The Elementary Particles as Quantum Knots in Electroweak Theory

We explore a knot model of the elementary particles that is compatible with electroweak physics. The knots are quantized and their kinematic states are labelled by $D^j_{mm'}$, irreducible representations of $SU_q(2)$, where j = N/2, m = w/2, m' = (r+1)/2 and (N,w,r) designate respectively the number of crossings, the writhe, and the rotation of the knot. The knot quantum numbers (N,w,r) are related to the standard isotopic spin quantum numbers $(t,t_3,t_0)$ by $(t=N/6,t_3=-w/6,t_0=-(r+1)/6)$, where $t_0$ is the hypercharge. In this model the elementary fermions are low lying states of the quantum trefoil (N=3) and the gauge bosons are ditrefoils (N=6). The fermionic knots interact by the emission and absorption of bosonic knots. In this framework we have explored a slightly modified standard electroweak Lagrangian with a slightly modified gauge group which agrees closely but not entirely with standard electroweak theory.Comment: 29 pages; LaTex fil

Masses and Interactions of q-Fermionic Knots

The q-electroweak theory suggests a description of elementary particles as solitons labelled by the irreducible representations of SU_q(2). Since knots may also be labelled by the irreducible representations of SU_q(2), we study a model of elementary particles based on a one-to-one correspondence between the four families of Fermions (leptons, neutrinos, (-1/3) quarks, (2/3) quarks) and the four simplest knots (trefoils). In this model the three particles of each family are identified with the ground and first two excited states of their common trefoil. Guided by the standard electroweak theory we calculate conditions restricting the masses of the fermions and the interactions between them. In its present form the model predicts a fourth generation of fermions as well as a neutrino spectrum. The same model with q almost equal to 1 is compatible with the Kobayashi-Maskawa matrix. Depending on the test of these predictions, the model may be refined.Comment: 40 pages, 2 figures, latex forma