Saturday, September 08, 2018

Woit and probability in quantum mechanics

Peter Woit is a veteran physics hater. His website has regularly served the most idiotic anti-physics garbage and it has attracted some of the world's šittiest mammals to its comment section for almost 15 years.

It would have been natural for Woit to be a leader of the anti-quantum zealots, too. String theory is the power of quantum mechanics applied to the realm of gravity. If one is claimed to be "bad", the other must be "bad", too. But for some reasons, Woit has avoided the discussions about the foundations of quantum mechanics. He wasn't inclined to join various branches of the anti-quantum zealots.

Was it because of some surprising relative wisdom hiding in the void of the skull of the crackpot-in-chief? Or was it because of his utter cluelessness? A new text on his blog,
Is Quantum Mechanics a Probabilistic Theory?,
shows that the second answer is correct. And spectacularly so. Woit has clearly never spent a second by thinking about the foundations of quantum mechanics – so he understands it less than the average embryo.

He says that he was led to realize this point while watching a recent talk by Weinberg. Woit's claims get ludicrous very early on:
To explain why, note that I wrote a long book about quantum mechanics, one that delved deeply into a range of topics at the fundamentals of the subject. Probability made no appearance at all, ...
Oh, really? So a probability made "no appearance at all" in a "long book" about quantum mechanics? One that "delved deeply at the fundamentals"? That's juicy, indeed, because all connections between the mathematics of quantum mechanics and the real world are made through probabilities and existentially depend on probabilities.

Writing a "long book about quantum mechanics" with no probabilities in it is just like writing a long book about religion without God or the concept of divinity, about Christianity without Jesus Christ, or about Tesla Inc. without Elon Musk. It just doesn't add up. It's a contradiction. A book that avoids probabilities cannot be a book about quantum mechanics at all, let alone one that "delves deeply at the fundamentals".

Quantum mechanics is a scientific theory and their very purpose is to make predictions about Nature. All the generic and elementary predictions that quantum mechanics can make are propositions about probabilities. The probability of one outcome is \(P_1\), of another outcome is \(P_2\). The uncertainty principle – an underlying fundamental feature of quantum mechanics – guarantees that probabilities that are strictly between 0 and 100 percent are unavoidable. If you avoid discussing probabilities, you avoid discussing any physics. You may mathematically masturbate without mentioning probabilities but the masturbations will have nothing whatever to do with natural science.

Thankfully, the ludicrous sentence continues and the rest of the sentence softens it a little bit:
...other than in comments at the beginning that it appeared when you had to come up with a “measurement theory” and relate elements of the quantum theory to expected measurement results.
So probabilities appear in the "long book about quantum mechanics", after all. But they are confined to "comments at the beginning" that say some things about the "measurement theory". It's still ludicrous because probabilities appear whenever we physically interpret anything that we have calculated in quantum mechanics. All the complex numbers calculated by quantum mechanics (e.g. by path integrals) are probability amplitudes or factors in expressions for probability amplitudes. But at least, a small comment at the beginning of his "long book" could have something to do with physics, unlike the rest of the "long book".

In the remainder of Woit's short blog entry and the comments, we see the standard game of anti-quantum zealots. He demands a precise definition of the measurement. It needs to have some properties [that are normally called classical physics]. So people like Aaronson informed Woit that with those conditions, he is bound to accept Bohmian mechanics or something very close to it. What a shock, Woit is another full-blown anti-quantum zealot.

OK, his demands for a "definition of measurements" and the "right interpretation of quantum mechanics" are the main content of that blog post:
One difficulty here is that you need to precisely define what a “measurement” is, before you can think about “deriving” the Born rule [...]

So, my question for experts is whether they can point to good discussions of this topic. If this is a well-known possibility for “interpreting” QM, what is the name of this interpretation?
The last paragraph starting with "So" is really cute. A guy who has pretended to know something about cutting-edge theoretical physics asks "experts" to explain the rudiments of modern science to him. Cute. It's flabbergasting that some people may consider themselves "experts" when it's so obvious that they don't have the slightest clue about the field. You don't really need an "expert" to answer these basic questions about quantum mechanics (Woit calls for an "expert" to make him look important). A competent instructor in an undergraduate course is enough.

Now, let's answer those elementary questions. Yes, to apply quantum mechanics in a completely controllable way, "you" need to precisely define what a measurement is. But an important question is who is the "you" in that sentence. It is you, the observer. It is the "duty" of the observer to precisely define what events are the measurements and what aren't. And if he wants the predictions to be reliable, he must consider the input – including the values of the initial measurements – to be reliable, too.

