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David Deutsch's circular argument on the Born rule

Brian Greene's The Hidden Reality presents various kinds of "parallel universes" and "multiverses" that have appeared in physics – a large universe, an inflating universe, bubble universes, braneworlds with many branes, stringy multiverse based on the tunneling in a landscape, simulated multiverses, many worlds of quantum mechanics, and others. Some of them have really nothing to do with others but they were combined to a nice and meaningful book.

When I was translating it to Czech, I was happy about Brian's treatment of the cosmological issues – despite the fact that he is clearly more anthropic than your humble correspondent and most of his wishful thinking about the conversion of the anthropic rules into quantitative arguments in the future is clearly unrealistic (and may be excluded by pure thought even today).

However, I must admit that his discussion of the "many worlds" in quantum mechanics, which Brian considers "inevitable", along with all of his thoroughly irrational attacks against the proper Copenhagen quantum mechanics, was driving me up the wall. Of course, I just professionally decoupled from the validity of the content, avoided any temptation to comment on those things (too many things would have to be corrected), and carefully translated what the author wanted to say including all the physics blunders.

One of the papers that Brian mentioned in the book was this 1999 article by David Deutsch,

Quantum Theory of Probability and Decisions
Sean Carroll uncritically promoted this paper a few days ago. I was pleased that a large portion of the Cosmic Variance readers understood that the paper is completely nonsensical.




What Deutsch claimed to do was to derive the Born rule – the claim that the probabilities are given by the squared absolute values of the complex probability amplitudes in quantum mechanics – without any assumptions about probabilities, just by the maximization of utility in game theory or decision making or whatever.

This is preposterously wrong at many levels.

First of all, probabilities are clearly essential for a proper understanding of the predictions of any quantum theory, so if your "school of thought" denies that they're fundamental, you simply can't use quantum mechanics properly. A fundamentally probabilistic theory such as quantum mechanics simply can't suddenly emerge out of a non-probabilistic one. If you insist on non-probabilistic starting points, you will either make statements that are downright wrong or you won't be able to make any statements about the probabilistically predicted phenomena at all.

Second of all, the very idea that one should talk about "utility" is a pure linguistic sleight of hand without any content. The utility is just the expectation value of the profit (or less, if the sign is negative) – the profit in various scenarios weighted by the probabilities. What we're really interested in is e.g. the expectation value of the position. Utility only differs by having the "profit in dollars" instead of the "position in meters". Otherwise the maths is completely identical. The weighting by the probabilities works exactly in the same way. One learns exactly nothing if he talks about game theory, decisions, and utilities. It's amazing if someone is unable to see that these things have nothing to do with the problem to "derive the Born rule".

Third of all, Deutsch claims to prove that the probabilities are given by the second powers and phases are irrelevant. None of these things can really be proved out of nothing. A legitimate justification of the rule may look as follows.

It's just true that the squared formula for the probabilities is very natural because the evolution of the state vectors by the unitary operators – which may be obtained as exponentials of anti-Hermitian operators – naturally preserves the "quadratic norm" but doesn't preserve anything else. Because we want and Nature wanted to conserve the "total probability of all mutually exclusive possibilities" – the sum should be 100 percent at all times – it's not shocking that the most natural invariant under continuous transformations (unitary ones), i.e. the quadratic norm, was used by Nature to express the probabilities. This rule penetrates all of quantum mechanics and makes it a pretty and consistent structure.

But if you don't discuss the preservation of the total probability or the independence of the total probability on the choice of a basis, the natural embedding of evolution operators into the group \(U(N)\) which may be defined by a quadratic invariant, and so on, then of course you won't be able to find anything wrong with the non-Born rules. As an example, imagine that the probabilities are given by the 2012th powers of the absolute values of the complex probability amplitudes:\[

P_n = |c_n|^{2012},\qquad P_n \neq |c_n|^2

\] How and where does Deutsch "prove" that this modified rule, the Born2012 rule (which is just an example of infinitely many wrong rules one could propose), is wrong? Well, it's very easy to see how he "proves" it's wrong: he assumes it is wrong. And if one is allowed to assume something, even "intellectual giants" of Deutsch's caliber are capable of proving the same thing.

Where does Deutsch assume that the the probabilities are given by the squared formula? Open the PDF file on page 7 and look at equation 7. It talks about an equal-amplitude superposition of two \(X\)-eigenstates which he instantly identifies with this expression:\[

\ket\psi = \frac{1}{\sqrt{2}} \zav{\ket{x_1}+\ket{x_2}}

\] Now, note that the factor of \(1/\sqrt{2}\) appears for the first time in the paper and it is totally unjustified by anything. Of course, the actual justification is that we want the state to be normalized to one. Why? Because the squared norm is the "total probability of all mutually exclusive possibilities" and we want the total probability to be one. But note that in the previous two sentences, we have already used that the probabilities are the squared amplitudes.

