A week ago, Scott Aaronson wrote that he is a champion of the "Many Worlds Interpretation" (MWI) even though MWI is slightly more frail than heliocentrism. That's what I call an understatement on steroids.
The term "MWI" is notoriously ill-defined, it may mean everything or nothing or something in between and there is no actual theory of physics that would deserve this name and that would work. But let's assume that the proponents of MWI mean that there exist many worlds and different mutually exclusive properties of a physical system are realized simultaneously.
In the following 40 seconds, let's see that it ain't the case.
Let's take an electron and measure its spin component \(j_z\) via the Stern-Gerlach apparatus i.e. via a magnetic field.
The initial state of the electron is prepared to be "up" with respect to a particular tilted axis – every state of the spin in 3 dimensions is "up" with respect to a semi-axis – so that we have\[
\ket\psi = 0.6 \ket{\rm up} + 0.8 \ket{\rm down}.
\] So the electron will have a 36% chance to have the spin "up" and 64% chance to have the spin "down". Note that it's not just the absolute values of the amplitudes that matter. The relative phase matters, too. If we changed the relative phase of the two terms by the factor of \(\exp(i\alpha)\), it would mean that the axis with respect to which the electron is polarized "up" would rotate by the angle \(\alpha\). Such a rotation may be inconsequential for our measurement of \(j_z\) but it would matter for the measurement of all other components of the spin.
Now, let's ask the key MWI question: will there be an electron with spin "up" as well as an electron with spin "down"?
The MWI proponents say "Yes". They imagine that different possibilities "really occur" in different universes, and so on. So this is the main question that decides about the validity of the MWI. Stupid monkeys are obsessed by questions whether MWI and other things are "not even wrong", "politically correct", "obeying Occam's razor", "pretty", and all such irrational adjectives, but no one seems to care about the question whether it is scientifically false or true.
Quantum mechanics offers a universal rule to answer all Yes/No questions that have any physical meaning, that are in principle observable. For the given question, we identify the projection operator \(P\), i.e. a Hermitian operator \(P=P^\dagger\) obeying \(P^2=P\) (which is why its eigenvalue have to obey \(p^2=p\) as well and they must belong to the set \(\{0,1\}\) i.e. {No, Yes}). The expectation value\[
{\rm Prob} = \bra \psi P \ket \psi
\] is interpreted as the probability that the answer is Yes. Quantum mechanics doesn't allow us to predict anything else than probabilities. So there's always some uncertainty about the answer to the question. The only exceptions are projection operators whose expectation values are equal to \(0\) or \(1\): these values correspond to "certainly No" or "certainly Yes" and there's no uncertainty left.
We will see that the "key question of MWI" is of this sort. The projection operator for a question "A and B" is constructed as\[
P = P_A \cdot P_B.
\] When it comes to operators, "and" is multiplication. That's why Logical AND i.e. conjunction is also known as "binary multiplication". And that's also why the probabilities of two independent questions' having answers "Yes" is equal to the product of probabilities.
Fine, what are \(P_A\) and \(P_B\)? They are projection operators on the subspaces for which the answers to questions A and B are "Yes". In particular, we have\[
P_A = \ket{\rm up}\bra{\rm up}, \quad P_B=\ket{\rm down}\bra{\rm down}.
\] They're projection operators on the "up" and "down" states of the electron, respectively. There are just no other states in the Hilbert space for which the statement "there is an isolated electron with the spin up" or similarly "...down" would be valid. Now,\[
\braket{\rm up}{\rm down} = 0
\] and therefore\[
P = P_A P_B = \ket{\rm up}\bra{\rm up}\cdot \ket{\rm down}\bra{\rm down} = 0.
\] Therefore, the probability that there will be both an electron "up" and an electron "down" is\[
\bra\psi P \ket \psi = \bra \psi 0 \ket\psi = 0 \braket\psi\psi = 0.
