Possibly, although I am pretty sure that there are quantum spectra that are continuous. And there may also be spectra where the individual values, though countable, are irrationally spaced on a consistent scale.Structure seeker said:I think @PeroK is thinking in the context of quantum physics, where the eigenvalues of a Hamiltonian have a countable spectrum of values and you measure one of these. Since any system is a countable heap of quantum systems, this applies to all measurements.
Yes, but that's not the issue. The question is whether you could design a spectrometer that could produce an uncountable number of measurement outcomes.Dale said:Possibly, although I am pretty sure that there are quantum spectra that are continuous. And there may also be spectra where the individual values, though countable, are irrationally spaced on a consistent scale.
I disagree. I think that is fundamental to the issue. If we can get a measurement with a continuous spectrum then ...PeroK said:Yes, but that's not the issue.
even if a single measuring device can only produce a finite number of outcomes, two different measuring devices may have irrationally related outcomes.PeroK said:The question is whether you could design a spectrometer that could produce an uncountable number of measurement outcomes.
Not if they cannot produce irrational measurements. If two devices produce a measurement with finitely many digits, then everything, including their ratio is rational.Dale said:... even if a single measuring device can only produce a finite number of outcomes, two different measuring devices may have irrationally related outcomes.
You cannot even store the outcome of an uncountable number of measurement outcomes. Thus That's overly speculative.PeroK said:Yes, but that's not the issue. The question is whether you could design a spectrometer that could produce an uncountable number of measurement outcomes.
If experimentalists had waited until the philosophers have solved their pseudo-measurement-problem, they'd not have measured anything since 1926 ;-)). Measurements simply work in practice, as does their analysis and interpretation in terms of QT.PeroK said:If we consider QM then we have to tackle the measurement problem. If our device is macroscopic, then it has a predefined scale of resolution. And, if the device itself becomes too small, then it behaves like part of the QM system that it's supposed to be measuring. And no measurement takes place.
Of course, any measurement has a finite accuracy, and there are always statistical and systematical errors. Take time measurements which I think one can consider the most accurate measurements one can do nowadays. Even with an ideal clock, there's a finite uncertainty due to the finite line width of any transition of a quantum system you might use to define your time unit. You'll never be able to pin down a time or frequency at an accuracy where it makes a difference whether you use real or rational numbers. The right thing to do is to give the finite number of significant digits for the "measurement result" with as good an estimate of the "error" of this result.PeroK said:But, even at the microscopic scale, things have a finite scale. Whether we have atoms or EM radiation as part of our device, these things cannot be used to probe indefinitely down to the infinite resolution required to distinguish real numbers. Or, even a countable infinity of rational numbers within a fixed range.
That's the issue, IMO. Or, at least, one of the issues.
The issue is the cardinality of the set of measurement outcomes. You can measure an angle, which can be converted into a fraction of ##\pi##. But, only finitely many angles are possible from any given measurement device.vanhees71 said:The debate, whether you measure real or rational numbers of some quantity is entirely academic.
Not so. If one device measures a frequency as ##1.000 \ 10^6 \mathrm{\ Hz}## and another device measures a frequency as ##6.283 \ 10^6 \mathrm{\ rad/s}## then their ratio is $$\frac{6283}{2000 \pi}$$ which is irrational. You cannot place both measurements on the same scale with the rationals. Either the ##\mathrm{Hz}## measurements must be converted to an irrational number of ##\mathrm{rad/s}## or vice versa.fresh_42 said:Not if they cannot produce irrational measurements. If two devices produce a measurement with finitely many digits, then everything, including their ratio is rational.
I agree, that probably should be more important.fresh_42 said:Why is it important to concentrate on the number system if the outcome of every single measurement is always a random interval? Shouldn't the randomness be in focus rather than the numbers?
This isn't true irrationality. It is merely printed on the device.Dale said:Not so. If one device measures a frequency as ##1.000 \ 10^6 \mathrm{\ Hz}## and another device measures a frequency as ##6.283 \ 10^6 \mathrm{\ rad/s}## then their ratio is $$\frac{6283}{2000 \pi}$$ which is irrational. You cannot place both measurements on the same scale with the rationals. Either the ##\mathrm{Hz}## measurements must be converted to an irrational number of ##\mathrm{rad/s}## or vice versa.
And here is where we disagree. To me, it is a matter of convenience, whereas you seem to search for a fundamental truth behind it. I'm comfortable leaving this to the Wittgensteins, Russells, and Gödels in the world.Dale said:I guess I don't like the assertion that all measurements are rationals primarily because, if we take it seriously, we would need to scrap the use of calculus in our theories. That seems like too high a price to pay.
Agree if you take the entire density matrix rather than just probabilities.fresh_42 said:Shouldn't the randomness be in focus rather than the numbers?
This is not, and never has been the debate. The theory has continuous variables, but measurement devices do not have a continuous (or even infinite) set of possible outcomes.Dale said:I guess I don't like the assertion that all measurements are rationals primarily because, if we take it seriously, we would need to scrap the use of calculus in our theories. That seems like too high a price to pay.
I thought of the distribution function, yes.Structure seeker said:Agree if you take the entire density matrix rather than just probabilities.
