Friday, March 12, 2021

Edwin's understanding of QM: Part II, uncertainty and the weirdness of classical physics

If you haven't watched the first part of Edwin Steiner's online lectures on quantum mechanics, you are encouraged to click and listen to that first part because (like in serious physics), your new knowledge depends on the familiarity with the previous one.

At the beginning, Edwin returns to the very late 19th and early 20th century when the most important physics papers would be written in German. Sadly for Germany, around 1945, the European country lost its best Jewish physicists as well as the non-Jewish, i.e. Nazi physicists (I hope that this internal classification of the German-speaking physicists up to 1945 is accurate and detailed enough for the demanding readers!), and that's why Edwin switched to English as the lingua franca of physics, too.

His second video in the series is 22-minute-long.

At the beginning, you may see some pretty pictures of the founding physicists of the new, quantum mechanical era of physics. Many of them spoke German, the papers often looked like papers written with the latest version of \({\rm \LaTeX}\), and some of the nice pictures show the dark clouds that were flying above physics of that time.

The general topic of this second video in the series is "what was wrong about the classical physics' statements about our world".

Some of the old pictures and illustrations beneath Edwin are cool and they of course convey lots of the extra information that isn't hiding in his words. He talks about the ludicrously incorrect predictions about the atoms that classical physics did. It was unusable, falsified at some points, but quite simultaneously, physicists were learning about some statements that a better theory should be making instead. Some of these statements were observational and looked like statements that a classical theory of physics could have been capable of making. But the actual "classical theory of everything" that they tended to believe because it was the conclusion of centuries or research, as well as any remotely similar theory, actually sharply contradicted these statements.

Edwin says that classical physics was falsified but even in 2021, many people actually believe it. They believe that by giving classical physics a newer name that is designed to be impossible to disagree with, like "realism", they are making it true again. Many pundits don't want to acknowledge that the very general class of these theories, i.e. all of classical physics, has really been falsified by the observations and especially by the pretty clearly valid principles that may be extracted from some classes of observations.

You should subscribe to his channel.

He is going to talk about theories dealing with the location \(X\) and momentum \(P\) of a particle. Because those can be measured, up to the positive uncertainties \(\Delta X\) and \(\Delta P\), any viable theory must accommodate these quantities. The uncertainties represent our inability to distinguish the states. It's not quite a quantitative, precise definition of the uncertainty because our inability to distinguish the states grows gradually as \(\Delta X\to 0\) but his comments are true qualitatively or as order-of-magnitude estimates. The nonzero uncertainty is unavoidable because the apparatuses cannot be perfect and they only convey a finite amount of information (and of digits).

Now, these statements cuold be presented in many generic lectures but he adds something important that we have discussed frequently here – although I would bet that he had also understood it before I wrote it to him. ;-) And the statement is that the concept of the uncertainty also precedes the choice of our theory to describe the observations. It is a big mistake for pundits to claim that quantum mechanics has introduced something of a completely new kind that didn't exist before, a specific quantum uncertainty. It is still the same concept that has to exist when we describe our observations and classical physics had to accommodate the uncertainty, and it is fundamentally the same uncertainty, too. Quantum mechanics, a new theory, just makes some different statements (and statements that may disagree with those in classical physics or other hypothetical theories) about what the uncertainty can be and cannot be, how it affect or doesn't affect other things etc.

OK, around 4:00, he did return to the quantitative definition of the uncertainty but he increases our impatience by stating that "because of basic principles, we can't be certain about our uncertainty". I agree with that but it is not a sufficiently good starting point to make things quantitative. That is why he also does something more practical, and instead of talking about the "uncertainty of uncertainty of uncertainty ... infinitely many times ... of uncertainty", he introduces the continuous measure of our confidence that \(X\) does belong or doesn't belong to an interval, a probabilistic distribution for \(X\) whose standard deviation is \(\Delta X\). He has a very nice animation of some vertical Gaussian distribution, showing 68%. If he created these animations etc., it's impressive, I would need about 100 times more time to make these animations than to record the monologue LOL.

At any rate, because the error margins and inability to distinguish states is unavoidable and because the distinguishability kicks in gradually, we simply need the probability and more generally, we need probabilities to discuss the laws of Nature and phenomena in the world in general. Amen to that. And it is also unavoidable that probabilities (of the validity of statements about the future) don't describe anything objective about the external world and nothing else; probabilities always describe something about the union of the external world and "us", or about our (observer's) perspective to look at the external world. The appearance of probabilities in the physical analysis of a situation isn't a characteritic consequence of a theory of a special kind; it is unavoidable regardless of the choice of a theory due to our limited capacity to observe.

So the question isn't whether we need probabilities to discuss observations and their relationships at all – because we surely do need the probability in that general sense. Instead, the only open question is whether the presence of the notion of probability may always be reduced to our human and technological imperfections; or whether the correct theory implies some uncertainty that doesn't go away even in the hypothetical limit of maximally improved humans and technologies.

And this is the question in which classical physics and quantum mechanics offer opposite answers to each other. In classical physics, the uncertainty may always be explained as an artifact of imperfections of humans and technologies, imperfections that may be assumed to be arbitrarily small; quantum mechanics implies that the uncertainties or some increasing functions of them can't drop below a bound and that's true regardless of the choice of observers and regardless of any human or technological progress (as long as such progress is compatible with the laws of Nature).

