Time is not an observable, but....

[Demystifier]

Ok, so you take the subset of coherent states whose electric field is zero at all but one pixel $x$. But for each $x$ this still leaves an infinity of coherent states with different intensity; which ones do you pick for $|x\rangle$?

It is an experimental question. Those the expectation values of fields best describe the measured "classical" fields in the LCD screen.

 

[A. Neumaier]

It is an experimental question. Those the expectation values of fields best describe the measured "classical" fields in the LCD screen.

Ok, so you pick one coherent state per pixel with the experimental intensity. Now coherent states corresponding to neighboring pixels will substantially overlap. Thus the eigenstates of your position operator will have significant support in a number of pixels close to the intended one. This means that measuring $X$ will produce a superposition of pixels with probabilities that are maximal at the intended pixel but significantly less than 1. This means that you always get blurred measurements of the pixels, not true position measurements. This is sufficient for recognizing whether the digit 3 or 4 appeared, but not for a reduction of the digital time measurement process to one with true position measurements.

 

[Demystifier]

Now coherent states corresponding to neighboring pixels will substantially overlap. Thus the eigenstates of your position operator will have significant support in a number of pixels close to the intended one.

My intuition is that the overlap will be small, but I guess we are both guilty of not quantifying our intuitions.

But when I think again about all this, I realize that we are actually discussing irrelevant issues. What is relevant is that even though there is no time observable, there is an observable associated with a clock. In the case of LCD screen with digits, a relevant observable is the electric field in the screen. I believe you would agree with that.

 

[A. Neumaier]

a relevant observable is the electric field in the screen. I believe you would agree with that.

Yes, and the electric field changes with time. Thus time is not an observable but a parameter that tells which value $E(t)$ of the electric field applies in a particular instant. Similar for position. To talk about position you need to say at which time you mean it.

A measured time interval is usually the difference of the times at which two particular events happen. For example, you may ask about the time it takes between two adjacent sign changes of some component of the electric field. There is no operator corresponding to this measurement, only operators corresponding to the triggering events.