\] I've written the derivation really, really slowly so that at least 10% of the stupid monkeys have a chance to follow it. At any rate, we may prove that the probability that the electron exists in both mutually exclusive states simultaneously is zero. It can't happen. The derivation is identical for any other mutually excluding alternative properties of any physical system.
Note that the operators \(P_A,P_B\) commute with one another, i.e. \(P_A P_B=P_B P_A=0\), which means that both questions may have an answer at the same moment (the uncertainty principle adds no extra hassle). That allows us to avoid some discussions.
The simple conclusion is that there aren't many worlds. QED. Get used to it, monkeys. ;-)
Let me now spend some time by discussing how indefensible various "loopholes" would be and why there are many other ways to see that the answer to the question "Are there many worlds?" had to be "No". And I want to mention several likely fundamental and rudimentary errors that prevent MWI advocates from deriving the right answer to this simple question and from seeing that this is truly a kindergarten stuff and not something that they should be confused by for days, weeks, months, years, decades, or centuries.
First, let me discuss the interpretation of the "plus" sign.
As I already suggested, it's important to distinguish addition and multiplication. (If you don't know what multiplication is, watch 0:40-0:45 Miss USA on maths.) The key fact is that the wave function composed of several mutually exclusive pieces such as\[
\ket\psi = 0.6 \ket{\rm up} + 0.8 \ket{\rm down}
\] has a plus sign that roughly means "OR", not "AND" as many people apparently think. When we care about the \(j_z\) component of the spin, the formula above says that the state \(\ket\psi\) allows the electron to be either "up" OR "down". It doesn't say that there is both a spin "up" AND a spin "down".
If we need to say "AND" in quantum mechanics, either "one proposition/question AND another proposition/question" (as discussed with the \(P=P_A P_B\) relationship above) or "one object added on top of another object", we need multiplication, not addition. For the case of the two propositions, we have already discussed an example, the \(P=P_A P_B\) relationship above. If we discussed physical systems composed of several pieces, e.g. a group of 2 apples and a group of 3 apples, we would need another kind of a product, the tensor product,\[
\ket{\text{5 apples}} = \ket{\text{2 apples here}} \otimes \ket{\text{3 apples there}}.
\] The matrix elements extracted from similar "tensor products" are products of the matrix elements for the individual subsystems and the same thing therefore holds for the probabilities, too.
Some people may be thinking that it almost looks like I am suggesting that the MWI advocates are complete idiots with the IQ of a retarded third-grader because they can't distinguish addition from multiplication. The reason why it looks so is that this is exactly what I am trying to say. In fact, it's pretty obvious that my attempts to say such a thing are successful and I am actually saying this thing. ;-)
Why is there so much confusion about the meaning of addition and multiplication here?
Because people with common sense – as it evolved for millions of years – and no genuine knowledge of the pillars of modern physics (which includes the MWI advocates) always think in terms of objects, e.g. apples. So when you're adding two apples and three apples, place the two groups next to one another, you're adding apples. Similar addition more or less applies to lengths of sticks, momenta and other conserved quantities, and even quantities such as voltages, currents, charges, and many others.
But this "combining objects that exist simultaneously is addition" is fundamentally and completely wrong for wave functions in quantum mechanics. In quantum mechanics, addition of wave functions or density matrices roughly corresponds to "OR", not "AND", and "AND" must be expressed by multiplication. How can we understand the origin of this flagrant difference between the classical thinking and quantum mechanics?
The primary reason is that quantum mechanics just isn't describing the objects themselves. It describes propositions we can make about objects. As Niels Bohr used to say, Physics is not a tool to describe how the reality is. Physics is a tool to say right things about what we can see. The basic building blocks such as the wave functions and projection operators don't describe and count objects but encode propositions, knowledge, information.
For propositions and their probabilities (expectation values of the projection operators), addition is simply not "AND", addition is "OR". The right mathematical expression for "AND" is another operation, namely multiplication rather than addition.
