum numbers are almost standard.
Quantum computation like activities, copying of information, and communication of if might occur at level of generalized Feynman diagrams.
Braid replication and its reverse at vertices of generalized Feynman diagrams where lightlike partonic 3-surfaces meet along their ends at vertices could be seen as a copying of information.
The counterpart of DNA replication would occur at fundamental level and DNA replication would be space-like image of this.
Hi Kea,
Your comment about discrete braids relates to a problem which I have been working with.
The cobordisms define also a braiding or tangle - number theoretic in my case. The question concerns about the exact definition of this braiding. One can imagine many possibilities and I have been fluctuating between them.
a) The flows defined by Noether currents associated with various isometries are obvious candidates. Non-uniqueness is a problem here.
b) The flows defined by the topological currents associated with induced Kahler field lines at partonic 3-surface closely related to Chern-Simons action can be considered and their 4-D variants play key role in the construction of extremals of Kahler action. The problem is which one to choose among various candidates. The Hodge dual of the induced Kahler field, density of Kahler magnetic flux defining magnetic field lines in lightlike direction, is perhaps the most natural candidate.
What is nice that in this case the restriction of cobordism to a discrete set of slices would unavoidably result from the requirement that imbedding space points are algebraic at each slice.
c) One can also ask whether one should one perform continuous transformation of the imbedding space coordinates (a one parameter family of isometries of imbedding space is perhaps too restricted transformation) so that coordinates of the points of braid would remain constant and algebraicity would not be lost? This flow would induce a continuous braiding. But is there any hope of making this braiding unique? For a moment I was excited about this option but now I feel myself rather skeptic.
Matti,
While the standard model symmetries are written in terms of continuous groups, the actual observed symmetries are discrete. For example, isospin is a continuous SU(2) symmetry, but no one ever observes a particle which is half neutron and half proton.
Carl,
the notion of quantization axes and freedom to choose the direction of quantization axis tells about the continuity of symmetry. Also the dimensions of irreducible representations, Glebsc-Gordan rules in tensor products of representation of rotation group follow from Lie group theory and are the basic predictions following from symmetry. In the case of mere discrete symmetry or "topological quantum numbers" you lose much of this structure.
I do not deny that the idea about braids as characterizers of elementary particles or more complex systems might have something in it. In my own framework the braidings could characterize zero energy states since braid in time direction (patterns at dance floor is excellent metaphor for this) would connect positive and zero energy parts of the state. Different braidings might characterize different time evolutions (S-matrices) but I would not explain standard model quantum numbers from braid patterns but see braidings as something genuinely new. One can of course ask whether braid patterns could correlate with the standard model quantum numbers of the state.
I regard the interpretation inspired by DNA more promising: in case of DNA braiding would be like quantum computer program coding the topological characteristics for the evolution of part of organism: at level of elementary particle reactions you would speak about S-matrix as coder of this time evolution.
Matti, "the notion of quantization axes and freedom to choose the direction of quantization axis tells about the continuity of symmetry."
Yes, but these are theoretical concerns. The experimental observations are no particles that are half neutron and half proton.
In my physics, isospin is related to spin orientation. But for any given particle, its spin orientation defines its isospin orientation. That is, you only get to choose one spin orientation, and from this you automatically choose the isospin orientation.
So I think that there is something to the continuous symmetries, but in a different way than usual (which requires that spin and isospin be disconnected by way of special relativity). I would prefer to see isospin derived the same way as spin.
Carl Brannen
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