You can't make controllable (or precise) predictions of the future if the data about the problem are uncontrollable (or imprecise). What a shock.
The measurement is the change of the observer's knowledge about an observable \(L\), a physical quantity that is mathematically represented by a Hermitian linear operator. Mathematically, the change of the knowledge is represented by an update of the wave function \[

\ket\psi \mapsto \ket{\psi_{\rm after}} = P_{L=\lambda}\ket\psi

\] described as an action by a Hermitian projection operator \(P_{L=\lambda}\). The resulting state vector is an eigenstate of \(L\) i.e. \[

L\ket{\psi_{\rm after}} = \lambda \ket{\psi_{\rm after}}.

\] Only eigenvalues \(\lambda\) of \(L\) are possible outcomes and the probability of a particular \(L=\lambda\) outcome is given by Born's rule:\[

Prob_{L=\lambda} = \frac{ \bra\psi P_{L=\lambda} \ket \psi }{ \langle\psi\ket\psi }

\] The projected post-measurement wave function \(P_{L=\lambda}\ket\psi\) should be substituted as the initial state \(\ket\psi\) to similar predictions of subsequent measurements.
That's it. Nothing is missing here. It's the complete "measurement theory". One may give extra pedagogic comments but they are not "needed" as a part of the definition. For example, one may explain that the abrupt change of the wave function is a complex generalization of the step in Bayesian inference where the prior probabilities \(P(H)\) get replaced by posterior probabilities \(P(H|E)\). The collapse plays the same role – it's a quantitative description of the change of the observer's knowledge. For this reason, it's correct to say that the "collapse" occurs in the observer's mind, not in "reality".

The rule in the blockquote above may be named the "Copenhagen interpretation" but the interpretation is a silly word – one that Heisenberg coined in the 1950s. Before his book was out, he correctly predicted that the word "interpretation" would be abused by tons of crackpots who have their "alternative interpretations". Well, maybe Heisenberg encouraged them by this "advise". At any rate, it has happened. The popular book market and the philosophy departments are flooded by crackpots with their own "interpretations". But the correct rules, the "Copenhagen interpretation", aren't optional. They are an essential part of quantum mechanics – in fact, they are the very heart of the theory. Without such rules, the mathematical formalism of quantum mechanics cannot be connected with Nature or natural science at all! Well, every proposed connection of the mathematical formalism with Nature that avoids probabilities can be quickly shown to be a wrong theory of Nature.

Whenever we answer a question, like the question "what is the right and complete measurement theory", it's always possible to keep on asking "why" but it's not always productive to keep asking "why". In my definition, I said that the measurement is the "change of the observer's knowledge". So we may demand a definition of "the observer" and "his knowledge", too.

Be free to demand such a definition. But they're elementary building blocks in quantum mechanics – and in most of the human thought, too. In axioms of set theory, we don't indefinitely answer the question "what a set is" and "what is the thing we have used in the previous answer". Instead, we write down the rules that the entities assumed to exist – sets – satisfy. If you don't know what your knowledge is and whether it has changed by your perception, then you cannot do science. If you don't know whether you have found some information about Nature, it's your fault, and a debilitating one, not a fault or incompleteness of quantum mechanics.

Even if the theory added some extra pages that would be "defining what a measurement is", these pages would be totally useless if the observer didn't know whether the conditions were actually met in his real-life situation. To apply any scientific theory, the observer must know something about the events in his real-world situation. Quantum mechanics demands that the observer knows whether he has learned the value of an observable – it's the elementary thing he must be able to distinguish.

In classical physics, the problem would exist as well. Classical physics may predict the locations of planets \(\vec x_a(t_2)\) at a later time from the locations and speeds of the planets at an earlier time, \(\vec x_a(t_1),\vec v_a(t_1)\). If you aren't sure whether you have found the (correct) values of the information about the initial time, e.g. because you don't know whether you should trust your new Chinese telescope, you can't be sure about the predictions about the later time, either. Is it shocking? It's not. There can't be any universal definition that would permanently remove all uncertainty that may arise in the real-world situations. The case of quantum mechanics is analogous; quantum mechanics isn't "less satisfying" or "less complete" in this sense at all. Quantum mechanics, to be applied, also demands that the observer knows whether he measured the initial state and what the outcome was, just like classical physics does in my planetary example. The information about the initial or final state is described as knowledge about observables that is obtained by observations. In classical physics, the observables are coordinates on the phase space. In quantum mechanics, they are Hermitian linear operators on the Hilbert space. Those are mathematically different representations of observables but both of them are equally complete and logically consistent.

Indeed, this observer-dependent definition doesn't guarantee that the identification of "observations" will be universal and objective. And you know why? Because it is not. The very nature of the observer and his or her or its or their observation is subjective in general, at least in principle. There is no set of "objective answers about the objects in the real world". If there were objective answers, it would be classical physics, not quantum mechanics. Each observer is supposed to have his own subjective axiomatic framework to determine which propositions about the state of Nature are valid and which are not. Those truth values are given by his observations which are subjective in principle.

