TITLE: How would two charges interact through a scalar magnetic field in $2+1$ D theory?
QUESTION [0 upvotes]: In Maxwell Chern Simons theory we get a scalar magnetic field attached to local charge density. I want to know that how do two charges interact through this scalar field. 
Will it be repulsive for same charges or attractive? As for a scalar it does not make sense to talk about attractive or repulsive behavior. I can provide further details about my question if required.

REPLY [1 votes]: The question essentially boils down to asking what the correct generalization of the Lorentz force law to arbitrary dimensions is. In three dimensions we have
$$ \vec F = q(\vec E + \vec v \times \vec B)\tag{1}$$
and in terms of the field strength $F_{\mu\nu}$ we have $B^i = \epsilon^{ijk} F_{jk}$, where the Latin indices indicate we only take the spatial components, i.e. the magnetic field is the spatial Hodge dual of the spatial part of the field strength tensor, and therefore a $(d-3)$-dimensional object in $d$ spacetime dimensions. The generalization of $\vec v \times \vec B$ is ${\star}_s (v \wedge B)$ (where ${\star}_s$ is the spatial Hodge dual), which is always a 1-dimensional object, i.e. a vector/1-form and therefore has the right dimension to give a force. So the Lorentz force law in arbitrary dimensions is
$$ F = q(E + {\star}_s (v \wedge B)),$$
and in three dimensions this becomes
$$ F = q(\vec E + {\star}_s B \vec v) = q(\vec E + B {\star}_s \vec v) = q (\vec E + B \vec v^\perp),$$
where $v^\perp$ is the vector of the same length perpendicular to $\vec v$. We should observe that this is also consistent with the original 3d law (1), if we embed the 2d space here into 3d space. That is, charges in 2d do not react "differently" to fields than they would in 3d, the major difference is that fields generated by monopoles fall off as $1/r$ rather than as $1/r^2$.
Note that the natural Lorentz force law for magnetic charges, $\vec F = q_m (\vec B - \vec v \times \vec E)$ (the sign is somewhat arbitrary), has no such straightforward generalization. Some might conclude from this that four dimensions are the only dimension in which magnetic and electric charges, if both exist, can actually interact with each other. It also calls into question the practice of calling the spatial part of $F_{\mu\nu}$ the "magnetic field" in dimensions other than four.
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