Manu Mathur

Coherent states for SU(3)

We define coherent states for SU(3) using six bosonic creation and annihilation operators. These coherent states are explicitly characterized by six complex numbers with constraints. For the completely symmetric representations (n,0) and (0,m), only three of the bosonic operators are required. For mixed representations (n,m), all six operators are required. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties. We introduce an explicit parametrization of the group SU(3) and the corresponding integration measure. Finally, we discuss the path integral formalism for a problem in which the Hamiltonian is a function of SU(3) operators at each site.

Color invariant additive fluxes for SU (3) gauge theory

Abstract The analogue of the triangle rule for the addition of angular momenta is constructed for the SU (3) group. This is used to obtain color invariant additive fluxes for SU (3) gauge theory. Pure SU (3) lattice gauge theory on a square lattice is shown to be equivalent to a certain abelian gauge theiry with a U(1)×U(1) local gauge invariance on a Kagome lattice. Degrees of freedom which generate Y-type string interactions emerge.

SU(N) irreducible Schwinger bosons

We construct SU(N) irreducible Schwinger bosons satisfying certain U(N-1) constraints which implement the symmetries of SU(N) Young tableaues. As a result all SU(N) irreducible representations are simple monomials of (N−1) types of SU(N) irreducible Schwinger bosons. Further, we show that these representations are free of multiplicity problems. Thus, all SU(N) representations are made as simple as SU(2).

Magnetic monopoles, gauge invariant dynamical variables and Georgi-Glashow model

Abstract We investigate the Georgi-Glashow model in terms of a set of explicitly SO(3) gauge invariant dynamical variables. In the new description a novel compact abelian gauge invariance emerges naturally which is not a subgroup of the original gauge group. The magnetic monopoles occur as point-like “defects” in spacetime and couple to the abelian gauge field with coupling ( 1 e ). Their non-perturbative contribution to the partition function is made explicit. This procedure corresponds to dynamical “abelian projection” without gauge fixing. The effect of the θ term in the abelian version of the theory is also studied.

Prepotential formulation of SU(3) lattice gauge theory

The SU(3) lattice gauge theory is reformulated in terms of SU(3) prepotential harmonic oscillators. This reformulation has enlarged SU(3)⊗U(1)⊗U(1) gauge invariance under which the prepotential operators transform like matter fields. The Hilbert space of SU(3) lattice gauge theory is shown to be equivalent to the Hilbert space of the prepotential formulation satisfying certain color invariant Sp(2, R) constraints. The SU(3) irreducible prepotential operators which solve these Sp(2, R) constraints are used to construct SU(3) gauge invariant Hilbert spaces at every lattice site in terms of SU(3) gauge invariant vertex operators. The electric fields and the link operators are reconstructed in terms of these SU(3) irreducible prepotential operators. We show that all the SU(3) Mandelstam constraints become local and take a very simple form within this approach. We also discuss the construction of all possible linearly independent SU(3) loop states which solve the Mandelstam constraints. The techniques can be easily generalized to SU(N).

Harmonic oscillator pre-potentials in SU(2) lattice gauge theory

We write the SU(2) lattice gauge theory Hamiltonian in d dimensions in terms of pre-potentials which are SU(2) fundamental doublets of harmonic oscillators. The Hamiltonian in terms of pre-potentials has SU(2) ? U(1) local gauge invariance. These pre-potentials enable us to solve the SU(2) Gauss law and characterize the SU(2) gauge invariant Hilbert space in terms of a set of integers. We discuss the consequences of the additional U(1) gauge invariance. The extension to SU(N) lattice gauge theory is discussed.

Irreducible SU(3) Schwinger bosons

We develop simple computational techniques for constructing all possible SU(3) representations in terms of irreducible SU(3) Schwinger bosons. We show that these irreducible Schwinger oscillators make SU(3) representation theory as simple as SU(2). The new Schwinger oscillators satisfy certain Sp(2,R) constraints and solve the multiplicity problem as well. These SU(3) techniques can be generalized to SU(N).

Loop approach to lattice gauge theories

Abstract We solve the Gauss law and the corresponding Mandelstam constraints in the loop Hilbert space H L using the prepotential formulation of ( d + 1 ) -dimensional SU ( 2 ) lattice gauge theory. The resulting orthonormal and complete loop basis, explicitly constructed in terms of the d ( 2 d − 1 ) prepotential intertwining operators, is used to transcribe the gauge dynamics directly in H L without any redundant gauge and loop degrees of freedom. Using generalized Wigner–Eckart theorem and Biedenharn–Elliot identity in H L , we show that the above loop dynamics for pure SU(2) lattice gauge theory in arbitrary dimension, is given by real and symmetric 3 n j coefficients of the second kind (e.g., n = 6 , 10 for d = 2 , 3 respectively). The corresponding “ribbon diagrams” representing SU(2) loop dynamics are constructed. The prepotential techniques are trivially extended to include fundamental matter fields leading to a description in terms of loops and strings. The SU(N) gauge group is briefly discussed.

Abelianization of SU(N) gauge theory with gauge invariant dynamical variables and magnetic monopoles

Abstract It is shown that SU ( N ) gauge theory coupled to adjoint Higgs can be explicitly re-written in terms of SU(N) gauge invariant dynamical variables with local physical interactions. The resultant theory has a novel compact abelian U (1) ( N -1) gauge invariance. The above abelian gauge invariance is related to the adjoint Higgs field and not to the gauge group SU(N) . In this abelianized version the magnetic monopoles carrying the magnetic charges of ( N −1) types have a natural origin and therefore appear explicitly in the partition function as Dirac monopoles along with their strings. The gauge invariant electric and magnetic charges with respect to U (1) ( N −1) gauge groups are shown to be vectors in root and co-root lattices of SU ( N ) respectively. Therefore, the Dirac quantization condition corresponds to SU ( N ) Cartan matrix elements being integers. We also study the effect of the θ term in the abelian version of the theory.

SU(N) coherent states and irreducible Schwinger bosons

We exploit the SU(N) irreducible Schwinger boson to construct SU(N) coherent states. This construction of an SU(N) coherent state is analogous to the construction of the simplest Heisenberg–Weyl coherent states. The coherent states belonging to irreducible representations of SU(N) are labeled by the eigenvalues of (N − 1) SU(N) Casimir operators and characterized by (N − 1) complex orthonormal vectors describing the SU(N) group manifold.