It covers physics from the viewpoint of parallel worlds. This term may have many different meanings that are related to each other in some cases and that are mostly unrelated (and shouldn't be confused) in other cases. But they still make a nice unifying theme for a book that allows the author to cover many topics.
In the initial chapters, Greene explains the modern cosmology beautifully. Is the Universe finite or infinite? Is the expansion accelerating? In a very large Universe, regions may be causally separated and they form a "quilted multiverse" if the total volume is infinite. He adds inflation with its "inflationary multiverse" and explains how the bubbles expand and how they look infinite from inside and finite from outside.
When the topics get more stringy, there are beautiful explanations of the quantum tunneling, with bubbles inside bubbles (which didn't occur in the pre-stringy inflationary cosmology) and the landscape of the stringy vacua. Greene talks about the "brane multiverse", "cyclic multiverse" (selling its marketing points but admitting that everyone thinks that it's babbling), and "landscape multiverse". Later in the book, he would discuss holography (which isn't really a multiverse, but he uses the word here as well), a Platonic multiverse of all mathematical structures, and a few other things.
The landscape multiverse
Of course, the things get controversial when he turns his attention to the anthropic principle. It's still mostly balanced but I think that he vastly underestimates the extent to which the "principle of mediocrity" is ill-defined and probably incorrect. In the anthropic principle as clearly described by Greene, we should expect the probability "P" that a parameter of Nature has value "X" to be the fraction of the observers "P" in the whole multiverse that see the value "X" of the parameter in their surrounding environment.
Fine. Greene thinks that the only problem with this prescription - aside from the difficulty of the relevant maths - is the problem of "infinity over infinity" (illustrated by the question whether the set of even integers is bigger than the set of all integers). There are infinitely many observers and each subset is infinite as well and infinite numbers can't be canonically divided by each other because you don't know how to order the infinite sets so that they could be compared.
I don't think it's the only problem. I don't even think it's the main difficulty of the "measure problem". Even if the multiverse and its history were finite, there would be serious problems with the anthropic reasoning. The main problem is that one doesn't know and cannot know what he counts as an "observer" - is that just humans, or all their cells? Do smarter observers have a bigger weight? Is their measure multiplied by their lifetime and/or the number of thoughts they can make in one life? Are the basic measures proportional to volume of space (at some time) or the volume of spacetime? And so on.
More importantly, he doesn't really admit the key point that the "mediocrity principle" in any form is almost certainly wrong. It's just one particular measure - an attempt to define uniform priors (which can't exist) - but a valid theory can use any other, non-uniform measure as well and chances are that if a measure on the landscape is relevant, it's a highly non-uniform one, whether you view it as a measure on the universes or the observers. There is absolutely no rational reason why the uniform measure should be the right one (someone's being a Marxist who loves egalitarianism everywhere is only an utterly irrational reason).
I don't want to talk about the anthropic principle too much here. At least, I am happy that as far as I can see, The Hidden Reality contains nothing of the Sean-Carroll-style crackpottery about the duty of the early Universe to have a high entropy that have plagued some chapters in Greene's second book, The Fabric of the Cosmos.
But right now I am going through the quantum chapter and I can't believe my eyes. In The Elegant Universe, Chapter 4 was actually my most favorite chapter. In my opinion, Greene did a superb job in explaining basics of quantum mechanics over there. And most importantly, there was nothing conceptually wrong about what he wrote, despite the technical simplifications that were needed in a popular book. (The chapter largely avoided the subtleties of "interpretations".) In fact, as far as I remember, that could be said about the whole book.
In this case, it's much more accurate to say that there is nothing right about what Greene wrote about the interpretation of quantum mechanics. It contains pretty much all the laymen's misconceptions one can routinely hear and read in various popular and sometimes even "not so popular" books. All this stuff is frustrating because as far as I can see, there doesn't exist a single popular book about the interpretation of quantum mechanics that would be basically right. Correct me if I am wrong. (Maybe Zeilinger has written something that makes sense?)
So the chapter about the "quantum" or "many worlds multiverse" is a sequence of dozens of pages filled with irrational criticisms of Niels Bohr - who is painted as a really bad guy - and against conventional quantum mechanics. Pretty much every argument is wrong - it's really upside down. Of course, I am professionally translating exactly what the author wrote which doesn't mean that I don't suffer. ;-)
One may say that Greene is squarely in the camp of anti-quantum zealots (I think that this is what he means the "realist camp"), no doubts about that. How the same physicist could have written a totally OK introduction to quantum mechanics in The Elegant Universe is not clear to me. ;-)
Everett's and other misconceptions
So the reader learns about Hugh Everett III, a grad student who wrote a heroic thesis in which he outlined his opinion that Niels Bohr was an idiot and that the probabilistic interpretation of quantum mechanics had to be replaced by something else. John Wheeler, his adviser, liked it, but Wheeler was bullied by the evil Niels Bohr whom Wheeler had to worship, Greene essentially writes, so Bohr and Wheeler forced poor Everett to censor and dilute his thesis. :-)
The reality is, of course, that Everett's original thesis was full of complete junk about the non-existent problems of proper quantum mechanics and Niels Bohr kindly explained to John Wheeler why this stuff was junk. So Wheeler made it sure that Everett wouldn't write this junk into the final draft of his thesis because no person should get a physics PhD if he thinks, in the 1950s or later, that proper quantum mechanics is inconsistent.
