Re: Time, Parameter, Operator



Seratend wrote:

In classical QM, time is an observable but not of the configuration
hilbert space. In other words, the operator associated to the time
parameter commutes with all the operators on the configuration space.


There is just one Hilbert space where state vectors and operators of
observables are defined. You can choose different bases in this
Hilbert space, e.g., position (or configuration) representation or
momentum representation. The choice of the basis set does not affect
state vectors or operators. If time is an operator, then it ought to act
in this Hilbert space.

There is nothing contradictory with such an affirmation, all the
current classical QM experiments are based on this property.
This is why we may consider the time as a parameter in non relativistic
QM: we obtain a measurement "a" result @time t of the observable A =3D>
(a,t) is the measurement result of the commuting set of observables
(time observable,A).


Suppose that we constructed such a "time operator" that has continuous
spectrum from -infinity to +infinity and commutes with all other
observables. Then, according to postulates of quantum mechanics, for
each state |Psi> of a physical system we should be able to
calculate/measure the expectation value of time <Psi|T|Psi>.
We also should be able to prepare linear combinations of systems
with different values of time. All this does not make any sense to me.

Moreover, if T commutes with all operators, including the generator
of boosts K, then the time of events should not depend on the
velocity of observer. How would you explain the Lorentz transformations
for time and position in such a theory?

In my view, it is better not to use words "time" and "operator"
in one sentence. Time is just a numerical label assigned to
each measurement by looking at the wall-clock.

Eugene.


.

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