I'm reading that in EM theory, in hamiltonian formalism, we choose a specific reference frame with a specific time, and that this breaks covariance.
Why? Surely it's simple because it's just stated everywhere, so I must be missing something about covariance definition.
Lorentz boost is actually a rotation in time-space coordinates: it mixes time components and space components. When we move to Hamiltonian formalism this freedom to rotate time axis is lost, as time coordinate t and spatial coordinates are not treated on a equal footing. While time coordinate parametrises the evolution of the system, spatial coordinates are dynamical variables that determine the system's position in phase space.
Hamiltonian formalism singles out the notion of time. This is related to the fact that Hamiltonian is the generator of infinitesimal time translations and the space and time are treated separately unlike the Lagrangian formalism (think about action) where it is manifestly covariant. But, there are theories of gravity which have been formulated entirely on the basis of hamiltonian formalism. It involves reconstruction of space-time and restoration of general covariance after the full theory has been constructed.
