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mitchell porter
Aug25-11, 12:47 AM
P: 753
After staring at the weak isospin and weak hypercharge assignments for a while, I have devised a new approach to this whole idea. I haven't even tried to get the right numbers of particles, I just want to mention it as a mutant form of the hypothesis which might assist its analysis.

Alejandro's idea involves pairing (anti)quarks, adding the electric charges, and then supposing that these pairings have superpartners, and is called the super-bootstrap. I do the same, except that I add the ordered pairs (weak hypercharge, weak isospin), so I call it the "hyper-bootstrap".

To add ordered pairs, the rule is (a0,b0)+(a1,b1)=(a0+a1,b0+b1). There is also a secondary "rule" that you can add two ordered pairs which both have nonzero weak isospin, only if one has isospin +1/2 and the other has isospin -1/2. Also, you only add quarks; leptons are an exit point. (This is "because" only quarks feel color, and the strong force is the rationale for all the pairings.) And finally, you only add two ordered pairs at a time.

To begin with, we suppose we only have left-handed quarks and right-handed antiquarks to work with; so we have ordered pairs of the form (+/- 1/3, +/- 1/2). Because of the secondary rule about only adding nonzero isospins of opposite signs, the only ordered pairs we can make from these are (0,0) and (+/- 2/3, 0). That is, left-handed neutrino / right-handed antineutrino, and left-handed down-type antiquark / right-handed down-type quark.

Next, suppose we are adding ordered pairs of the form (+/- 2/3, 0). From this we can again get (0,0), and we can also now get (+/- 4/3, 0), i.e. right-handed up-type quark / left-handed up-type antiquark.

Next, suppose we are adding ordered pairs (+/- 2/3, 0) and (+/- 4/3, 0). This allows us to get (+/- 2/3, 0) and (+/- 2, 0). So here the hyper-bootstrap offers an additional way to obtain (+/- 2/3, 0), as well as putting right-handed electrons / left-handed positrons (and their muon and tauon counterparts) within reach.

Finally, suppose we add (+/- 1/3, +/- 1/2) and (+/- 2/3, 0). This allows us to obtain (+/- 1, +/- 1/2) ... left-handed leptons and right-handed antileptons ... and (+/- 1/3, +/- 1/2) ... left-handed quarks and right-handed antiquarks again, the hyper-bootstrap feeding into itself again.

As happens for Alejandro, I don't have a rule that prevents me from combining (+/- 4/3, 0) with itself, so I also get the annoying extra combination (+/- 8/3, 0). edit: Nor do I have a rule against adding (+/- 4/3, 0) with (+/- 1/3, +/- 1/2), which produces (+/- 1, +/- 1/2) as above, and another nonexistent assignment (+/- 5/3, +/- 1/2).

Obviously the hyper-bootstrap and the super-bootstrap have considerable similarities - including the leftover at the end! And we need to examine whether the actual multiplicities, of quark fields and their combinations, work at all. But it's interesting that even at the slightly finer-grained level which considers isospin and hypercharge quantum numbers separately, you can still define a similar scheme.