I guess another interpretation of that statement is more along the lines of "We can't measure how much we are currently unaware of", which seems to make more sense.
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> But one should say: There is a regime where Newtonian mechanics breaks down and Quantum mechanics governs, which is determined by the Planck constant.
I find this completely unsatisfying. I used to think that a theory is either proven wrong or not proven wrong, and Newtonian mechanics is proven wrong. However, I got a new confidence in Newtonian mechanics when I had explained to me that Quantum mechanics yields Newtonian mechanics in a make-believe world where Plack's constant is exactly zero. So, while Newtonian mechanics is proven wrong, it is also (nearly) identical to our best current theory in many circumstances.
That is: According to Quantum mechanics, Newtonian mechanics is correct within the bounds of this exact forumla: insert super complicated formula here
In yet simpler terms: Newtonian mechanics is Quantum mechanics (except that it is always off by a miniscule, totally ignorable, amount. Except in extreme circumstances)
> In yet simpler terms: Newtonian mechanics is Quantum mechanics (except that it is always off by a miniscule, totally ignorable, amount. Except in extreme circumstances)
I believe we are saying the same thing.
Your "as Planck's constant approaces zero" is the same as me saying "in the regime where Planck's constant is irrelevant" (not quote from above, but those are the words I would use).
As you say, one can think of it as an expansion like:
QM = NM + \hbar C_1 + \hbar^2 C_2 + ...
(QM = Quantum mechanics, NM = Newtonian Mechanics, \hbar = Planck's constant, and C_i are correction factors (your "insert super complicated formula here")). Take \hbar to zero and you are get what you just said. But if you go into a regime where \hbar is significant, "Newtonian Mechanics breaks down", e.g. corrections are of same size (or perhaps even larger).
It might be "unsatisfying" how I try to describe it, but I tried to say the same thing as you did (I think). :-)
The Newtonian model is wrong in some very important ways. Not just 'not quite right' but 'fundamentally in error.'
The fact that gives good-enough answers for a whole class of problems is misleading. So did the old theory of planetary epicycles.
Relativity and Quantum Field Theory use different models which give accurate predictions for a much wider class of problems. But the models make the difference. The math comes afterwards.
It's true if you make some simplifying assumptions in the equations they reduce to Newtonian mechanics. (Well - relatively does. QFT is weirder. Part of it reduces to plain vanilla Maxwell, but it also includes forces and interactions Maxwell never had to worry about.)
That doesn't mean the Newtonian model is just as good. It means it's been demoted to a simpler toy model you can use for certain problem - as long as you understand it's just a toy.
Science is really about model making, not math or approximation. The math just gives you ways to use a conceptual model for practical predictions.
The concepts behind the model always come first. And the concepts underlying GR and QFT are much richer and more explanatory than the concepts underlying Newtonian mechanics.
Sooner or later there's going to be another update, and relativity and QFT will be demoted in turn. But they'll be demoted by another conceptual revolution expressed in math, not by more math based on the same concepts.
Not completely sure of what you are saying, but I will try and answer some points.
> Well - relatively does. QFT is weirder. Part of it reduces to plain vanilla Maxwell, but it also includes forces [...]
Indeed. QFT is usually referred to as the more general framework, the actual models that explain our physics are usually called Electroweak theory (QED (Quantum electro dynamics) as a special case) and QCD (Quantum chromo dynamics). These are all QFTs with associated gauge groups (a.k.a. Gauge theories). QED reduces to Maxwell, more or less, in a classical limit.
> That doesn't mean the Newtonian model is just as good.
Oh, absolutely not. But for certain experiments it is preferable to use. The same goes with every theory so far, say GR or QFT, non of them are the final "theory of everything", but I wouldn't say that they are "wrong" or "misleading".
(Disclaimer: I don't know Special relativity, so I might make mistakes that betray that fact ;)
We are certainly in agreement about the facts of the matter :)
The thing I find unsatisfying is specifically the phrasing "Newtonian mechanics breaks down". Breaks down how? Breaks down why? "There is a regime (...), which is determined by the Planck constant." to me reads like, and I am adding color here for effect, "The governor has decreed that on Mondays, Quantum Mechanics shall be in effect. All other days are to be executed with Newtonian mechanics only, except when Max Planck has a belly ache!"