Unfortunately, as of 2011, Brian Greene - and all other authors of popular books about quantum mechanics, to be sure that I am not singling him out - still misunderstands why the criticism of quantum mechanics has always been and remains rubbish.
The main problem that shows that "Bohr is in trouble" is that one needs to get from the ambiguous wave function to the definite outcomes. But Schrödinger's equation doesn't allow such a collapse, so "Bohr is in trouble", we repeatedly read. Holy cow. It's really Greene et al. who is in trouble because he doesn't want to admit that the wave function is just a probability, not a real wave, even though it's clearly needed to explain the experiments that yield clear outcomes.
Greene constantly uses the term "probability wave" for the wave function but he doesn't mean it. It is clear that from almost every sentence that he doesn't believe that the wave function only contains the information about the probabilities. He is still imagining a classical wave.
This is highlighted by pretty much all the details. For example, in the book's discussion, all particles have their "waves" (plural). However, there's only one state vector - or a wave in a higher-dimensional space - that describes the whole world. This is not really a detail. This is a totally essential thing for the discussion of any interactions in a quantum mechanical theory and the measurements in particular. The wave function remembers and has to remember all the correlations between the individual particles.
All these things were beautifully described by Sidney Coleman in his Quantum Mechanics In Your Face lecture. Around 2005, Brian Greene and his friends at Columbia asked me to make the talk located somewhere in Harvard's archives available to them (they wanted to use it somewhere) - which I/we did, if I remember well - but Greene had to be very disappointed because this talk by Coleman proves very clearly why exactly the opinions about quantum mechanics held by Greene are fundamentally wrong.
Does normal quantum mechanics predict that we will see a "mixed haze" of different outcomes? The answer is a clear No.
As Coleman emphasized, to discuss these questions, we have to describe not only the microscopic particle but also the apparatus - or the humans - in terms of quantum mechanics. When we do so correctly, we may easily prove that the humans always perceive a definite outcome.
The first proof of this kind was pointed out in the early days of quantum mechanics by my English great granduncle Nevill Mott. He considered a bubble chamber and asked why the particles leave clear tracks going in one direction instead of some diluted isotropic fog.
Well, one may prove - and he did prove - that there is a correlation between the bubbles' angular direction. So if the particle creates the first bubble in the Northern direction, one may show that the momentum points into this direction and the second bubble will also be in the Northern direction. This is a statement that can be clearly shown in a quantum mechanical description of the particle plus the whole bubble chamber.
The same thing holds for another basis vector in which the particle moves into another direction, and so on. So if we define the operator "P" that has eigenvalues 1 or 0 if the first two bubbles appear in the same direction or not, respectively, then we can show that for any initial state of the particle in the bubble chamber, the eigenvalue is 1. It follows that the eigenvalue is 1 for any linear superposition, too.
Brian Greene probably uses a nonlinear operator whose eigenvalue is 1 ("sharp track/answer") when acting on basis vectors North or South, but 0 ("fuzzy dizzying track/answer") when acting on any nontrivial linear superposition of North and South. But such a "Greene operator" violates a basic postulate of quantum mechanics that all observables and properties of physical systems (and all perceptions and any other phenomena and questions with answers that may be talked about) are and must be encoded in linear operators. If you're answering questions about physics with something else than linear operators, you're not doing quantum mechanics, at least not correctly!
In other words, particles will leave a clear, straight track regardless of their initial state. The particle may be in an s-wave which is spread all over the sphere. But that won't change the fact that the angular directions of the individual bubbles stay correlated.
In the same way, one may ask whether Schrödinger will have mixed or sharp feelings after he observes his cat. The cat is either alive, in which case Schrödinger will be happy, or dead, in which case he will be sad. Both happy and sad states have eigenvalue of the operator "Schrödinger has a well-defined feeling" equal to 1, i.e. yes. (I could use the same argument for the cat itself but I chose its owner. Brian Greene explicitly contradicts this simple calculation and says that the humans will have "confused feelings" - seeing a display showing "Grant's Tomb" and "Strawberry Fields" at the same moment - which is just unambiguously and demonstrably wrong.) So any linear superposition has the same eigenvalue. It follows that observers always have well-defined feelings about the cat. I don't need any collapse. There is no collapse.
