What is the difference between manifest Lorentz invariance and canonical Lorentz invariance? I often read that the Lorentz symmetry is manifest in the path integral formulation but is not in the canonical quantization - what does this really mean?
 A: Manifest Lorentz symmetry means that one can see Lorentz invariance directly from the way the theory is formulated; typically when space and time are treated on the same footing as components of a 4-vector. In these cases, the Lorentz group generators are represented in a simple way (hence the ''manifest'' symmetry), but it is far less trivial to find a corresponding Hilbert space of state vectors on which the interacting energy-momentum 4-vector acts. 
However, a theory can be Lorentz invariant in a more indirect way, such as in the canonical formalism, where a Hilbert space and an associated Hamiltonian is specified directly. Then Lorentz invariance is established by proving the (then far less trivial) existence of 6 generators satisfying the commutation relations for the Lorentz generators, such that the interacting Hamiltonian and the free momentum generators transform jointly as a 4-vector. 
A: The canonical formulation is based on a Hamiltonian framework which requires the definition of a time coordinate.  So, all of the quantities you calculate depend on this choice of time, and so it is not obvious that everything is Lorentz invariant.  
Path integrals respect lorentz invariance from start to finish, as they are based on a Lagrangian framework.
A: In Lorentz invariant theories:


*

*The Lagrangian density is a Lorentz scalar.

*The Hamiltonian is the generator of time translations (as Jerry Schirmer correctly says, one needs a specific time variable to begin with) and thus transforms as the zero component of a four vector. And it is less obvious to say if a quantity is a zero component of a four vector than to say if it is a scalar. The best way to say if a field theory with a given Hamiltonian is Lorentz invariant is to work out the commutators (or Poisson brackets if the theory is  classical) of the Hamiltonian with the boost and angular momentum generators to check if they close the Lorentz algebra. 
