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Consistent histories aren't inconsistent

The prohibited inconsistency of histories in the formalism is synonymous with Bohr's complementarity

The de Broglie-Bohm pilot wave theory and the many worlds interpretation are the two most widespread "alternative axiomatic systems" that are claimed to compete with the proper, Copenhagen or neo-Copenhagen, quantum mechanics. The Ghirardi-Rimini-Weber "objective spontaneous random collapse" theories are a distant third and other "frameworks" meant to replace the postulates of quantum mechanics are pretty much incoherent even at the level of the grammar.

Both the pilot wave theory and the many worlds are irrational and both of them ultimately contradict important and well-established facts about the physical world. Both of them are motivated by the champions' attachment to "realism" – a euphemism for the observer-independent i.e. classical physics. If I had to choose, I would choose the Bohmists as the much worse physicists among the two. They're in a much deeper denial of modern physics.

You know, one may divide those confused (and/or bigoted) people's efforts to deviate from quantum mechanics as formulated in Copenhagen into two levels:

  1. Dissatisfaction with the philosophical conceptual "words" that Heisenberg, Bohr, Pauli, Dirac, Wigner, von Neumann, and others have been teaching us
  2. Disagreement with some universal properties of the mathematical formalism of quantum mechanics
Both Bohmists and many-worldists suffer from (1). But only the Bohmists commit the sin number (2). In practice, most many-worldists prefer to say that the mathematical foundations of quantum mechanics are right and here to stay. They "just" believe that it may be and should be supplemented with some "more realist" set of words and more visualized "ways to imagine" what's going on.




What do I mean by the fundamental mathematical properties of the quantum theory that the many-worldists usually accept while the Bohmists deny it?
  • Observables (and the evolution and other transformation operations) are represented by linear but non-commuting operators.
  • The theory ultimately computes probabilities and complex probability amplitudes are an intermediate step in the calculations. They are combined into sesquilinear expressions of the type \(c_1^*c_2\).
Many-worldists whom we know usually agree with these statements. Many of them are particle physics or condensed matter physics practical men who compute correlators of linear operators all the time. They don't really expect themselves to mess up with the "complex linear" spirit of quantum mechanics.

They're just thinking (more precisely, they are victims of a wishful thinking) that all these expressions could be employed by some other philosophy – supplemented by different conceptual words – in which the observer wouldn't be needed, wave functions wouldn't collapse in any sense, the wave functions could be split into sums of many parts that behave as "many worlds", and the probabilities could be interpreted as some subjective belief about "where we are".




There exists no consistent way to satisfy these conditions and the many worlds approach therefore fails as soon as an intelligent person looks carefully. But at least, the many-worldists aren't in the full denial of the change of the spirit of calculations that was forced upon the people by the quantum revolution.

On the other hand, Bohmists deny everything – not only the "philosophical" need for the observers, the intrinsically probabilistic description etc. But they also deny the importance of linear operators, complexity of some probability amplitudes in the mathematical formalism etc. They want to return physics to the epoch of classical physics not only when it comes to the philosophy, words, and concepts; but also when it comes to the kind of mathematical structures that should be used to calculate the physical predictions.

In the pilot wave theory, linearity of the operators – and the evolution of the wave function (rebranded as the pilot wave) – plays no role. It seems completely coincidental. The pilot wave is a classical field (a multilocal field in the case of many particles) and classical fields generically evolve according to nonlinear equations. Bohmists should really admit that they're predicting that the equations governing the evolution of the wave function should be nonlinear because it's infinitely unlikely that all the nonlinear terms vanish (there is no reason for such a vanishing in the Bohmian picture, and anything that can happen will happen). This prediction clearly contradicts the observations.

Moreover, the pilot wave theory postulates the objective existence of the Bohmian particle positions, real classical degrees of freedom that are neither quantum nor complex. None of these degrees of freedom were ever useful to explain any quantum phenomenon but the Bohmists don't care.

I was looking at various Bohmist sources in recent days. Ilja Schmelzer, a Bohmist who once wrote a TRF guest blog, had a discussion forum where he promoted Bohmian mechanics and reported every event of the type "Luboš Motl wrote this and that which was critical of Bohmian mechanics". This forum was hijacked by spamming machines. Every minute, a new comment (with 0 views) is promoting another product – usually a pharmacological product. The transition of a Bohmist propagandist forum to an unregulated pile of spam is rather characteristic.

Also, I spent more time by reading Jean Bricmont's book promoting Bohmism (which has appeared in previous blog posts). More than before, I was amazed by the amount of political ideology in the book. Bricmont is a Marxist defending all sort of far left things. That could be fine for a physicist if he could defend himself against being affected by the ideological stuff. However, in his book claiming to be about quantum mechanics, the root "Marx" appears a whopping 33 times. Not bad for a political guy who died in 1883, 42 years before quantum mechanics was first formulated. ;-)

Among other things, Bricmont criticizes Leon Rosenfeld – a Marxist who was however a close friend and soulmate of Bohr's and a vehement critic of Bohmian mechanics and the many worlds. One may see in between the lines that Bricmont considers comrade Rosenfeld a traitor. Marxists should be obliged to worship the Bohmist crap, shouldn't they?