An MWI advocate could start to spread fog. It may be debatable which one it is, the difference between "AND" and "OR" isn't that important, anyway, and it may be up to centennial deep philosophical discussions which way it goes. Well, all these statements are pure rubbish. There isn't any ambiguity, confusion, or room for modifications. Addition and multiplication are completely different operations so you should better not confuse them. The right theory that is tested is the theory that says the same thing about the interpretation of addition and multiplication as I did. Be sure that if you modify its rules, the rules of quantum mechanics, by randomly replacing addition by multiplication and vice versa at various places, you will get a completely, qualitatively different theory that will yield a totally different description of the reality and it will disagree with almost all observations, including some extremely elementary ones.
There just isn't any room for confusions and debates. Just like a 7-year-old schoolkid who invents arrogant excuses why she cannot learn the difference between the addition and multiplication (note that I am politically correct so I sometimes include "she" in similar sentences, especially if it increases the degree of realism), the MWI proponents should be given a failing grade and should be spanked.
Be sure that any "technical" modification of my proof that there aren't many worlds will damage the theory so that it will become totally incompatible with the experimental tests. For example, if you suggested that the projection operator for "A and B" should be \(P_A+P_B\) rather than \(P_A P_B\), you will easily find out that the same rule used for any experimentally testable situation will lead to wrong predictions. In fact, pure thinking is enough to see that "AND" must be expressed by the product of the projection operators and not the sum.
Using charge conservation to prove there aren't many worlds
The fact that one electron can't suddenly be split to two electrons so that it would be both "here" and "there" may also be derived from charge conservation, angular momentum conservation, mass conservation, or other conservation laws. In quantum mechanics, such laws still hold.
If the initial state \(\ket\psi\) is an eigenstate of the electric charge operator \(Q\),\[
Q\ket\psi = q\ket\psi,
\] then, because \(QH=HQ\) i.e. the charge is conserved i.e. the symmetry generated by it is a symmetry of the Hamiltonian i.e. of the laws of physics, the final state will obey the same relationship with the same value of \(q\). But if there were an electron on both places, the electric charge could be shown to be doubled and different than the original one. That would conflict with the conservation law.
Inflating the Hilbert space along the way
Some people could say that my derivations are missing the point that there is an "Everett multiverse". I should have increased the size of the Hilbert space before the measurement etc.
There are many wrong things about such a potential objection.
First, the constancy of the dimension of the Hilbert space is a mathematical necessity. Especially because some MWI proponents including Brian Greene say that they want to be led by the most natural interpretation of the equations of quantum mechanics, it's totally indefensible to actually change the dimension of the Hilbert space along the way. It's surely not what quantum mechanics tells us to do. In fact, one may easily show that such a proliferation of the degrees of freedom couldn't lead to an internally consistent theory.
It may be explained in many ways, e.g. by the quantum xerox no-go theorem. There can't be any evolution of a state in \({\mathcal H}\) to a state in a larger Hilbert space such as \({\mathcal H}\otimes {\mathcal H}\) because the evolution of the state vector in quantum mechanics is linear while the map \[
\ket\psi\to \ket\psi\otimes \ket\psi
\] is not linear; it is bilinear or quadratic. If \(\ket x\) and \(\ket y\) were evolving to \(\ket{xx}\) and \(\ket{yy}\), respectively, then linearity would dictate that \(\ket{x+y}\) evolves to \(\ket{xx+yy}\) while the universal squaring formula would say that it should evolve to \(\ket{(x+y)^2}=\ket{xx+xy+yx+yy}\). These are different ket vectors on the larger Hilbert space because there are extra mixed terms. At any rate, it's a contradiction: in a quantum world, there can't be any gadget that creates two exact copies out of the arbitrary initial state.
