I understand that the topic of a time operator in quantum mechanics has come up more than a few times so forgive me if this is a repeat question but I couldn't find anything specific to my question.
My question is more to do with the relationship between general relativity and quantum mechanics.
My understanding is that according to GR, time and space are elements of the same thing, the spacetime manifold. So I would have thought that (up to a point at least) the mathematics of time would be the same as for position. Yet in QM, there is no operator for time... Why is this? I am prepared to accept that time is a parameter in QM, not a variable, but then why is the same not true for space? General relativity allows me to calculate a spacetime metric but QM tells me time is not a variable...
To be clear, my question is: Why does general relativity tell me that space and time are very similar, but quantum mechanics tells me they are very different?
Furthermore, is this in any way related to the incompatibility between QM and GR?