At 6:48, readers with amazing observational skills noticed that the color of Edwin's shirt jumped from violet to grey-blue. If you haven't noticed, you should train your skills using a similar Škoda Fabia attention test (a small Czech car produced by a subsidiary of the German Volkswagen Group).

OK, in classical physics, it was primarily possible to assume that situations with simultaneous limits \(\Delta X\to 0\) and \(\Delta p\to 0\) were allowed, that we could know \(X,P\) at the same time. If it were possible, the 2D phase space spanned by the coordinates \(X,P\) behaved as a two-dimensional plane whose arbitrarily small regions if not points were in principle fully distinguishable from each other. As you may expect, quantum mechanics doesn't allow this perfect high-resolution separation of the points in the phase space, not even in principle. Still, \(X,P\) come from the measurement so they must be represented by some mathematical objects within any theory. What these objects are must be left arbitrary. A condition is that after any observation of \(X,P\), there must be a way to reduce the (possibly complicated) mathematical objects representing these quantities to be reduced to real numbers \(x\in\RR\), \(p\in\RR\). Well, the real numbers aren't quite correct because real numbers are dimensionless but let's not be too annoying.

The possibly complex mathematical objects (QM spoilers: operators \(X,P\)) may be identified with the questions that we ask about the physical system; the real numbers \(x,p\) that they get reduced to upon the observation represent the answers. We add the time dependence. In classical physics, \(X(t)\) and \(P(t)\) were commuting, basically real-valued functions of time, and those evolved according to the differential equations that could have been derived from the classical Hamiltonian and Poisson brackets, among other ways to derive them. Edwin nicely pronounces "Poisson" to prove that he belongs to a yet another group of German-speaking physicists. With the Poisson brackets that are quickly reviewed, the differential equations are first-order. The knowledge of initial conditions in classical physics makes the evolution deterministic and unique.

The conclusion is that classical physics assumed that the real world had the objective properties like \(X(t)\) and \(P(t)\) and all the observations and related stuff were just an irrelevant cloning of the real world to our sensors, a one-way street in which the real world affected our sensors but our observations didn't have to influence the external world. This assumption really summarizes what classical (non-quantum) physics means. It doesn't matter how you call the degrees of freedom in the external world, how many of them there are, or what are the exact rules for their evolution in time. If they exist regardless of the observer and the observations may be reduced to an inconsequential copying of the information, it is a theory in classical physics.

But this independence of the external world's reality of the observers and observations is an assumption, not a tautology, and this assumption may be wrong. And be sure that it is wrong in the world we inhabit. Again: this key problematic assumption of classical physics (that was made silently) is that all the quantities that may be measured by some apparatuses separately may also be assumed to have some particular real values simultaneously at every point of time. Was the music at 15:33 the X-Files theme music? The shirt color will jump back to violet now.

To recapitulate: Uncertainties are needed to describe an experimental process but classical physics also assumed that they are not fundamental needed in the theoretical description of what is going on and they may be sent to zero e.g. assumed that they are only nonzero due to some apparatus-related imperfections that may be simultaneously made arbitrarily small. The points in the phase space which has coordinates \(X,P\) were assumed to be reliably distinguishable from each other in principle. Determinism follows: everything that can be measured at a later time is completely determined by \(x(t),p(t)\) at the initial time. Classical physics therefore told us that there was no need to distinguish between known and unknown quantities – and a simple corollary is that the observation is irrelevant for the description because it just transfers some quantities from "unknown" to "known".

Around 17:40, Edwin pointed out that classical physics was bizarre because it made the consciousness of ours unnecessary – in some soft sense, it predicted that consciousness shouldn't really exist. Observations were irrelevant but in this setup, what else the value of quantity should mean, except for the result of an observation, and why should we exist that something this unrelated to observations should exist at all? Also, with the idealized certainty, there is no difference between the future and the past, both are equally precisely determined by the \(t=0\) state at the present.

Classical physics allowed a picture whose information was exactly equivalent to the history of the Universe but that was on par with pictures of mathematical functions. Many people got used to the identification of "having such a function-like mental picture" with "understanding the world" and that's why they don't even want to separate these two different phrases. But there is no reason why "pictures" and "understanding" should be the same thing. But this identification is unnatural because it leads us to assume that our own consciousness is basically non-existent.

Both classical physics and quantum mechanics make predictions about the results of observations. The classical predictions contradict the experimental results. The quantum mechanical predictions are confirmed empirically; classical physics is wrong. More general classes of classical theories, e.g. local realist theories (but even classes of non-local realist theories), have been rather rigorously proven to disagree with the experiments. It is really all the theories which promise a "complete observer-independent picture of the world" that are in conflict with observations. The complete picture doesn't exist and we must abandon this idea to understand modern physics. We must switch to a more general interpretation of the word "understanding" which is about the mapping of the relationships between observations.

In the next, third episode, Edwin will show how observable quantities are represented mathematically and how this representation changes our assumptions about the observations and knowledge. If you have questions, post them under the YouTube videos. If you don't have any questions, post them under the video, too. ;-) Please, send this video to the list of all your contacts, write the YouTube URL on every toilet, and see you next time.

I don't plan to proofread this real time review of Edwin's wonderful video.

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