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There is a form of the Dirac equation that treats these symmetrically:

[itex]i \gamma^\mu \partial_\mu \Psi = m \Psi[/itex]

However, at least in nonrelativistic QM, time is not treated symmetrically with the spatial coordinates, in that the Hilbert space consists of square-integrable functions of space, not space and time. So in NRQM, [itex]\partial_x[/itex] is a Hilbert-space operator, but [itex]\partial_t[/itex] is not, since the elements of the Hilbert space are not functions of time. Another way of putting the distinction is that [itex]t[/itex] is a parameter, while [itex]\hat{x}, \hat{y}, \hat{z}[/itex] are (Hilbert space) operators.

I'm assuming that the Dirac equation has a similar notion of a Hilbert space that is square-integrable functions of space, so that would mean that despite the symmetry between spatial and time coordinates in the covariant form of the Dirac equation, the meaning of space and time coordinates is different. So I would assume that it's true for the Dirac equation, as well, that [itex]t[/itex] is a parameter, while [itex]\hat{x}[/itex] is an operator? Similarly, [itex]\partial_x[/itex] is a Hilbert space operator, but [itex]\partial_t[/itex] is not?

Can somebody clarify this for me?