Another problem with the objection is that I actually haven't done any assumption about the non-existence of the "Everett multiverse". For example, in the fast "charge conservation" proof, \(Q\) could have meant the total electric charge in "all branches" of the world you could ever hypothesize. Clearly, if the number of worlds is being multiplied, the charge won't be conserved. That will be a problem because the symmetry generated by \(Q\) won't be a symmetry of the laws that control the "Everett multiverse" anymore. It won't be able to be exact at a fundamental level, you won't be able to use it to constrain the laws of physics, and so on. This "demise" will be fate of all the symmetries in physics (translations, rotations, Lorentz boost, parity, etc.) because all symmetries are related to a conservation law.
One more problem with the "splitting of the Universes along the way" is that there can't possibly exist any justifiable rule about "when this splitting takes place". There aren't any sharp qualitative boundaries between phenomena in Nature. It's clear that there can't be any splitting during a sensitive interference experiment – because such an "elephant in china" converting the fuzzy quantum information into the classical one would surely destroy the interference pattern.
The problem is that in principle, we may say the same thing about 2 particles, 3 particles, 100 particles, or \(10^{26}\) particles. In principle, the interference pattern involving an arbitrarily large system may be measured so the Universe is just not allowed to "split" into possibilities where different classical outcomes are realized because such a splitting would make the "reinterference" permanently impossible while it is arguably always possible in principle.
In practice, there's a lot of irreversibility, "decoherence", but this process always depends on our inability to manipulate with the elementary building blocks of information too finely. Decoherence is an emergent phenomenon and it isn't sharp, either. There is no point during the decoherence process when you could say "now it's the right time for the universes to split into many worlds". Decoherence is just a continuous process in which the off-diagonal elements of the density matrix gradually decrease. They decrease faster and faster but they're never "quite" zero.
Shannon told us that Brian Greene thinks that he and your humble correspondent have a "little disagreement" about a physics question. ;-)
The little disagreement is about the existence of a paradigm shift in the 20th century science that would invalidate the previous framework of classical physics. I am sure it has happened in the 1920s; Brian Greene thinks that it hasn't happened so it is still possible to think about Nature in the "realist" way. Of course, I could also be saying it is a little disagreement, I have also been taught how to be diplomatic, polite, hypocritical, and dishonest. But I just don't think it's right to behave in this way. The disagreement is clearly about a major question, about the very existence of modern physics as something that is outside the box of classical physics. Brian Greene is really denying the existence of quantum mechanics; instead, he is suggesting that what we need are new theories (e.g. nonlocal ones or multiverse ones) within classical physics (although he and others prefer more obscure ways to describe the very same thing, ways that make the naked Emperor's new clothes look more fashionable and decent).
The MWI chapter of The Hidden Reality by Brian Greene (whose Czech translation by me will be in the bookstores on Monday) really drove me up the wall many times because most of it is literally upside down. One repeatedly "learns" that if we want to describe the whole world in a uniform fashion, we must adopt the MWI ideology. Bohr et al. were incapable of doing so, so they preferred to live in their messy, marginally inconsistent system of ideas, and use behind-the-scene tricks to fight against the true messengers of the truth such as Hugh Everett III.
This uses the right words except that the content is exactly the opposite of the truth.
Bohr et al. always used legitimate, official, and transparent channels to discuss similar physics questions – e.g. in the Bohr-Einstein debates – and it is the MWI advocates who are using non-standard channels such as popular books to spread misconceptions. Equally importantly, the "universal validity of the laws for small and large objects" is an important consideration, indeed. But it unambiguously says that MWI is wrong and QM as understood by Bohr et al. and the followers – modern physicists – is the only plausible right answer.
I have already mentioned why it is so. There just can't be any splitting of the worlds when one quantum particle is coherently and peacefully propagating through an experimental apparatus. The same comment applies to 2 or 3 particles so if we're using the laws of physics coherently for small as well as large systems, there can't ever be any "splitting of the Universes".
