Loop approach to lattice gauge theories

2004

We solve the Gauss law of SU(2) lattice gauge theory using the harmonic oscillator prepotential formulation. We construct a generating function of a manifestly gauge invariant and orthonormal basis in the physical Hilbert space of (d+1) dimensional SU(2) lattice gauge theory. The resulting orthonormal physical states are given in closed form. The generalization to SU(N) gauge group is discussed.

Physics Letters B, 1994

It is showed that the very recently introduced Lagrangian loop formulation of the lattice Maxwell theory is equivalent to the Villain form in 2+1 dimensions. A transparent description of the classical loop action is given in pure geometrical terms for the 2 + 1 and 3 + 1 dimensional cases.

Zeitschrift für Physik C Particles and Fields, 1982

As a first step towards a duality transformation for the SU(2) lattice gauge theory in 3 dimensions, the integration over all gauge variant variables is performed explicitly after introducing gauge invariant auxiliary variables. The resulting new Hamiltonian is complex and involves a sum over closed loops. Each of these loops is confined to an elementary cube of a dual lattice. Like in a previous investigation for the 0(4) symmetric Heisenberg ferromagnet Riihl's boson representation is used to derive the result.

Classical and Quantum Gravity, 2004

We present an interpretation of loop quantization in the framework of lattice gauge theory. Within this context the lack of appropriate notions of effective theories and renormalization group flow exhibit loop quantization as an incomplete framework. This interpretation includes a construction of embedded spin foam models which does not rely on the choice of any auxiliary structure (e.g. triangulation) and has the following straightforward consequences: The values of the coupling constants need to be those of an UV-attractive fixed point. The kinematics of canonical loop quantization and embedded spin foam models are compatible. The weights assigned to embedded spin foams are independent of the 2-polyhedron used to regularize the path integral, |J| x = |J| x′. An area spectrum with edge contributions proportional to l PL 2(j + 1/2) is not compatible with embedded spin foam models and/or canonical loop quantization.

Physics Letters B, 1999

The abelian-projected monopole loop distribution is extracted from maximal abelian gauge simulations. The number of loops of a given length falls as a power nearly independent of lattice size. This power increases with β = 4/g 2 , reaching five around β = 2.85, beyond which, it is shown, loops any finite fraction of the lattice size vanish in the infinite lattice limit.

Journal of Physics A: Mathematical and Theoretical, 2009

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 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).

Physical Review D, 1997

A gauge invariant Hamiltonian representation for SU(2) in terms of a spin network basis is introduced. The vectors of the spin network basis are independent and the electric part of the Hamiltonian is diagonal in this representation. The corresponding path integral for SU(2) lattice gauge theory is expressed as a sum over colored surfaces, i.e. only involving the $j_p$ attached to the lattice plaquettes. This surfaces may be interpreted as the world sheets of the spin networks In 2+1 dimensions, this can be accomplished by working in a lattice dual to a tetrahedral lattice constructed on a face centered cubic Bravais lattice. On such a lattice, the integral of gauge variables over boundaries or singular lines -- which now always bound three coloured surfaces -- only contributes when four singular lines intersect at one vertex and can be explicitly computed producing a 6-j or Racah symbol. We performed a strong coupling expansion for the free energy. The convergence of the series expansions is quite different from the series expansions which were performed in ordinary cubic lattices. In the case of ordinary cubic lattices the strong coupling expansions up to the considered truncation number of plaquettes have the great majority of their coefficients positive, while in our case we have almost equal number of contributions with both signs. Finally, it is discused the connection in the naive coupling limit between this action and that of the B-F topological field theory and also with the pure gravity action.

We introduce in a natural and straigthforward way the loop (Lagrangian) representation for the partition function of pure compact lattice QED. The corresponding classical lattice loop action is proportional to the quadratic area of the loop world sheets. We discuss the parallelism between the loop description of this model in terms of world sheets of loops and the topological representation of the Higgs (broken) phase for the non-compact lattice QED in terms of world sheets of Nielsen-Olesen strings.

Physical Review D, 1996

We show how the Hamiltonian lattice loop representation can be cast straightforwardly in the path integral formalism. The procedure is general for any gauge theory. Here we present in detail the simplest case: pure compact QED. The lattice loop path integral approach allows to knit together the power of statistical algorithms with the transparency of the gauge invariant loop description. The results produced by numerical simulations with the loop classical action for different lattice models are discused. We also analyze the lattice path integral in terms of loops for the non-Abelian theory.

Physics Letters B, 1986

The Schwmger-Dyson equation of a Wilson loop is used to study the phase structure of the SU(2) lattice gauge system. It is shown that there is no evidence of a phase transition in the four-dimensional case, while there is a deconfmement phase transition in the five-dimensional case