I can just calculate the probability that the answer is Yes according to the probabilistic rules of quantum mechanics and the results is 100 percent. So the debate is over. 100 percent means that the answer is settled. 100 percent is certainty. Saying that something else is needed by quantum mechanics to answer this question is just pure bullshit. Readers (and writers) of popular books love to paint quantum mechanics as a realm where everything is fuzzy and diluted; it's spread everywhere. But that's not the case. Despite the probabilistic character, proper quantum mechanics perfectly remembers all the correlations that exist - and predicted correlations may often be higher than what any classical theory could achieve (as constrained by Bell's inequalities and their generalizations). In this case, predictions of quantum mechanics are often sharper and more certain than predictions of classical physics. In particular, some statements may be calculated to have 100-percent probabilities. Those statements are certain.
A similar discussion holds for the question why the results of the repeated experiments are random. Design any quantity that measures some non-randomness of the sample. For example, repeat the same experiment measuring a spin (up or down) 1 million times and try to find a correlation between the two measurements that follow after each other - that would prove that the sequence isn't quite random. But you may calculate this correlation from quantum mechanics and it's zero in the limit of infinitely many repetitions. Experiments confirm it. The results are random - any other test will also show that - and this fact can be proved by the basic framework of QM; nothing else is needed.
Again, I don't need to talk about any collapse. This has nothing whatsoever to do with collapse. There is no physical process that could be called the collapse. It's about the validity of a proposition and quantum mechanics has a totally well-defined recipe to calculate the probability that a proposition is right. In these textbook cases (Do observers have sharp feelings? Are the results of independent repetitions of the same experiment uncorrelated?), the calculated probability is 100 percent, which means that quantum mechanics unambiguously implies that these things have answers and they're the answers we know.
Also, Coleman has discussed the alleged "nonlocality" of quantum mechanics. There's nothing nonlocal about entanglement - it's just a damn correlation. In the simplest cases, there isn't even an interaction Lagrangian, so the interactions surely can't be nonlocal. Even in interacting theories, we usually have a strict locality - that's why relativistic quantum field theories are also called local quantum field theories. Phenomena's (and especially decisions') impact on spatially separated other phenomena is strictly zero. The idea of a "nonlocality" in EPR setups is always an artifact of the confused person's attempt to imagine that the wave function is an observable that has to be "remotely modified" to avoid problems. But the wave function is not [an] observable.
Saying that quantum mechanics is incomplete (or even invalid) in its answers to any of these questions is clearly and self-evidently incorrect. Pretty much all the authors of popular books on quantum mechanics want to promote these questions to some huge, religion-scale philosophical mysteries that can never be fully grasped or solved and that break Niels Bohr's teeth and render his theory inconsistent or incomplete. In reality, they're easy homework exercises that an undergraduate student who wants an A from quantum mechanics should be able to calculate within minutes.
Now, if someone faces severe psychological problems in accepting that the fundamental laws of physics make probabilistic predictions, he may try to invent silly ways how to "visualize" what's going on - from imagining the wave function as a "classical wave" (that has to "collapse") to imagining many worlds of Everett. But these psychological problems of the person can in no way justify the statement that there is something incomplete or wrong about proper quantum mechanics.
There's nothing wrong or incomplete about quantum mechanics - and on the contrary, there's a lot of wrong things about all the "theories" of "not quite quantum mechanics". In fact, none of them works as a description of all the known things about the world. Also, as Coleman emphasizes, the very notion of a field looking for "interpretation of quantum mechanics" is based on a misunderstanding. Everywhere in the history of physics, only the previous phenomena described by a simpler approximate theory may be interpreted within a more complete theory. So it makes sense to talk about the "interpretation of classical physics" within quantum mechanics - but not in the other way because quantum mechanics is the more correct, complete, and accurate theory while classical physics is the less correct, incomplete, and approximate one!
Anti-quantum zealots and geocentrism
For those who haven't heard the punch line of Sidney Coleman's lecture, I can't resist to recall it.
At the very end, Sidney paraphrases a wise comment by philosopher Ludwig Wittgenstein. People used to believe geocentrism. Wittgenstein asked why people usually considered such a belief in the past "natural". His friend told him that it was because it "looks like" the Sun is revolving around the Earth. Wittgenstein replied with this key point:
Well, how would it look like if it had looked as if the Earth were rotating? :-)Obviously, it would look exactly like the world around us. ;-)
[It's natural for things to move - and people could have known for quite some time that "free motion" is indistiguishable from the rest (the old principle of relativity, perhaps combined with the equivalence of inertial and gravitational masses).]
In the same way, authors of popular books on quantum phenomena still find it "natural" to think that there is a "realist" or classical picture behind quantum mechanics because it "looks like" there is a "realist" or classical picture in the quantum mechanics. Oh, really?
Well, what would it look like if it looked like that the world is really following the causal laws of quantum physics without any reductions of the wave packets - and not any laws of classical physics - at the fundamental level?Needless to say, it would look exactly as our ordinary everyday life. Welcome home. :-) And thank you for your patience.