Cool anniversary (1/25): CERN discovered the W-boson (UA1 experiment) exactly 30 years ago, two months after their first W candidate; there was a press conference. Via Joseph S.Lectures on Quantum Mechanics by one of the world's most achieved living physicists may be grabbed from the bookshelves; click at the amazon.com link on the left side.
A pulsar with a button periodically switched by ET aliens in between two regimes, to broadcast a binary message to us, was found. This answers the question "Where are they?" The new question is "What are they talking about?" Via The Register.
Aside from the Weinbergization of lots of the usual technical topics you expect in similar textbooks, there is also a section, Section 3.7, dedicated to the interpretations of quantum mechanics.
One may see that Weinberg's views have changed. Unfortunately, the direction of the change may be associated with the word "aging".
Lots of web pages such as a Facebook Weinberg fan page and John Preskill's blog (comments) quoted the most characteristic sentences in the book:
My own conclusion (not universally shared) is that today there is no interpretation of quantum mechanics that does not have serious flaws, and that we ought to take seriously the possibility of finding some more satisfactory other theory, to which quantum mechanics is merely a good approximation.I think that I remember the Czech translation of Weinberg's "Dreams of a Final Theory" that I bought in 1999. At that time, I was already completely certain that quantum mechanics – in the founders' (refined or unrefined) interpretation – made a complete sense and it was a complete theory linking observations to mathematical objects and able to make (probabilistic) predictions.
And my memory indicates that Weinberg just confirmed my conclusion that the universal postulates of quantum mechanics were exact. They had to be a final answer to the "foundational questions" and they couldn't be deformed.
By now, Steven Weinberg has revised his opinions and – because "no interpretation looks good enough" to him – he believes that quantum mechanics isn't exact, isn't the final story, and so on. Too bad. The basic axioms and postulates of quantum mechanics are rather crisp and easy and if there is a problem with them, I wonder why Steven Weinberg didn't analyze them (rather important questions in science) and didn't articulate their hypothetical problems about 30 or 40 years ago when no one had doubts that his brain was among the 10 most penetrating and reliable physics brains in the world.
However, there are also comments about the foundations of quantum mechanics in his new book that are right on the money. In particular, Weinberg says that all attempts to derive Born's rule for the probabilities out of something "more fundamental" seem to involve circular reasoning.
Circular reasoning is found everywhere – for example, in the very meme that the proper Copenhagen-like interpretations of quantum mechanics are "incomplete". Whenever people say such a thing, it's because they first convince themselves that there must be a whole skyscraper of mechanisms that explain the rules of quantum mechanics using "something more fundamental". Then they search for possible forms of this "more fundamental skyscraper" and when they do it well, they find out that none of the candidates really works. Therefore, they conclude that there's a problem with the interpretations of quantum mechanics – even though their failure only actually proves that there is a problem with the assumption that there is something else and "deeper" to be found about the foundations of quantum mechanics.
But let me return to Weinberg's more specific claim, "published derivations of Born's rule from something else involve circular reasoning". I want to mention one example I was forced to get familiar with, Brian Greene's "The Hidden Reality" that I translated to Czech. In general, it's a very good book about all types of parallel universes one may encounter in physics. The book has many flaws, too. In particular, the whole quantum mechanical chapter is a deeply misguided promotion of the Many Worlds Interpretation and "realism" in quantum mechanics where pretty much every sentence is invalid even though all the misconceptions are presented with Brian's extraordinary clarity. But there are also some would-be "new discoveries" in the chapter that aren't new at all and that don't prove or derive what Brian claims to be proven or derived. I hope it's OK to copy a part of the end note #9 for Chapter 8:
Over the years, a number of researchers including Neill Graham; Bryce DeWitt; James Hartle; Edward Farhi, Jeffrey Goldstone, and Sam Gutmann; David Deutsch; Sidney Coleman; David Albert; and others, including me, have independently come upon a striking mathematical fact that seems central to understanding the nature of probability in quantum mechanics.As I will discuss in some detail, this list is totally misleading because the "finding" is a general property of all probabilities and it was totally comprehensible to the founding fathers of quantum mechanics, too. For example, if you watch the lecture Quantum Mechanics In Your Face by Sidney Coleman – one of the physicists in Brian's list – he explains this very same argument (without some of the philosophical conclusions that don't follow from the argument) but he also tells you, at the very beginning, that nothing in his lecture is new – except for his style of presentation of these things.
