International Journal of Modern Physics A

By way of paying tribute to Abdus Salam, I recall the ideas of higher unification that he and I initiated. I discuss the current status of those ideas in the light of recent developments, including those of: (a) gauge coupling unification, (b) discovery of neutrino-oscillation at SuperKamiokande, and (c) ongoing searches for proton decay. It is remarked that the mass of ν_τ(~ 1/20 eV) suggested by the SuperK result seems to provide clear support for an underlying unity of forces based on the ideas of (i) SU(4)-color, (ii) left–right symmetry and (iii) supersymmetry. The change in perspective, pertaining to both gauge coupling unification and proton decay, brought forth by supersymmetry and superstrings is presented. The beneficial roles of string-symmetries in addressing certain naturalness problems of supersymmetry, including that of rapid proton decay, are noted. In the last section, attention is drawn to the recent joint works with K. Babu and F. Wilczek, where the influence of neutrino masses and thus of the new SuperK result on proton decay are noted. In this context, it is remarked that with neutrino masses and coupling unification revealed, the discovery of proton decay, that remains as the missing link, should not be far behind.

Quantum mechanics on manifolds is not unique, and in general an infinite number of inequivalent quantizations can be considered. These are specified by the induced spin and the induced gauge structures on the manifold. The configuration space of the collective mode in the Skyrme model can be identified with S³ and thus the quantization is not unique. This leads to the different predictions for the physical observables.

Starting from a one-particle quasi-exactly solvable system, which is characterized by an intrinsic sl(2) algebraic structure and the energy-reflection symmetry, we construct a daughter N-body Hamiltonian presenting a deformation of the Calogero model. The features of this Hamiltonian are (i) it reduces to a quadratic combination of the generators of sl(N+1); (ii) the interaction potential contains two-body terms and interaction with the force center at the origin; (iii) for quantized values of a certain cohomology parameter n it is quasi-exactly solvable, the multiplicity of states in the algebraic sector is (N+n)!/(N!n!); (iv) the energy-reflection symmetry of the parent system is preserved.

The spin fractions and deep inelastic lepton structure functions of the proton are analyzed using chiral field theory involving Goldstone bosons. A detailed comparison with recent chiral models sheds light on their successful description of the spin fractions of the proton as being due to neglecting helicity nonflip chiral transitions. This approximation is valid for zero mass but not for constituent quarks. Since the chiral spin fraction models with the pure spin-flip approximation reproduce the measured spin fractions of the proton, axialvector constituent-quark-Goldstone boson couplings are found to be inconsistent with the proton spin data. Initial quark valence distributions are then constructed using quark counting constraints at Bjorken x→1 and Regge behavior at x→0. Sea quark distributions predicted by chiral field theory on this basis have the correct order of magnitude and shape. The spin fractions also agree with the data.

The Yang–Baxter system related to quantum doubles is introduced and a large class of both continuous and discrete symmetries of the solution manifold are determined. A strategy for solution of the system based on the symmetries is suggested and accomplished in dimension 2. A complete list of invertible solutions of the system is presented.

Extending early work, we formulate the large N matrix mechanics of general bosonic, fermionic and supersymmetric matrix models, including Matrix theory: the Hamiltonian framework of large N matrix mechanics provides a natural setting in which to study the algebras of the large N limit, including (reduced) Lie algebras, (reduced) supersymmetry algebras and free algebras. We find in particular a broad array of new free algebras which we call symmetric Cuntz algebras, interacting symmetric Cuntz algebras, symmetric Bose/Fermi/Cuntz algebras and symmetric Cuntz superalgebras, and we discuss the role of these algebras in solving the large N theory. Most important, the interacting Cuntz algebras are associated to a set of new (hidden!) local quantities which are generically conserved only at large N. A number of other new large N phenomena are also observed, including the intrinsic nonlocality of the (reduced) trace class operators of the theory and a closely related large N field identification phenomenon which is associated to another set (this time nonlocal) of new conserved quantities at large N.