I repeatedly say "in principle" because in practice, most or many or all observers will accept the same (or very similar) truths about the observables, especially about properties describing macroscopic objects. And quantum mechanics indeed does predict the agreement between the observers' conclusions that we normally experience. So when we say that the observations are "subjective", it does not mean that the propositions about Nature made by two observers are completely random, chaotic, or uncorrelated with each other. They are heavily correlated in most everyday situations – because the observers may observe each other which links their knowledge about the rest of the world, too. But the two observers' knowledge simply can't be assumed to be precisely identical or arising from a single, objective description. That's not a disadvantage, inconsistency, ugly feature, sign of incompleteness, or anything "bad" like that. Instead, it is a novel, consistent, and paramount feature of quantum mechanics, a framework of physics that has superseded classical physics.

Quantum mechanics was born 93 years ago but it's still normal for people who essentially or literally claim to be theoretical physicists to admit that they misunderstand even the most basic questions about the field. As a kid, I was shocked that people could have doubted heliocentrism and other things pretty much a century after these things were convincingly justified. But in recent years, I saw it would be totally unfair to dismiss those folks as medieval morons. The "modern morons" (or perhaps "postmodern morons") keep on overlooking and denying the basic scientific discoveries for a century, too! And this centennial delay is arguably more embarrassing today because there exist faster tools to spread the knowledge than the tools in the Middle Ages.

One more point. Lots of readers at Woit's blog propose that "this lack of knowledge about the initial state [can] explain why your predictions about measurements have to be probabilistic". Joking is the only reader who gives the only correct response:
No. Probabilities and Born’s rule apply even when you have complete knowledge of the entire pure state. So unitary evolution forbids this kind of mechanism. The environment is necessary to make the quantum mechanical probabilities behave like classical probabilities (with high probability, and for the pointer states).
Yup, it's the point of the uncertainty principle that uncertainty about the final measurement is unavoidable even if you measure the initial state as accurately as you can. Joking also correctly adds:
Indeed, there is no non-circular derivation of the probabilistic interpretation of quantum mechanics. The standard fallacy is to say that a probabilistic description follows if we neglect terms that are small in the norm. But neglecting small norms because they correspond to low probability is exactly assuming what one is trying to derive!

[...] There’s nothing wrong here. Assuming the relationship between physical measurement and mathematical theory to be probabilistic, Born’s rule is the only possibility.
Yup. Born's rule cannot be derived without assuming something (almost) equivalent – because some claim about probabilities in the theory has to appear somewhere for the first time, it can't appear out of a thin air, so it must be an axiom. And incidentally, the right to "neglect negligible probabilities" wouldn't be a good axiom for precise calculations because you don't know "how much is still negligible". So you should better assume well-defined quantitative statements and the precise Born's rule that quantifies the probabilities is such a quantitative axiom that you need.

And the last paragraph is true while slightly nontrivial. The probabilities have to be proportional to the second power of the absolute values of the amplitudes because when your measurement projects the state on a 2-dimensional space, the amplitudes \(c_1,c_2\) are left to describe the post-measurement vector and \(|c_1|^2+|c_2|^2\) is the only function of the amplitudes that behaves additively, as probabilities for mutually exclusive options 1,2 should.

The sum of squares – the norm – is what is preserved by the allowed evolution or by the allowed changes of the bases which are given by unitary operators. So the unitary change of the allowed transformations (unitary means preserving the sesquilinear norm, like complexified rotations that respect a Pythagorean theorem) is more or less equivalent to the second power's being the only consistent one for Born's rule assuming that something like Born's rule exists at all.

But the unitary evolution of the wave function – without an interpretation of the wave function – makes no statement about physics yet. It only constrains what interpretations can be given to the wave function i.e. constrains on the possible detailed choice of Born's rule. To connect the mathematical objects with physics at all, one has to interpret the wave function. It's done by Born's rule, it's the only one that is consistent with the generally needed additive properties of probabilities given the legal status of the unitary transformations. (The unitary transformations underlying QM are also sort of mathematically unavoidable, forming the There’s nothing wrong here. Assuming the relationship between physical measurement and mathematical theory to be probabilistic, Born’s rule is the only possibility.only large or generic enough closed-under-composition group of transformations that may meaningfully act on an infinite-dimensional linear space.)

So Born's rule is consistent, nice, and rigid but it still needs to be added as an axiom, otherwise the mathematical formalism of quantum mechanics would have no relationship with Nature and science at all.

No comments:

Post a Comment