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Gigi

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'For example, the Schrödinger equation does not keep its written form under the coordinate transformations of special relativity; thus one might say that the Schrödinger equation is not covariant. By contrast, the Klein-Gordon equation and the Dirac equation take the same form in any coordinate frame of special relativity: thus, one might say that these equations are covariant. More properly, one should really say that the Klein-Gordon and Dirac equations are invariant, and that the Schrödinger equation is not, but this is not the dominant usage. '

Thus I have a question on the transition from quantum mechanics to relativistic quantum mechanics:

In non-relativistic quantum mechanics, if the Hamiltonian commutes with J, the angular momentum operator that is a generator of the SU(2) group, it will be rotationally invariant and after a rotation the system will still obey Schroedinger’s equation.

In relativistic quantum mechanics Schroedingers equation is replaced by the Klein Gordon and Dirac equations. Why?

Is it because the corresponding relativistic Hamiltonians in relativistic QM only commute with generators of the Lorentz group, while they do not commute with generators of the SU(2) group---leading to a lack of rotational invariance?

Is that why when we want to combine relativity with quantum mechanics, only a relativistic hamiltonian would work?

Please help---I am so confused :)!

Many thanks!