A Fundamental Theorem of Quantum Measurements

[Demystifier]

This is amusing. About a year or so ago you said something to the effect that reality was continuous (I am not able to find it). I replied, "No it's not, it's discrete. Prove me wrong." @Dale then deleted my comment because "Prove me wrong." wasn't up to PF standards. I actually meant it as a joke.

Nice to see you now know the TRUTH.
BTW, the term "an observable" is almost always used in the context of an operator in QM, which may be what is bothering @Neumaier.

Only a fool never changes his mind. My views evolve, see e.g. https://www.physicsforums.com/insights/stopped-worrying-learned-love-orthodox-quantum-mechanics/

That being said, I have to say that I still think that objective reality may be continuous, but it just can't be measured.

 

[rubi]

Continuity in the context of observables doesn't mean that we mustn't bin the measurement outcomes, but rather that for every choice of binning, we can achieve a finer binning by using a more accurate measurement device, i.e. there is no physical limitation on the possible accuracies of future measurement devices. The limitations are only of technological nature and can be overcome by using more advanced technology. At the moment, we have no experimental evidence for the existence of a physically finest possible binning of position or momentum, so the state of our knowledge is that position and momentum are continuous observables.

 

[Zafa Pi]

But page 42 looks pretty much like the definition of a mathematical model to me, but I accept opinions could vary.

Ballentine makes it clear that physics consists of both a mathematical model and a correspondence to the physical world.
Claiming "physics is a mathematical model" is likely to hurt the feelings of experimentalists.

 

[Zafa Pi]

Only a fool never changes his mind.

You've held that position for quite a while now. Perhaps it's time for a change.

I still think that objective reality may be continuous, but it just can't be measured.

there is no physical limitation on the possible accuracies of future measurement devices.

Those positions are far from universally accepted (e.g., Planck length limitation), and sufficiently untestable to be "not even wrong".

With a slightly amended set theory beyond ZFC all of analysis can be done with infinitesimals and thus countable. If you were raised with that orientation, then continuous would mean something different.
The usual real numbers are merely "a thinking tool", and should be kept from reality with likes of the tooth fairy.

 

[rubi]

Those positions are far from universally accepted (e.g., Planck length limitation), and sufficiently untestable to be "not even wrong".

Planck length is just a unit of length and not a smallest length scale. There's no problem with having half a Planck length or things like this and there is no indication today that there is such a thing as a smallest length scale. The currently accepted view is that observables such as position and momentum are indeed continuous. Of course, this may change in the future, like all scientific knowledge.

With a slightly amended set theory beyond ZFC all of analysis can be done with infinitesimals and thus countable. If you were raised with that orientation, then continuous would mean something different.
The usual real numbers are merely "a thinking tool", and should be kept from reality with likes of the tooth fairy.

I don't see what you mean by that. Non-standard analysis can already be done in ZFC and any extension of ZFC (and also super weak subsystems like second order arithmetic) will have uncountably many real numbers as well (hyperreals may have an even larger cardinality). But I don't see what your argument is. Already the rationals have the property that any interval can be split up into even smaller intervals.

 

[Zafa Pi]

Planck length is just a unit of length and not a smallest length scale. There's no problem with having half a Planck length or things like this and there is no indication today that there is such a thing as a smallest length and the currently accepted view is that observables such as position and momentum are indeed continuous.

The above is about QM, which is theory.

there is no physical limitation on the possible accuracies of future measurement devices.

This is about reality. You are mixing metaphors.

Already the rationals have the property that any interval can be split up into even smaller intervals.

The rationals are not continuous in the usual sense (Dedekind).

 

[rubi]

The above is about QM, which is theory.

It's true in all our accepted scientific theories.

This is about reality. You are mixing metaphors.

It is also about theories, because of course we propose only theories that model reality. If we have no experimental evidence for a limitation of possible measurement accuracy, we wouldn't arbitrarily include such a limitation in our models.

The rationals are not continuous in the usual sense (Dedekind).

Again, I don't see your point. The question is whether there is a smallest length scale and even if we were to reject the real numbers for whatever reason and resort to the rationals, there would still not be a smallest length scale in a model based on the rationals. But I don't understand your criticism of the reals anyway.

 

[Zafa Pi]

I don't know how to make my position clearer.

there is no physical limitation on the possible accuracies of future measurement devices.

Would you please give a reference to justify the above.

I don't understand your criticism of the reals anyway.

You're in good company, my girl friend doesn't understand my criticism of the tooth fairy.

 

[rubi]

I don't know how to make my position clearer.

Probably, because it cannot be made clear in the first place.

Would you please give a reference to justify the above.