An impressive song about the Higgs, a new genre of music.
There is one more aspect of the unity that could be violated by the MWI advocates to defend the indefensible. They could say that the question "is there an electron here as well as an electron there", the question whose probability we calculated to be zero, shouldn't be answered by the rules of quantum mechanics i.e. by identifying the right projection operator and by computing its expectation value (interpreted as the probability of "Yes"). They could say that this is a question "above the system" that should be answered by some philosophical dogmas.
But that's not how physics works or should work. Quantum mechanics has a way to answer all physically meaningful i.e. in principle observable questions and it is the same way for all the questions. In fact, there is nothing unusual about asking whether there are electrons at two places. This is the kind of questions that all of physics is composed of. If you were free (or even eager) to abandon your standardized theory and methodology to answer such questions and if you switched to some metaphysical dogmas just because this question about the many worlds is "ideologically sensitive", it would prove that the theory you may still be using for other questions isn't something you are taking seriously, isn't something to answer really important questions in physics. It would surely show that you have double standards and the technical theory you're using isn't universal and uniformly applicable because you often replace it by metaphysical dogmas.
Your attitude would be completely analogous to the attitude of a fundamentalist Christian physicist who just chooses to believe that Jesus Christ could walk on the sea because the laws of gravity and hydrodynamics didn't have to apply and the non-nuclear conservation of carbon atoms could have been invalidated when he was converting water into wine. And I don't mention many other Jesus' hypothetical crimes against the laws of physics that such a physicist could be eager to overlook for political reasons. ;-)
The MWI advocates prefer metaphysical dogmas and their naive classical intuition over the standardized quantum mechanical "shut up and calculate" approach to answer such questions about the electron on two places (or pretty much any other question in physics) because they haven't started to think in the quantum way yet. To think in the quantum way is to be deciding about the validity of propositions (or the probabilities of their being valid) and the procedure is always the same. One constructs the projection operator related to the proposition and calculates its expectation value in the quantum state. It's the probability and if the result is \(0\) or \(1\), we may be certain that the answer is "No" or "Yes", respectively.
(The detailed arguments or calculations may proceed differently and avoid concepts such as "projection operators" but they must still agree with the general rules of quantum mechanics.)
When we follow this totally universal quantum procedure – valid for questions about microscopic systems as well as macroscopic systems – carefully and rigorously, we will find out that quantum mechanics as it stands, in the same Copenhagen form as it has been known since the 1920s, answers all questions, including those that "look philosophically tainted", correctly i.e. in agreement with the experiments. Sidney Coleman gave many examples in his lecture Quantum Mechanics In Your Face.
For example, it's often vaguely suggested by the MWI champions and other "Copenhagen deniers" that the experimenter could feel "both outcomes at the same moment". However, by the correct quantum procedure whose essence is absolutely identical to my discussion of the two positions of the electron at the beginning, we may actually find the answer to the question "whether the experimenter feels both outcomes at the same moment". We will convert the proposition to a projection operator, it has the form \(P=P_AP_B\) again, and because its expectation value is zero for totally analogous reasons as those at the top, it follows that according to quantum mechanics, the experimenter doesn't perceive both outcomes at the same moment. This is a completely physical question, not a metaphysical one, and quantum mechanics allows one to calculate the answer. It's just not the answer that the anti-Copenhagen bigots would like to see.
Quantum mechanics doesn't predict "unambiguously" which of the outcomes will be perceived by the experimenter (spin is "up" or "down"?) but this uncertainty is something totally different than saying that he will perceive two outcomes. The number of outcomes he will perceive may be calculated unambiguously by the standard rules of quantum mechanics and the number is one. There is no room for "two worlds" or "two perceptions at the same moment". Which outcome will be felt has probabilities strictly between 0 and 100 percent so the answer isn't unequivocal.
