R. Gambini

Gauge Dynamics in the C Representation

Hamiltonian dynamics is implemented in a simple non-perturbative framework of interacting loops. The use of the C-representation, which incorporates automatically the Mandelstam identities, makes it possible to discuss the theory for general N.

Quantum mechanics for totally constrained dynamical systems and evolving hilbert spaces

We analyze the quantization of dynamical systems that do not involve any background notion of space and time. We give a set of conditions for the introduction of an intrinsic time in quantum mechanics. We show that these conditions are a generalization of the usual procedure of deparametrization of relational theories with Hamiltonian constraint that allow one to include systems with an evolving Hilbert space. We apply our quantization procedure to the parametrized free particle and to some explicit examples of dynamical systems with an evolving Hilbert space. Finally, we conclude with some considerations concerning the quantum gravity case.

On confinement in pure Yang-Mills theories

Abstract By using a gauge-invariant quantization procedure in the loop-space previously proposed by the authors, a simple realization of t Hoofts algebra is obtained which suggests that the dynamics of the system singles out the permanent confinement region in t Hoofts phase diagram.

Intersecting Braids and Intersecting Knot Theory

An extension of the Artin Braid Group is considered, with the introduction of new operatores that generate double and triple intersections. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple intersections, and a counter example is given for the case of quadruple intersections. Intersecting knot invariants are also constructed via Markov traces defined on the intersecting braid algebra representations, and the extended Turaev representation is discussed as an example.

Link invariants for intersecting loops

We generalize the braid algebra to the case of loops with intersections. We introduce the Reidemeister moves for 4 and 6-valent vertices to have a theory of rigid vertex equivalence. By considering representations of the extended braid algebra, we derive skein relations for link polynomials, which allow us to generalize any link Polynomial to the intersecting case. We perturbatively show that the HOMFLY Polynomials for intersecting links correspond to the vacuum expectation value of the Wilson line operator of the Chern Simon’s Theory. We make contact with quantum gravity by showing that these polynomials are simply related with some solutions of the complete set of constraints with cosmological constant �,for loops including triple self intersections. Previous derivations of this result were restricted to the 4-valent case.

Classical loop actions of gauge theories

Since the first attempt to quantize gauge theories and gravity in the loop representation, the problem of the determination of the corresponding classical actions has been raised. Here we propose a general procedure to determine these actions and we explicitly apply to the case of electromagnetism. Going to the lattice we show that the electromagnetic action in terms of loops is equivalent to the Wilson action, allowing us to do Monte Carlo calculations in a gauge-invariant way. In the continuum these actions need to be regularized and they are the natural candidates to describe the theory in a ``confining phase.

Interaction between pure gauge theories

Abstract A standard procedure is proposed for inducing interactions between pure gauge theories. To exhibit the method the Kalb-Ramond theory and the Green-Schwarz superstring theory are considered. When “big” gauge transformations are considered a quantization condition arises in disagreement with the values of the constants of the superstring theory.

Canonical Quantum Gravity on the Space of Vassiliev Invariants

I review some recent results on canonicalquantum gravity in the spin network representation. Aset of ambient isotopy spin network invariants isintroduced. These invariants are the natural extensionto spin networks of the Vassiliev invariants. Itis shown that this set is loop differentiable in thesense of distributions. The quantum gravity constraintsare written in terms of loop derivatives. It is explicitly shown that Vassiliev invariantssolve the diffeomorphism constraint. The regularizedAshtekar Hamiltonian constraint is studied, and itsaction on valence-four Vassiliev invariantsdiscussed.

Non-parametric strings

Abstract A formulation to relate the free bosonic string to the geometry of the group of loops is discussed. A neat implementation of the quantum dynamics and the constraints is found in terms of contact loop deformations.