How about you quote me correctly? This sentence was part of a definition, not a claim about reality: "Continuity in the context of observables doesn't mean that we mustn't bin the measurement outcomes, but rather that for every choice of binning, we can achieve a finer binning by using a more accurate measurement device, i.e. there is no physical limitation on the possible accuracies of future measurement devices." Please don't try to hide your lack of arguments behind this kind of dishonesty. The claim in my post was: "At the moment, we have no experimental evidence for the existence of a physically finest possible binning of position or momentum." Apparently, you can't prove me wrong here without resorting to dishonest methods.

You're in good company, my girl friend doesn't understand my criticism of the tooth fairy.

Well, I have explained, why your reasoning is wrong and you haven't responded to my criticism, so the situation is quite different.

 

[bhobba]

Ballentine makes it clear that physics consists of both a mathematical model and a correspondence to the physical world.
Claiming "physics is a mathematical model" is likely to hurt the feelings of experimentalists.

Get your issue now - yes that's right. But I don't think those into mathematical modelling as a discipline are that unconcerned about experimental verification of their models

Thanks
Bill

 

[Zafa Pi]

Get your issue now - yes that's right. But I don't think those into mathematical modelling as a discipline are that unconcerned about experimental verification of their models

Thanks
Bill

If what you mean by "unconcerned" is concerned, then that certainly seems to be the case for string theorists.

 

[bhobba]

If what you mean by "unconcerned" is concerned, then that certainly seems to be the case for string theorists.

We are getting off topic here, but IMHO, and the opinion of others, for what you said, and other reasons, string theory has morphed a bit:
https://www.quantamagazine.org/string-theorys-strange-second-life-20160915/

THanks
Bill

 

[Zafa Pi]

We are getting off topic here, but IMHO, and the opinion of others, for what you said, and other reasons, string theory has morphed a bit:
https://www.quantamagazine.org/string-theorys-strange-second-life-20160915/

THanks
Bill

Rather than saying we are getting off topic, why not say the topic is evolving/morphing in interesting ways?
That was a fun article, thanks, but not much about experimental confirmation of string theory.

From the article: "Using the physical intuition offered by strings, physicists produced a powerful formula for getting the answer to the embedded sphere question, and much more. “They got at these formulas using tools that mathematicians don’t allow,” Córdova said. Then, after string theorists found an answer, the mathematicians proved it on their own terms."
Many years ago I came across a simple version of this: If P is a convex polyhedron, with a distribution of mass and a center of mass C, then there exists a face F such that the line passing through C perpendicular to the plane of F lands in F.
The physics justification is that P can't roll for ever, so where ever it rests must be such an F. Even as a mathematician I was convinced, others were not.
The math proof is messier.

 

[Demystifier]

You've held that position for quite a while now. Perhaps it's time for a change.Those positions are far from universally accepted (e.g., Planck length limitation), and sufficiently untestable to be "not even wrong".

With a slightly amended set theory beyond ZFC all of analysis can be done with infinitesimals and thus countable. If you were raised with that orientation, then continuous would mean something different.
The usual real numbers are merely "a thinking tool", and should be kept from reality with likes of the tooth fairy.

Have you seen this? https://arxiv.org/abs/1609.01421

Speaking of set theory beyond ZFC, perhaps you could also add something here: https://www.physicsforums.com/threads/is-there-a-decidable-set-theory.939216/

 

[Zafa Pi]

Have you seen this? https://arxiv.org/abs/1609.01421

I dig Baez, so I'll look it over, thanks

Speaking of set theory beyond ZFC, perhaps you could also add something here: https://www.physicsforums.com/threads/is-there-a-decidable-set-theory.939216/

I'm a fan of undecidability so I'm not going search for a decidable set theory. I'd keep it secret if I found one, why spoil the fun.

 

[A. Neumaier]

objective reality may be continuous, but it just can't be measured.

Objective reality might also be discrete, but it just can't be decided by measurement. See my answer at https://physics.stackexchange.com/a/35676/7924

But almost all of physics assumes continuity, for very good reasons: The discrete case is essentially untractable since analysis (the tool created by Newton, in a sense the father of modern physics) can no longer be applied.

 

[stevendaryl]

With a slightly amended set theory beyond ZFC all of analysis can be done with infinitesimals and thus countable. If you were raised with that orientation, then continuous would mean something different.

Nonstandard analysis is another way of proving the same theorems as ordinary analysis. The elements are not really "countable" in the sense that you can count them using finite numbers---you have to go to hyperfinite numbers.

It is nice that nonstandard analysis allows us to prove things using more intuitive (to many people) reasoning about infinitesimals, rather than limits, but it's really equivalent.

 

[vanhees71]

Ballentine makes it clear that physics consists of both a mathematical model and a correspondence to the physical world.
Claiming "physics is a mathematical model" is likely to hurt the feelings of experimentalists.

As a theorist, without hurting myself or my colleagues, claim the opposite: Physics is an empirical science in the fortunate position to have an astonishingly far-reaching theoretical description of the empirical findings ;-))).

