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Is there a notion of measurement, which doesn't correspond to a yes/no question or with the idea of the comparison of two real world objects, which produces a real number?

And does at least one of the object which are compared to produce the measurement result have to be modeled within the theory?

(One of my motivations for that question is that I wonder if a human can make a measurement involving light without a length measurement and it also seems to me that almost all measurements involve looking at data, which therefore effectively makes them length measurements. And then I speculate if there are even other mearuements than these. For example, I can't test the quantum wave values unless I collapse it - its reality is hidden (this question is certainly not only about QM though.) and therefore the specific configuration is not real is a strict sense. Similarly there is no time mearuesment, is there? The configuration of the clock is just infered by a mearuement of light.)

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1 Answer

Please consider the following before you speculate further:

We routinely find out the color (and hence make a spectral measurement, though not a very accurate one) of light without a length measurement. Also the intensity (and hence a field strength measurement).

Instead of looking at data you can have them read to you by automatic equipment for the blind.

Most real measurements of nontrivial complexity (almost all beyond those of simple introductory textbook examples) are only inferences from raw measurements based on , hence have very little to do with the kind of measurements discussed in the context of the Born rule.

For example, first principles distance measurements as done in GPS rely on complicated computations involving general relativity.

Query to test your quantum understanding: Did the wave function collapse
(a) when the signals arrived on which the computations were based,
(b) when the calculation is completed,
(c) when the result has been communicated to the device on your car and was looked at by you?
(d) in none of these cases?

Reading time from a digital clock also meant a lot of inference going on inside the clock before you could look at the display and (is this a length measurement?) deduce from the form of the black and white distribution the current time. Same problem for all digital electronic equipment.

Measurements such as that of a particle lifetime, a reaction rate, or the integral cross section of a particular reaction do not even have a natural associated operator of which the measurement result would be an eigenvalue.

The idealized textbook measurement theory based on Born's rule is appropriate only for the measurement of spin and related variables that result in recording decisions between a small number of cases.

The measurement process as described by von Neumann (and copied from there to numerous textbooks) is an unrealistic idealization compared with many (and probably most) real measurements. The latter are usually much better described by suitable POVMs (positive operator valued measures) rather than by Born's rule, which corresponds to PVMs (projection-valued measures), a special case of POVMs in which the positive operators are in fact projections.

And no, measurements in general do not cause any collapse, but the dissipative processes happen all the time, and measuring something means taking mental or automatic notice of something that has been completed and already existed as a fairly stable record in the detector. A record of something that happens to be stable in the detector, not necessarily a length. (In electronic equipment, typically, the stable things are magnetic or ooptical states.)

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I think I have to disentangle (badum tss) the parts of this answer which are about QM. I was actually motivated by thermodynamics here (As said, it's not really about QM for me and I didn't add the "measurement-problem" tag). In any case, I think I'm not sure what e.g. a spectral measurement is. I intepreted using your eyes as a length measurement, but it's just the problem that I don't really know where to draw the line if the observer is not really part of the theory. I can't really physically conceptualize how one observes a system, like by hearing. Time measurement is especially confusing. –  NiftyKitty95 Mar 25 '12 at 19:50
getting the color right is measuring a frequency, hence a spectral measurement. - Observation is the macroscopic respons of a system to another system. A macroscopic change in my ears or eyes is an observation of the interaction that caused it, independent of quantum or classical dynamics. Time measurement is matching coincidences with fractions of a more or less periodic process. –  Arnold Neumaier Mar 25 '12 at 20:15
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