Gibbs States of the Chern-Simons Charged Particle System in the Mean-Field Type Limit

Abstract

Topological electrodynamics is a theory describing an interaction of a U(1) gauge field A(x, t), a vector-valued function on three-dimensional space, with a charged matter field, characterized by a current j(x, t), a vector-valued measure with a discrete support. The Lagrangian is given by

$$ L(A,j) = \frac{k}{2}\int {{{\varepsilon }^{{\alpha \beta v}}}{{A}_{v}}(x,t){{\partial }_{\alpha }}{{A}_{\beta }}(x,t){{d}^{2}}x - \frac{1}{c}\int {{{A}_{\alpha }}(x,t){{j}^{\alpha }}(x,t){{d}^{2}}x + \frac{1}{{2{{m}_{0}}}}{{{\sum\limits_{{s = 1}}^{n} {\left\| {{{v}_{s}}} \right\|} }}^{2}}} .} $$

where greek indices run over the set (0,1,2), repeated indices are summed over and ε ^αβv is the antisymmetric tensor. The integral is taken over 2-d space, ‖v _j ‖ is the Euclidean norm of the two-dimensional velocity vector of a particle with two-dimensional position vector x _j and charge σ _j ; t is the time (x ^o = ct), c is the velocity of light,

$$ {{j}^{\alpha }}(x,t) = \sum\limits_{{k = 1}}^{n} {v_{k}^{\alpha }{{\sigma }_{k}}\delta } (x - {{x}_{k}}(t)),\alpha = 1,2,{{j}^{0}}(x,t) = c\rho (x,t) = c\sum\limits_{{k = 1}}^{n} {{{\sigma }_{k}}\delta } ( - {{x}_{k}}(t)). $$