 

[Danny Boy]

@Demystifier Just one more thing. Am I correct that if we consider the measurement on a pure state $\rho = | \psi \rangle \langle \psi |$ which results in $$\rho \rightarrow \tilde{\rho} := \frac{\int_{\mathcal{M}}A_C \rho A_C^{\dagger}dC}{Tr[\int_{\mathcal{M}}A_C \rho A_{C}^{\dagger}dC]}$$ Then if $\tilde{\rho}$ is a pure state as well then we can consider the change of the state $| \psi \rangle$ after measurement to be $$|\psi \rangle \rightarrow |\tilde{\psi} \rangle = \frac{1}{\mathcal{N}} \int_{\mathcal{M}} A_C | \psi \rangle dC?$$ where $\tilde{\rho} = | \tilde{\psi} \rangle \langle \tilde{\psi} |$.

 

[Demystifier]

If $A_C$ are projectors, then yes.

 

[vanhees71]

Well, the final statement in #70 is the socalled "collapse postulate". One should be aware that the validity of this assumption at best holds in a simplified sense for very special kinds of measurements, named von Neumann filter measurements, named after John von Neumann, who wrote a partially brillant book on the mathematical foundations of QT, establishing the Hilbert-space formalism, clarifying the somewhat delicate issue of unbound operators with continuous spectra. The physical part of this book, however, is partially misleading, precisely because of the issues with the collapse, quantum-classical cut, of some naive flavors of the Copenhagen interpretation.

 

[Zafa Pi]

Objective reality might also be discrete, but it just can't be decided by measurement. See my answer at https://physics.stackexchange.com/a/35676/7924

Agreed.

But almost all of physics assumes continuity, for very good reasons: The discrete case is essentially untractable since analysis (the tool created by Newton, in a sense the father of modern physics) can no longer be applied.

I would modify this with: "But almost all theories of physics ...
Physics in the lab is essentially discrete.
I've also read there is a program that simulates all of Newtonian mechanics (I'm still looking for it). Functions like cos and equations are subroutines. To get a definite answer the programs must halt giving discrete solutions. The real numbers never show up.

This article is interesting: https://www.wired.com/2009/04/Newtonai/

 

[vanhees71]

Hm, the stuff around me doesn't look very discrete to me. Also quantum theory, describing "discrete particles/quanta" is a theory described by continuum theories, i.e., (functional) calculus. What "nature really is" or what "objective reality" is, nobody can seriously say, and it's the realm of philosophical speculation rather than fact-based empirical natural science.

 

[Boing3000]

Hm, the stuff around me doesn't look very discrete to me.

I have never "seen" something non-discrete in my entire life. Even a "continuous" blue sky have to be distinguish from a "continuous" blue sheet of paper by its discrete borders...

Also quantum theory, describing "discrete particles/quanta" is a theory described by continuum theories, i.e., (functional) calculus.

Indeed, but the question is that maybe a another type of math would better suit the problem of describing quanta of Nature.

What "nature really is" or what "objective reality" is, nobody can seriously say

But do you agree that the goal of science is to deepen this knowledge ?. The stuff around me doesn't look very (im)probable to me. Yet I don't mind decreasing the scale of certain experiments until "probability" (or something like it) starts showing up.

and it's the realm of philosophical speculation rather than fact-based empirical natural science.

Isn't it an undisputed fact that all observable are discrete ? Isn't it another fact that space/time coordinate and probabilities are abstraction (not facts) ?

For example a photon falling into a gravitational field will exchange its energy with the "continuous" field in a discrete manner. Isn't it actually falling down some stairs more than some space ?

For the records when I begin my computer career I was a full believer in continuum, and was wondering if the "abstract information/bits" mentality wasn't polluting all kind of sciences. I have completely changed my mind. A fact I learned soon enough is that some abstract continuous thing like a "real" won't fit any set of (f)actual bits (whatever the number of them 8-16-32...)

The Planck constant is staring us in the face for more then a century, and the Universe "facts" are made of light and matter ... not field nor probability.

 

[Zafa Pi]

Hm, the stuff around me doesn't look very discrete to me.

You need to get refitted with glasses giving you 20/10³⁵ vision.

 

[OCR]

You need to get refitted with glasses giving you 20/10³⁵ vision.

Did you mean... 6.096/3.048³⁵ vision ?

 

[Zafa Pi]

Did you mean... 6.096/3.048³⁵ vision ?

Those would only cost half as much

 

[OCR]

Those would only cost half as much

No... !
They would cost two as much...

giving you 20/10³⁵

Did you mean... 6.096/3.048³⁵... ?

I think you are being contentious, and trying to make me look as a spectacle, simply because your unit is so short...

You need to get refitted with glasses giving you 20/10³⁵ vision.

Lol, I'll bet if I wore those glasses, and stood on your shoulders... I could really see alot ... .

.