For the mathematically inclined reader, here’s what it says: Let \(\ket\psi\) be the wavefunction for a quantum mechanical system, a vector that’s an element of the Hilbert space \(\HH\). The wavefunction for \(n\) identical copies of the system is thus \(\ket\psi^{\otimes n}\). Let \(A\) be any Hermitian operator with eigenvalues \(\alpha_k\), and eigenfunctions \(\ket{\lambda_k}\). Let \(F_k(A)\) be the “frequency” operator that counts the number of times \(\ket{\lambda_k}\) appears in a given state lying in \(\HH^{\otimes n}\). The mathematical result is that\[But this "discovery" is a trivial consequence of probability theory. It doesn't depend on any new fact about quantum mechanics. At most, it is one consistency check you can make if you want to verify that the probabilities predicted by quantum mechanics are consistent with the usual rules of probability theory.
\lim_{n\to \infty} [F_k(A) {\ket\psi}^{\otimes n}] = \abs{ \braket{\psi}{\lambda_k} }^2 {\ket\psi}^{\otimes n}.
\] That is, as the number of identical copies of the system grows without bound, the wavefunction of the composite system approaches an eigenfunction of the frequency operator, with eigenvalue \( \abs{ \braket{\psi}{\lambda_k} }^2\).
While the formalism above may look intimidating, its essence is completely simple. It says that if some property is predicted to appear with probability \(p\) and you repeat the same experiment \(n\) times, then the probability is nearly 100 percent that the property will appear in \(pn\pm 5\sqrt{pn}\) cases, i.e. in \(pn\) cases with an error margin that becomes tiny, relatively speaking, as you send \(n\to\infty\). The number \(5\) meant that I wanted a 5-sigma certainty that we will be inside the interval. So by combining \(n\to\infty\) independent propositions of the same form with the same probability to be true \(p\) and by counting, you may construct propositions about the number that are almost certainly true.
This fact is true pretty much by the definition of probabilities: it's what the notion of probabilities means according to the frequentists. In other words, you may prove it by deducing the binomial distribution simply from the distributive law applied to the power \([p+(1-p)]^n\) and from a separation of different powers of \(p\). And this simple claim that is true by the definition of probabilities – and that was true even in statistical physics applied to models of classical physics – was pretty much just translated to the notation of quantum mechanics. Claims about the numbers' having a value were translated to eigenvalue equations; the calculated probabilities were translated, via Born's rule, to the squared absolute values of the complex probability amplitudes.
This is a remarkable result.No, it's not. Again, it's at most one of the trivial consistency checks one can make to verify that the probabilities predicted by quantum mechanics are compatible with the rudimentary, general, frequentist properties of the notion of probability. Also, one doesn't have to make infinitely many consistency checks one by one. Instead, one can verify that all axioms of general probability theory are satisfied by the quantum-predicted probabilities which implies that all conceivable consistency checks like that would work.
Being an eigenfunction of the frequency operator means that, in the stated limit, the fractional number of times an observer measuring \(A\) will find \(\alpha_k\) is \( \abs{ \braket{\psi}{\lambda_k} }^2\) – which looks like the most straightforward derivation of the famous Born rule for quantum mechanical probability.As Weinberg correctly said about similar "derivations" in general, this derivation boils down to circular reasoning. Brian Greene concluded that an identity for the limit of some ket vector acted upon by some operator implies that we may say something certain about the number \(pn\) of repetitions of the same experiment, with \(n\) repetitions in total, when a property was satisfied.
But in this claim, it's being assumed that the (nonzero) deviations of the state vectors in the sequence from the ultimate limit of the sequence "don't matter" when \(n\) is large. However, for any finite \(n\), they do matter. If you want to calculate how large \(n\) has to be for your confidence level to exceed a certain threshold (e.g. to discuss the reasonable deviation away from \(pn\) that you may expect, and it is of order \(\sqrt{pn}\)), you will need to know and use Born's rule for the probabilities \(0\lt p\lt 1\) of individual repetitions of the experiment, anyway.
At most, Brian "derived" Born's rule for propositions whose \(p=1\) – by assuming a "simpler, more plausible, special form" of Born's rule for such \(p=1\) propositions – while he allowed himself to be sloppy about the quantification of the small differences of \(p\) from one and about the rigorously required calculation how small they actually are (and whether they are small). At any rate, the reasoning is circular. One may use the general Born's rule for any value of \(p\) and derive the same thing, or one may use the special Born's rule (eigenvalue equation) for propositions that happen to have \(p=1\) and derive the same thing (which is useless at the rigorous level, however, because no nontrivial propositions with \(p=1\) exactly may be constructed out of propositions with general values of \(p\)).
