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Sure, time evolution in quantum theory is continuous, also position and momentum observables are continuous. That doesn't mean that it makes sense to discuss, whether the measurements deliver real or rational numbers. You anyway always have a finite resolution for any continuous observable, even in principle, as the most simple example of the position and momentum uncertainty relation show. Time is somewhat special since it's not an observable but a parameter in QT (inherited from our classical space-time concepts). Also here you have, however, an energy-time uncertainty relation (with the careful analysis of its meaning given by, e.g., Tamm). The most accurate clocks are based on transitions between atomic states, used to define the unit second in the SI based on measurements of the transition frequencies. Any transition line, however, has a finite "natural line width" you cannot avoid in principle. So also here the question, whether time is measured with real or rational numbers is mute.Dale said:A Rolex, or even a sundial, or just the analog voltage in a LC oscillator. If a measurement is a physical process then all of those physical processes are continuous.

A galvanometer reading after all is based on position measurements of its pointer and again at least you have the position-momentum uncertainty relation, i.e., there's always a principle minimal limit of accuracy. This quantum limit is of course very hard to reach (although it's possible as the example of the LIGO mirrors shows). Macroscopic positions are much less accurate and the main source of noise is thermodynamical, but on the other hand that's accurate enough in the macro world, and that's the reason why macroscopic objects appear to behave according to classical physics. Again given this level of accuracy the question, whether you measure currents or voltages as real or rational numbers with your galvanometer is pretty meaningless.Dale said:There is also the galvanometer I mentioned in the OP, and other classical analog measurements. And there are many other QM measurements with continuous spectrums.

A measurement of course means to get "a number with an estimate of its accuracy" out. The (macroscopic) position already is in a sense the measurement, because it indeed consists of averaging over macroscopically small but microscopically large space-time intervals thus averaging out all the thermal (and of course also quantum) fluctuations.Dale said:One unresolved issue is whether you consider the position of the galvanometer needle to be the measurement, or whether you consider the number that you write down to be the measurement. I am still somewhat ambivalent although I tend toward the first, but the choice does have consequences. In the first case, the measurement is continuous, but cannot be easily written down. In the second case the measurement is not continuous, but it is more than just the physical process.

Of course photon detectors have, as any detector for "particles", a finite resolution of position, e.g., the pixels of a Si-pixel detector. You can only say that a photon was detected within a space-time interval of finite extent.Dale said:I am not accepting that onus. I have never made any claims about photon detectors that I would have any onus to either defend or retract.

I don't understand the latter statement. Measurement devices as any piece of matter obey the physical laws and their use for measurements needs a construction based on these physical laws.Dale said:A finite number of particles may still have an infinite number of possible arrangements or states.

Here is my current thinking. In QM there are measurements with continuous spectra and in classical mechanics there are system properties that vary continuously and which can be measured. So, if a measurement is the physical process, then those are continuous. On the other hand, if a measurement is the number obtained from a physical process then there is more than just the physical process involved.