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Lack of Time

Posted: Thu Dec 07, 2017 10:11 pm
by Jim
In quantum mechanics, there is no time operator. Time, they say, is a parameter, not an observable. There is, therefore, no time operator.

This means that in the quantum theory of measurement, where every observable is associated with an operator, time cannot be measured.

This also produces problems in the interpretation of the time-energy uncertainty relation. The position-momentum interpretation is straightforward: since the position and momentum operators don't commute, there is some residual uncertainty in their simultaneous measurement - if you measure both at the same time, which comes first? But if time is not an operator, what does this mean?

The energy is of course associated with an operator: the Hamiltonian. It is also the time component of the 4-momentum. So, it makes sense that there should be an energy-time uncertainty relation.

But, for some reason its conjugate, the time, isn't an operator, can't be measured quatum mechanically. So what does that mean for the uncertainty relation?

Does that make sense? Are all time measurements classical? (if so, what does that mean for universal interpretations of the wavefunction, e.g. Everett's?)

Why doesn't the symmetry hold up?

Re: Lack of Time

Posted: Fri Dec 15, 2017 1:25 am
by Morrison
I'd like to hear your thoughts on following treatment of this issue, where they derive a Hamiltonian-time uncertainty relation, which they say is a more rigorous expression than the energy-time uncertainty relation (though I don't understand why the difference is more than a symantic one):

Generalization: Energy-Time Uncertainty (last section on the page) ... principle/

Re: Lack of Time

Posted: Tue Jan 09, 2018 12:11 am
by Jim
I've seen a lot of proofs like that. I recently decided to work through Sakurai's Modern Quantum Mechanics over the break -- funny story, I'll put it up in the review post on the blog when I finish it -- but he gives yet another such proof at the start of Chapter 2, "Quantum Dynamics." In this case, he used the correlations between the state at different times to justify the statement. The thing is, proofs like this don't really explain the reason why position and time (or should that be displacement and duration?) are placed on different footing.

But, Sakurai did take the problem seriously. Seriously enough to explain what happens in QED. And apparently QED solves the different footing problem in a different way: it turns spatial variables into parameters, too.

But that still makes me wonder: what does it mean to be a parameter that you cannot measure in quantum mechanics? If you can't measure these quantities quantum mechanically, what are they? Is any quantum theory of the vacuum doomed?