2,054 research outputs found
Poisson Algebra of Wilson Loops and Derivations of Free Algebras
We describe a finite analogue of the Poisson algebra of Wilson loops in
Yang-Mills theory. It is shown that this algebra arises in an apparently
completely different context; as a Lie algebra of vector fields on a
non-commutative space. This suggests that non-commutative geometry plays a
fundamental role in the manifestly gauge invariant formulation of Yang-Mills
theory. We also construct the deformation of the loop algebra induced by
quantization, in the large N_c limit.Comment: 20 pages, no special macros necessar
The Geometry of Non-Ideal Fluids
Arnold showed that the Euler equations of an ideal fluid describe geodesics
on the Lie algebra of incompressible vector fields. We generalize this to
fluids with dissipation and Gaussian random forcing. The dynamics is determined
by the structure constants of a Lie algebra, along with inner products defining
kinetic energy, Ohmic dissipation and the covariance of the forces. This allows
us to construct tractable toy models for fluid mechanics with a finite number
of degrees of freedom. We solve one of them to show how symmetries can be
broken spontaneously.In another direction, we derive a deterministic equation
that describes the most likely path connecting two points in the phase space of
a randomly forced system: this is a WKB approximation to the
Fokker-Plank-Kramer equation, analogous to the instantons of quantum theory.
Applied to hydrodynamics, we derive a PDE system for Navier-Stokes instantons.Comment: Talk at the Quantum Theory and Symmetries 6 Conference at the
University of Kentuck
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