The spectrum of a quantum mechanical Hamiltonian with SU(N) symmetry is studied in the limit N→∞. A complete description is given for the states which transform according to (i) the scalar representation (singlet states) and (ii) the adjoint representation of SU(N) (’’adjoint’’ states). The eigenvalues of the singlet states are equally spaced with a finite gap ω (g) as N→∞. The spectrum of the adjoint states is equally spaced asymptotically for large excitations with the same gap ω (g). The first excitation of the adjoint states is lower than the first singlet excitation. An accidental degeneracy appears which is removed by 1/N corrections. We compute explicitly this splitting for the singlet states. In this formulation 1/N2 plays a role similar to h/ so that all computed quantities are related to the corresponding classical system. All the quantities which enter in the calculation of the spectrum are analytic near g=0 with the same radius of convergence.
We construct canonical transformations to reformulate SU(N) Kogut-Susskind lattice gauge theory in terms of a set of fundamental loop & string flux operators along with their canonically conjugate loop & string electric fields. The canonical relations between the initial SU(N) link operators and the final SU(N) loop & string operators, consistent with SU(N) gauge transformations, are explicitly constructed over the entire lattice. We show that as a consequence of SU(N) Gauss laws all SU(N) string degrees of freedom become cyclic and decouple from the physical Hilbert space H. The Kogut-Susskind Hamiltonian rewritten in terms of the fundamental physical loop operators has global SU(N) invariance. There are no gauge fields. We further show that the (1/g) magnetic field terms on plaquettes create and annihilate the fundamental plaquette loop fluxes while the (g) electric field terms describe all their interactions. In the weak coupling (g → 0) continuum limit the SU(N) loop dynamics is described by SU(N) spin Hamiltonian with nearest neighbour interactions. In the simplest SU(2) case, where the canonical transformations map the SU(2) loop Hilbert space into the Hilbert spaces of hydrogen atoms, we analyze the special role of the hydrogen atom dynamical symmetry group SO(4, 2) in the loop dynamics and the spectrum. A simple tensor network ansatz in the SU(2) gauge invariant hydrogen atom loop basis is discussed.
The generating function method that we had developing has various applications in physics and not only interress undergraduate students but also physicists. We solve simply difficult problems or unsolved commonly used in quantum, nuclear and group theory textbooks. We find simply: the generating function of the harmonic oscillator, the Feynman propagators of the oscillator and the oscillator in uniform magnetic field. We derive the invariants of SU(2) and the expressions of 3-j,6-j symbols. We find also the octonions or Hurwitz quadratic transformations. We show that the cross-product exist only in E3 and E7. We determine the {p} representation of hydrogen atom in three and n-dimensions. We generalize the Cramer's rule for the calculation of the rotational spectrum of the nucleus. We find the expression of the Hamiltonian in terms of quasi-bosons for study the collective vibration. We determine the basis and the expressions of 3-j symbols of SU (3) and SU(n).We find the Schrödinger equation from Hamilton-Jacobi formalism. We present these applications in independent chapters.