If we hadn't assumed this proposition, we couldn't have proved it. Indeed, one could write a 2012 version of equation 7 instead of Deutsch's equation 7:\[

\ket\psi = \frac{1}{\sqrt[2012]{2}} \zav{\ket{x_1}+\ket{x_2}}

\] It differs from the previous one because it has the 2012th root of two instead of the square root. The total probability of both outcomes is one if you define the probabilities as the 2012th powers of the amplitudes. This modification propagates to all other equations such as 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, and so on, you get the point, and one may easily write a modified Deutsch paper that "proves" that the probabilities have to be given by the 2012th power.

His argument has clearly nothing to do with the actual reasons why the Born rule uses the second power. The observation that he actually assumed the Born rule to start with isn't the only way to see that Deutsch isn't thinking rationally. In equations such as 18 where he adds another system with various eigenvalues of a new observable called \(Y\), he is using even more unjustifiable assumptions that result from quantum mechanics. All the normalization factors could be totally different in a "different version of quantum mechanics". Once again, he is just showing that the quantum formulae are compatible with themselves. He proves exactly as much as he assumes so the added value is zero. The only "added value" is his implicit suggestion that the paper has a nonzero value but this added proposition is clearly invalid.

The independence of the probabilities on the phases claimed in equation 25 isn't proved, either. He just says that the probabilities aren't affected by the phases because the unitary evolution may change phases of coefficients in the decomposition into eigenstates but not the probabilities of the individual eigenvalues. Well, but he talks about "unitary evolution". Unitary evolution is one that preserves the sum of \(|c_n|^2\), i.e. the sum of terms that don't depend on phases of the amplitudes. The conditions "unitary" and the "Born rule" are really equivalent; they store the same mathematical core, namely the quadratic prescription for the probabilities. If he replaced the preferred "unitary evolution" by a "Deutsch evolution" which preserves e.g.\[

P_{\rm Deutsch} = \sum_n \left[ |c_n|^{1938}+({\rm Re} \,c_n)^{1918} \right],

\] he would obtain a nontrivial dependence on the phase of the amplitudes \(c_n\). As I have said, the Deutsch evolution isn't as pretty as the unitary evolution (there are no nice continuous groups that only have higher-order invariants; \(U(N)\) with its quadratic invariant is really special) and can't be generated from a linear Hermitian Hamiltonian: that's the actual advantage of the quadratic probabilities relatively to various ad hoc "alternatives". But if you don't care about this simplicity and richness that allows you to produce continuous transformations out of linear Hermitian operators, the Born rule could clearly be replaced by other rules and all the proofs that it can't are therefore obviously wrong.

It seems incredible to me that and why some physicists aren't capable of seeing through similar extremely cheap tricks. Well, some of these people are cosmologists who may at least boast that they understand the arguments in favor of dark matter more correctly than a clown from TV screens. What an achievement for Caltech: congratulations!

And that's the memo.



P.S.: It's frustrating to look at the Cosmic Variance these days; well, it's been the case most of the time so I am not complaining about any detectable negative trend here.

Another article encourages "discussions" about physics' being a "dysfunctional science". Most of the people who love to say these things are crackpots who have no clue what physics is all about and most of the harm and postmodern deterioration in science is being done exactly by these people who love to emit hateful comments about whole traditional disciplines of science if not the scientific method as such and who would prefer to replace science by something "simpler" or "closer to their hormonal systems". Science has some very critical pre-requisites and demands a certain discipline and hard work and if you just don't respect these values, you are pretty much bound to be a pest that science and scientists should try to put down.

Indeed, the first commenter D.H. – followed by many other dimwits in the thread – repeated some nasty attacks against the core pillars of modern science and mentioned that it "seems to me I just read a book called 'The Trouble With Physics' by a renowned theoretical physicist...", after a couple of breathtakingly idiotic lies that Smolin wrote in his book.

I just wonder: How many pounds of excrements do you have to squeeze inside your hollow skull to think that crackpot Lee Smolin is a "renowned theoretical physicist"? Yes, I hate this brainwashed scum and I am stunned by the tolerance that many scientists have developed towards this scum which sometimes borders with the straight penetration of this scum into the scientific institutions themselves.

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