Is Time an Observable in Quantum Mechanics?

[masudr]

In the standard formulation of QM, time is a scalar parameter. I have seen time being treated as the 0th dimension of spacetime in a covariant Dirac equation, but is there any way of having time as an observable (i.e. associated with a Hermitian operator etc) which has a spectrum and so on?

 

[dextercioby]

Absolutely not.When speaking about dynamics in QM,in the II-nd chapter of his brilliant book,J.J.Sakurai asserts that time is just a parameter and no QM observable (no (...) operator) is associated to it.

Daniel.

 

[MiGUi]

I don't think so. Because operators are dynamic magnitudes, like in mechanics. So time can't be into this description. Mathematically, observables are hermitian operators which eigenvectors can perform a complete base of the state space... ¿Can time have eigenvectors or eigenvalues?

No, because when you measure time, you don't have a probability to measure different intervals... with an error...

 

[masudr]

I take Daniel's point; but this is an assertion. The problem of treating space and time on an equal footing still sticks out like a sore thumb to me.

In reply to MiGUi's comments: I'm not measuring time as such. When we measure position, we really measure the position between two things (i.e. our origin and the position of some particle). Why can we not measure, say the time a particle has spent since some other time (which we arbitrarily set as t=0, just as we do with space co-ordinates x=0)?

If a particle can be in a superposition of different position states corresponding to it being in different positions with respect to some origin, why can't a particle be in a superposition of different states corresponding to having spent different amounts of time with respect to some temporal origin?

 

[dextercioby]

If a particle can be in a superposition of different position states corresponding to it being in different positions with respect to some origin, why can't a particle be in a superposition of different states corresponding to having spent different amounts of time with respect to some temporal origin?

What states are you referring to...?Through what should we describe those states...And how would you define the dynamics of those states...?

Daniel.

 

[masudr]

Daniel, these are the exact questions that I am asking. I just wondered if someone else knew of such a thing; clearly they do not. Now I can think at length about this myself, resting assured that my thinking has not been done by someone else before.

Masud.

 

[dextercioby]

I could interpret your words as follows: "why can't a particle be in a superposition of different states corresponding to having spent different amounts of time with respect to some temporal origin" means that you're asking why a particle cannot be in an entagled state made up of pure states,nonstationary,ones the time evolved of others...?Is that right...?
The answer in this case is simple to give:it can be...I don't see a reason for that not be possible.I could be wrong though...Maybe someone else will contradict me or confirm my statement...

Daniel.

 

[quantumdude]

Masudr,

Please see the second page of the following document for an argument on why we do not associate operators with time.

http://xxx.lanl.gov/PS_cache/hep-th/pdf/9505/9505152.pdf

 

[dextercioby]

Thanks,Tom,great explanation,indeed...

So her name's Jessica,huh...? That spoils the mystery... :tongue2:

Daniel.