2009/12/17 by Robert J. Finkelstein, Finkelstein, Robert J.
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.0912.3552
22 pages, LaTex file
arxiv created 2010/11/30 · arxiv updated 2010/12/02
We examine a quantum group extension of the standard model with the symmetry SU(3) × SU(2) × U(1)× global SLq(2). The quantum fields of this extended model lie in the state space of the SLq(2) algebra. The normal modes or field quanta carry the factors Djmm^′ (q|abcd), which are irreducible representations of SLq(2) (which is also the knot algebra). We describe these field quanta as quantum knots and set (j,m,m^′)= 1/2 (N,w, ± r+1) where the (N,w,r) are restricted to be (the number of crossings, the writhe, the rotation) respectively, of a classical knot. There is an empirical one-to-one correspondence between the four quantum trefoils and the four families of elementary fermions, a correspondence that may be expressed as (j,m,m^′)=3(t,-t3, -t0), where the four quantum trefoils are labelled by (j,m,m^′) and where the four families are labelled in the standard model by the isotopic and hypercharge indices (t,t3,-t0). We propose extending this correlation to all representations by attaching D-3t-3t03t (q| abcd) to the field operator of every particle labelled by (t,t3, t0) in the standard model. Then the elementary fermions (t=1/2) belong to the j=3/2 representation of SLq(2). The elements of the fundamental representation j=1/2 will be called preons and D-3t,-3to3t may be interpreted as describing the creation operator of a composite particle composed of elementary preons. Dm m^′j also may be interpreted to describe a quantum knot when expressed as D_\frac w2 \frac± r+12 N/2 These complementary descriptions may be understood as describing a composite particle of N preons bound by a knotted boson field with N crossings.