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?


It is special relativity that tells us space and time are different aspects of the same thing--spacetime. And indeed quantum mechanics does not respect the covariance principle of special relativity. That's exactly one of the reasons of why we need to invent quantum field theory(QFT) when we already have QM at hand.

To make the space and time on the same footing, we have two paths. The first one is, like QFT, to lower the status of space and make both space and time to be parameters. Then spacetime constitutes a parameter space on which some observables are constructed. This theory turns out to be QFT. And the particles are simply excitations of different fields. Covariance is lost when we only consider non-relativistic phenomena and the QFT reduces to QM.

The second way is to lift the time to be an operator. Then the spacetime are observables on some parameter space. This parameter space can be one dimensional which may be an arbitrary parameter along the worldline of the particle. That is the relativistic QM. But relativistic QM turns out to be incomplete, for example, it has the negative probability problem. Instead, you may probably think the particles are strings at the very micro scale. Then the parameter space is on the worldsheet traced out by the string. That is the string theory. Viewing the spacetime as fields on the worldsheet, string theory is a 2-dimensional field theory.


Because time itself is the driving force of motion, whereas space is not, space is expanding as is time, but space doesn't measure motion, it's a measure of the area we occupy. If I have 80 billion units of space but no measurable time then the energy of motion would not exist, therefore nothing could move and space would remain a fixed location. Since time and space are relatively infinite, in each aspect a measurable distance, they couldn't get around how time and space operate so they made it the same entity.


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