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Deformed Chern-Simons interaction for nonrelativistic point particles
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Deformed Chern-Simons interaction for nonrelativistic
point particles
P.C. Stichel
An der Krebskuhle 21
D-33619 Bielefeld?
Abstract
We deform the interaction between nonrelativistic point particles on a plane and
a Chern-Simons field to obtain an action invariant with respect to time-dependent
area-preserving diffeomorphisms. The deformed and undeformed Lagrangians are
connected by a point transformation leading to a classical Seiberg-Witten map be-
tween the corresponding gauge fields. The Schroedinger equation derived by means
of Moyal-Weyl quantization from the effective two-particle interaction exhibits
– a singular metric, leading to a splitting of the plane into an interior (bag-) and
an exterior region,
– a singular potential (quantum correction) with singularities located at the
origin and at the edge of the bag.
We list some properties of the solutions of the radial Schroedinger equation.
1 Introduction
Two-dimensional incompressible fluids, in particular quantum-hall fluids (QHF’s) are well
known to be invariant with respect to time-independent area-preserving diffeomorphisms
(cp.[1]). In a particle picture a QHF is usually described by neglecting the kinetic energy
compared to the magnetic field term leading to noncommutative geometry and a reduced
phase space (cp.[2]). In the present paper we generalize this picture by allowing
– the particles to move in full phase space,
- the area-preserving diffeomorphisms to become time-dependent.
To do this we consider a deformation of the interaction of nonrelativistic charged point
particles on a plane coupled to a Chern-Simons (CS) field such that
– the deformed action is invariant with respect to time-dependent area-preserving dif-
feomorphisms ν2,t,
– the deformed and undeformed particle Lagrangians are connected by a point trans-
formation leading to the classical analogue of a Seiberg-Witten (SW) map between
the deformed and undeformed gauge f
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