What Lorentz symmetries do electric and magnetic fields break? When we turn on an external (non-dynamical) electric or magnetic field in (3+1)-dimensional Minkowski space we break rotational invariance because they pick out a special direction in spacetime. Does this also break boost invariance?
What about in (2+1)-dimensions when the magnetic field is a scalar? Now the magnetic field does not seem to break rotations. Does it break boosts?
How can I show this?
 A: Under boosts,the fields transform into each other in a prescribed way.  If we define the direction of the boost to be the $x$-direction, then we have
\begin{align*}
E'_x &= E_x & E'_y &= \gamma(E_y - \beta B_z) & E'_z = \gamma(E_z + \beta B_y) \\
B'_x &= B_x & B'_y &= \gamma(B_y + \beta E_z) & B'_z = \gamma(B_z + \beta E_y) 
\end{align*}
It is not hard to see from the equations there that an electric or a magnetic field is invariant under boosts in the direction of the field — i.e., if the field is in the $x$-direction in one frame, then any new frame moving in the $x$-direction with respect to the first frame will also observe the same field.  However, the fields change if they have any components perpendicular to the boost.
Presumably one could write down a set of field transformations for electric and magnetic fields in 2+1 dimensions.  These could be found by writing out the component-by-component transformation laws for the Faraday tensor in 2+1 dimensions:
$$
F'_{\mu \nu} = \Lambda_\mu {}^\rho \Lambda_\nu {}^\sigma F_{\rho \sigma}.
$$
A: The electromagnetic field is a bivector field. The components called $E_x,E_y,E_z$ are the $tx,ty,tz$ bivector components, and $B_x,B_y,B_z$ are the $yz,zx,xy$ components.
A component of a bivector is unchanged by a rotation in its plane or in any perpendicular plane (perpendicular in the sense that every vector lying in one plane is perpendicular to every vector in the other, which is only possible in 4 or more dimensions). So $E_x$ is unchanged by rotation in the $tx$ and $xy$ planes, and so is $B_x$. A rotation in the $tx$ plane is also known as a boost in the $x$ direction.
In three dimensions, the bivector space is spanned by $tx, ty, xy$. You can identify these by Hodge duality with vector components $y,x,t$, and the field breaks the continuous Lorentz symmetry the same way a vector does: the residual symmetry is rotation about the vector axis or in the plane of the bivector, which is spatial rotation for the $t$ axis or $xy$ plane, a boost in the $y$ direction for the $x$ axis or $ty$ plane, etc.
A: The electromagnetic field on spacetime is actually Lorentz invariant. It's this conflict between this symmetry group of electromagnetism and the symmetry group of classical mechanics, which is the Galilean group, that led Einstein to special relativity.
The electromagnetic field on spacetime is a single field, it can't be covariantly split into electric and magnetic fields.
It is when we choose a spitting of spacetime into space and time, which breaks Lorentz invariance, that we can split the electromagnetic field into an electric and magnetic field. Conversely, such a choice defines a splitting of spacetime into space and time, so breaking Lorentz invariance.
A: It is true that the electromagnetic field is not a Lorentz scalar and is not Lorentz invariant. However it is a second rank, Lorentz covariant, antisymmetric tensor. Such an object does not break Lorentz symmetry.