When the MWI-like folks are discussing these matters, they are constantly making lots of other totally rudimentary errors – and perhaps "deliberate errors" – aside from the confusion of addition and multiplication I mentioned above. A frequent one is to totally forget or deny that quantum mechanics predicts and remembers correlations (in their most general form known as entanglement) between any pairs, triplets, or larger groups of degrees of freedom and properties that may co-exist in the real world.
For example, Coleman mentioned the cloud chamber example by Nevill Mott. A particle leaves the source in the cloud chamber. It is in the \(s\)-wave: its wave function is spherically symmetric so it has the same chance to move to each direction. So why does it create a straight line of bubbles in one direction rather than a spherically symmetric array of bubbles?
Again, this may be interpreted as some super-deep metaphysical question that goes well beyond quantum mechanics and the Copenhagen interpretation may be claimed to be incapable of answering such questions. Except that there is nothing hard or metaphysical about this question at all. It is completely physical, quantum mechanics allows us to answer it using a very simple calculation, and the answer is right. There will be a straight line of bubbles because one may prove that due to some demonstrable entanglement between properties of the supersaturated water or alcohol at various points that the propagation of the charged particle causes, the direction of two newly created bubbles as seen from the source is always essentially the same.
(One may prove that the charged particle only creates bubbles in a small region around its location; and one may prove that the position of the charged particle goes like \(\vec x = \vec p \cdot t / m\) where the momentum \(\vec p\) is essentially conserved. That's enough to see that the bubbles will be aligned.)
So again, while quantum mechanics gives ambiguous predictions about the direction in which the "bubbly path" will be seen – all directions are equally likely – it actually does unambiguously predict that the bubbles will have a linear shape, they will only emerge along a straight semi-infinite track. There is absolutely no inconsistency between these two assertions. Any wrong idea that QM has to predict that the distribution of the bubbles is spherically symmetric boils down to a trivial error: the omission of the fact that the existence or absence of bubbles at a point is correlated with the existence or absence of bubbles at other points. In fact, the correlation is so tight that for each semi-infinite line, there are either bubbles everywhere along the line or there are no bubbles on it. And there is only one semi-infinite line.
As I said many times, the people who have trouble with proper, i.e. Copenhagen or neo-Copenhagen laws of quantum mechanics, are always "eager" to simplify the quantum rules of the game prematurely and convert the situation to some "real physical object" way too early (well, one should really never do so but if one does it too early it may be more damaging). But Nature never does such mistakes. It remembers the wave function which knows about all the possible correlations between all the degrees of freedom, which knows about all the relative phases because they could matter, and only when an observable question has to be answered, it just calculates the right answer. The right calculation looks very different than any kind of reasoning in a classical world but it isn't too hard; it's really straightforward and in all situations in which classical physics used to work, it still gives the same answer (with tiny corrections).
When the initial wave function for the charged particle in a cloud chamber is spherically symmetric, it doesn't imply that spherically asymmetric configurations of the bubbles at the end are forbidden i.e. predicted to have vanishing probabilities. On the contrary, we may prove (the right verb really is "calculate" because the proof boils down to the calculation of an expectation value of a projection operator) that the distribution of the bubbles will be spherically asymmetric – a semi-infinite line in a direction. There is no contradiction because the initial wave function isn't a real object such as a classical field, stupid. It's a quantum-generalized probabilistic distribution. A spherically symmetric probabilistic distribution (on a sphere) doesn't mean that the actual objects such as the particles (or, later, the bubbles they will create) are spherically symmetric. Instead, it means that the probability that the objects are found in one direction is the same as it is for another direction. But because the particle may be shown to be in a direction, we know that the actual measurements of positions will inevitably be spherically asymmetric.
Is that really so hard to understand that the wave function in quantum mechanics is a generalization of a probability distribution – and not a generalization of a classical field? It encodes the information about the physical system, not the shape of the object itself. It is not really difficult to learn these things but some people just don't want to.
And that's the memo.
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