But the density matrix also describes the correlations, mixedness and so on. If you focus merely on the randomness, there's no reason quantum mechanics should be different from general statistics. In fact the phase should also be included.fresh_42 said:I thought of the distribution function, yes
Yes, that's where a calculus comes into play. Physical quantities as distribution functions are only the basic entities, just like variables in ordinary calculus are. x alone doesn't mean a thing. Of course, we have to consider all kinds of functions and dependencies. My suggestion was to reconsider analysis from the point of view of pdf.Structure seeker said:But the density matrix also describes the correlations, phase and so on. If you focus merely on the randomness, there's no reason quantum mechanics should be different from general statistics.
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.PeroK said:you, or even @vanhees71, cannot design a clock that measures time continuously
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.PeroK said:the onus is on you to design a photon detector that could detect a photon as infinitely many different points in space
A finite number of particles may still have an infinite number of possible arrangements or states.PeroK said:The same applies to any other measurements - ultimately you have only a finite number of particles to work with.
I also would find that very interesting. If you ever do run into such a thing in the future, please post it!fresh_42 said:A calculus based on pdf, however, would be - as far as I know - a new perspective. Measure theory comes close, but it focuses too much on measurability and not on the calculus part. But it could also be that there is such a calculus based on pdf and I simply do not know it
If a measurement is the number you get from a physical process then the printing on the device is an essential part of the measurement.fresh_42 said:This isn't true irrationality. It is merely printed on the device
Report #1 (started searching):Dale said:I also would find that very interesting. If you ever do run into such a thing in the future, please post it!
I just realized that this doesn’t matter except for clarity of communication. Either way there is a physical interaction and the generation of a number. Whether you use the word measurement to refer only to the first step or to both steps is only semantics.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.
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.
Obviously measurement devices are matter and are based on physical laws. The point is that numbers are not part of nature. I can run a given current through a given galvanometer. That will produce some amount of deflection. The number that is generated by that deflection is not set by nature, but is a matter of convention. You could choose different units, you could choose a different dimensionality, or even a different quantity entirely.vanhees71 said: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.
So what I said earlier.vanhees71 said:Sure? So?
Report #2:Dale said:I also would find that very interesting. If you ever do run into such a thing in the future, please post it!
I found this on Wiki but I got lost pretty quickly https://en.wikipedia.org/wiki/Itô_calculusfresh_42 said:Report #2:
A second, closer look and a question on MO resulted in:
Seems nobody wanted to deal with the problem of how to get a hold of the dependencies. The book is unfortunately copyright-protected (and ridiculously expensive) so I cannot see what Springer did. Symbolism is of course not satisfactory, and the other answers were only an admission of lack of imagination. I see the difficulties, too, but one should expect a few more theoretical results on processes we countlessly perform every single day. I thought of velocity as an example of the quotient of distance and time randomness, coupled by the object we assign velocity to. We measure it all the time in our cars, and unfortunately, police officers do the same. I expected a bit more substance than ##\pm 5km/h## and Doppler.
- what do you mean?
- hopeless
- symbolism: https://en.wikipedia.org/wiki/Algebra_of_random_variables
- M.D. Springer, 1979, The Algebra of Random Variables
https://www.amazon.de/dp/0471014060/
This reminds me of my motto: "Look where the money goes!" If someone deals with randomness and wants to handle margins and risks, then it is finance.Dale said:I found this on Wiki but I got lost pretty quickly https://en.wikipedia.org/wiki/Itô_calculus
But spacetime is not discrete (to the best of our knowledge).fresh_42 said:This discussion reminds me of my professor in my ODE class who said: "The real world is discrete!" The rationals are already unphysical because they are dense, and the real world, well, let's stop at the nucleus size or for the idealists at Planck length, is discrete.
We cannot even decide this question. Or tell "what" it is! Why should the manifold spacetime be continuous and the manifold living room table be discrete?AndreasC said:But spacetime is not discrete (to the best of our knowledge).
A discrete spacetime will have a different group of symmetries compare to a continuous one, and different representations. This may result in different set of elementary particles.fresh_42 said:We cannot even decide this question. Or tell "what" it is! Why should the manifold spacetime be continuous and the manifold living room table be discrete?
Discrete spacetime messes up a bunch of symmetries we generally know to be true. It also can't be modeled with our current mathematical tools, which is in contrast to things such as a table being modelled as "discrete".fresh_42 said:We cannot even decide this question. Or tell "what" it is! Why should the manifold spacetime be continuous and the manifold living room table be discrete?
Why can't it be modeled with current maths tools?AndreasC said:Discrete spacetime messes up a bunch of symmetries we generally know to be true. It also can't be modeled with our current mathematical tools, which is in contrast to things such as a table being modelled as "discrete".
Wellll I probably phrased it kinda badly, you could model it with different mathematical apparati but you can't just use the same ones we use right now, because currently spacetime is formulated using the theory of smooth manifolds and if you make it discrete it's not smooth any more, so you lose things such as derivatives etc. Of course you could come up with a discrete theory (and I believe such theories do exist) but you would then have a completely different theory, that uses different sorts of mathematical constructs. I think loop quantum gravity does something like that, but I don't know much about it, and my understanding is that spacetime isn't exactly discrete even there.martinbn said:Why can't it be modeled with current maths tools?
but but but, while any given apparatus will yield one of two spin states, one only knows the direction of this apparatus to some finite precision.PeroK said:There are infinitely many spin states for a spin 1/2 particle, but only two results for a measurement.