You make it look like QM is a special case of NM, while in reality, QM describes all the phenomena that NM does, not the other way around; NM is a special case of QM. I'm trying to explain it that way instead:
Quantum mechanics is correct. Always, always, always[1] use Quantum mechanics!!! Newtonian mechanics is obsolete!
[1]: Well, this is where my lack of knowledge about Special relativity comes in.
PS: Did you know that in specific circumstances you can use these much simpler formulae instead: (...), and the error is within these totally acceptable bounds: ... ?
PPS: Did you know that these simplified formulae were already known under the silly name "Newtonian mechanics"? The more you know! :)
The reason that I am adding it as a comment is not to correct you -- you are already right -- but to maybe help other readers. QM vs NM did not sit right with me until I understood that NM's equations actually fall out of QM's equations in specific, nameable, circumstances :)
> Quantum mechanics is correct. Always, always, always use Quantum mechanics!!! Newtonian mechanics is obsolete!
Okay, so when we talk about use we are talking about tools. A lot of the tools we got from Newtonian mechanics are pretty reasonably accurate. If I see a ball bounces 3 feet when I drop it from 5 feet up, there's no reason to use anything particularly fancy to guess how high it will bounce if dropped from 10 feet. These are simpler tools I can use, and they may not be great (gravity is weaker at ten feet, for instance, so it's clear I'd be wrong to some extent, forget my ignoring just about every pertinent detail of the ball), but it works. Right tool for the job doesn't need to be fancy.
> You make it look like QM is a special case of NM, while in reality, QM describes all the phenomena that NM does, not the other way around; NM is a special case of QM.
Quantum Mechanics is continuous and differentiable at all points, including in descriptions of e.g. mass. To make Newtonian Mechanics a special case, you'd (at the very least) have to put some gnarly integrals in place of almost every variable in your formulas. They don't _really_ agree, one isn't a subset of another. One is an approximation (and not even specifically of Quantum Mechanics, just an approximation of things-seen-on-earth, some of which have compelling quantum mechanical stories, some of which, like gravity, don't). And sometimes approximations are useful.
I suppose my use of the word "regime" is unconventional. What I mean is a "parameter regime", or "for a certain range of a parameter".
For example, for the range in which \hbar is much smaller than one (in some units), this is the "parameter regime" in which QM and Newtonian mechanics agree quite/indistinguishably well. And for the range in which \hbar is close to one, "Newtonian mechanics breaks down", in the sense that QM effects are large -- i.e. where Newtonian mechanics is no longer a good approximation to QM.
> Quantum mechanics is correct. Always, always, always[1] use Quantum mechanics!!! Newtonian mechanics is obsolete!
I would not agree with phrasing it like that, and it is indeed Special relativity (or a part of it).
If quantum mechanics makes Newtonian mechanics (NM) "obsolete", then what does that imply for Special relativity (SR)?
QM makes corrections to NM with the parameter \hbar. SR makes corrections to NM with v/c (velocities you make experiments at over speed of light). So say that we are doing a mechanics experiment on our desk (say dropping something on a spring, or whatever). And we go with your "always use QM", it would take a long time to write down what happens, the same (although a bit faster) if we were to go with "always use SR".
Before you start writing down what is going on for that experiment, you make an assumption/approximation of which range of parameters is relevant for you. An apple dropped on a spring bed would not have relevant corrections from QM nor SR.
> The reason that I am adding it as a comment is not to correct you -- you are already right -- but to maybe help other readers.
Absolutely! Thats is also my goal in discussing these things. I do not always know how to make them "popular"/less technical, so I'm very interested in hearing other ways of explaining it.
I guess another interpretation of that statement is more along the lines of "We can't measure how much we are currently unaware of", which seems to make more sense.
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> But one should say: There is a regime where Newtonian mechanics breaks down and Quantum mechanics governs, which is determined by the Planck constant.
I find this completely unsatisfying. I used to think that a theory is either proven wrong or not proven wrong, and Newtonian mechanics is proven wrong. However, I got a new confidence in Newtonian mechanics when I had explained to me that Quantum mechanics yields Newtonian mechanics in a make-believe world where Plack's constant is exactly zero. So, while Newtonian mechanics is proven wrong, it is also (nearly) identical to our best current theory in many circumstances.
That is: According to Quantum mechanics, Newtonian mechanics is correct within the bounds of this exact forumla: insert super complicated formula here
In yet simpler terms: Newtonian mechanics is Quantum mechanics (except that it is always off by a miniscule, totally ignorable, amount. Except in extreme circumstances)