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?
Answer
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.
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