It shouldn't be surprising that all derivations of Born's rule for the probabilities have to be circular – simply because Born's rule is a fundamental and concise enough postulate of quantum mechanics. Probabilities play a fundamental role in quantum mechanics so they can't honestly be derived from anything simpler or more fundamental. If someone has managed to run through consistency checks such as Brian's consistency check, it should assure him that the way how quantum mechanics incorporates probabilities is totally smooth, natural, and internally consistent. It should weaken his or her attempts to "fight" against the foundations of quantum mechanics. And because things would fail to work if anything were "deformed" away from the rules of quantum mechanics, the consistency suggests that the universal laws of quantum mechanics can't be deformed at all. Too bad that so many people – including Brian – try to interpret the success of these consistency checks exactly in the opposite way!
From the Many Worlds perspective, it suggests that those worlds...I decided to terminate this quote because it's getting preposterous at this point. Brian effectively tries to argue that the trivial translation of the frequentist definition of probabilities to the formalism of quantum mechanics proves the Many Worlds Interpretation. It surely doesn't. At most, Brian attempted to present a "story" whose goal is to demonstrate the compatibility of the quantum mechanical probabilities with the Many Worlds paradigm; even if true, this compatibility would be very far from "proving" the Many Worlds paradigm.
But the truth is that these two things aren't even compatible. The Many Worlds paradigm isn't compatible with the very fact that individual questions usually have probabilities \(p\) that are strictly in between \(0\) and \(1\) and that are, by the way, almost always irrational numbers. This can't really be achieved with "many worlds" at all. For any finite number of many words that are "equally likely", the probabilities will be rational (repetitions of the same world are needed to allow a trivial and generic situation, namely that \(p\) differs from \(1/2\) at all). And if you pick infinitely many worlds, in an attempt to approximate an irrational value of \(p\), the probabilities will be indeterminate form of the type \(\infty/\infty\), so they will be ill-defined.
This has to be combined with all the other severe diseases of the Many Worlds paradigm – no sensible or natural rule "when" the worlds should split and why, failure to obey conservation laws, failure to acknowledge that an arbitrarily large yet finite system always has a nonzero chance to "recohere" so the apparently irreversible "splitting of the worlds" should really never occur, and so on. If I summarize the flaws, the Many Worlds Interpretation contradicts the fundamental fact that the conditions for two possibilities to be already "decohered" are intrinsically subjective conditions, depending on the observer's choice of questions (and her choice of the set of consistent histories), her desired accuracy and confidence level, and other things. A fundamental, undebatable goal of any version of the Many Worlds Interpretation is to make these intrinsically subjective and fuzzily defined "events" look objective and this basic confusion of subjective and objective facts about the real world makes all conceivable mutations of the Many World Interpretation deeply flawed.
In Brian's comments about "derivations of the probabilities", and many similar philosophically oriented remarks about quantum mechanics, the illogical and flawed steps appear pretty much in every step. If Brian were a student who would use so many sloppy steps and incorporated so many logical errors in a calculation of some technical issue that isn't connected with philosophy and widespread misconceptions, he would just fail the exam and that would be the end of the story.
But because the misconception that the probabilities in quantum mechanics (and, more generally, postulates of quantum mechanics) aren't fundamental and exact is so incredibly widespread, all these numerous errors just "don't matter". Brian's end notes – and tons of articles with similarly flawed content – are viewed as OK simply because there are always lots of prejudiced people who feel "certain" about the totally invalid assumption of "realism behind quantum mechanics" and who are therefore ready to forgive an arbitrary number of mistakes as long as the basic spirit of the conclusion agrees with their prejudices.
As I have written many times, this belief in realism is analogous to any other religious belief. People's rational thinking simply gets turned off as soon as they hit questions that could threaten some opinions and assumptions that they view as fundamental for their world view. That's a pity. The advocates of "realism behind quantum mechanics" are running an industry of arguments that is analogous to creationism and its claims that Darwin's evolution has lethal flaws.
Now, Steven Weinberg apparently knows and admits that none of the existing attempts to derive quantum mechanics from something "more fundamental" is more than an example of circular reasoning or, if you want one words instead of two, gibberish. But the older Weinberg is still prejudiced that there should be some "less quantum" foundations beneath the quantum phenomena although this more general prejudice is still scientifically unjustifiable and ultimately wrong.
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