Bricmont also attacks Margaret Thatcher at one point – because she liked to say "There Is No Alternative [to global capitalism]" (TINA). There are two problems: The truth values of Heisenberg's and Thatcher's statements aren't necessarily the same. And even if they were the same, it doesn't matter much because Bricmont is wrong both about politics+economics and about quantum physics. There is just no damn alternative to global capitalism and Copenhagen quantum mechanics – and there's no other game in town than string theory, either. You should better used to these facts, comrades.

Finally, I am getting to the topic announced in the title

Florin Moldoveanu wrote a bizarre blog post about the claims that the "consistent histories" approach to quantum mechanics is inconsistent. This claim is mostly copied from a paper by Shelly Goldstein – a Bohmian philosopher at Rutgers whom I was seeing every week while he was writing the paper in the late 1990s, too. Every good physicist should be able to see what's wrong with Goldstein's criticism of the consistent histories. Sadly, Bricmont and Moldoveanu aren't good physicists so they just can't see anything.

Bricmont's book describes the "Goldstein's proof of the inconsistency of consistent histories" around page 219. The setup chosen by Goldstein is nothing else than Hardy's paradox that I wrote about in 2011 – or at most a tiny generalization of it.



In Hardy's paradox, the initial state is a singlet – or any maximally entangled state – of two qubits. I choose to interpret them as two spins of spin-1/2 particles. Well, it's a singlet with an extra term that deforms it a little bit:\[

|\Psi\rangle = a |e_1\rangle|f_2\rangle + a|e_2\rangle|f_1\rangle - b |e_1\rangle|f_1\rangle

\] The normalization condition is \(2|a|^2+|b|^2=1\). The first two terms are the usual "up down" minus "down up" states of the singlet that imply the perfect anticorrelation of the two spins (the sign or phase isn't too important in this simple example but the Bohmists' persistent denial of the complexity of the amplitudes is a sign of their conflict with the mathematical apparatus of quantum mechanics that I mentioned above). The last, \(b\), term allows "up up" with a nonzero probability as well but "down down" remains forbidden.



Fine, so the particle A's spin and the particle B's spin may be measured with respect to the vertical (unprimed) axis or another (primed) axis. Now, in quantum mechanics, once you measure the particle A's spin with respect to one axis, you change the predictions for the other axis, so only one of them may be measured during one repetition of the experiment if you want to determine the pre-existing state of the spin.

Our initial state has the following three implications for the predictions of the spin measurements:
  1. After the doubly unprimed measurements, \(A=+1\) and \(B=+1\) sometimes occurs
  2. After the mixed measurements, \(A=+1\) and \(B'=+1\) never occurs, and \(A'=+1\) and \(B=+1\) never occurs, either
  3. After the doubly primed measurements, \(A'=-1\) and \(B'=-1\) never occurs.
The first implication, one with the word "sometimes", follows from the \(b\) term in the state. The other implications follow from the special form of the state, i.e. from the equality of the coefficients \(a\) for the first two term, and from the absence of the \(\ket{e_2}\ket{f_2}\) term.

In classical physics, the variables \(A,A',B,B'\) could be measured and their values would have to objectively exist prior to the measurement. So these four observables would have to be numbers from the set \(\{+1,-1\}\) that commute with each other. However, if you impose all the "never" conditions from the list, you may deduce that \(A=+1\) and \(B=+1\) never occurs, either, contradicting the "sometimes" claim in our list.

Fine. All of us know why quantum mechanics allows these three claims to be satisfied at the same moment: Quantum mechanics isn't classical physics, stupid. Mathematically, the point is that \(A,A',B,B'\) do not commute with each other. In particular, \(A\) doesn't commute with \(A'\) and \(B\) doesn't commute with \(B'\) – like different components of the spin \(\vec S\). The remaining commutators are zero but these two nonzero commutators make a difference.

The "never" claims may be written algebraically. Some "logical" expressions \(P_{2A},P_{2B},P_3\) involving the operators \(A,A',B,B'\) annihilate the initial state \(\ket\Psi\):\[

P_{2A}\ket\Psi =0, \quad P_{2B}\ket\Psi =0, \quad P_{3}\ket\Psi =0,

\] If the ket vectors were missing, and if the operators were commuting, like in classical physics, you could use those equations to derive \(P_1=0\) as well. But they don't commute and \(\ket\Psi\) isn't missing, so the implication (1) with "never" instead of "sometimes" doesn't follow from the remaining properties in quantum mechanics. You just can't derive \(P_1\ket\Psi=0\), the expected quantum counterpart of \(P_1=0\), using the algebraic manipulations from the three displayed equations above.

Fine, simple, I spent more time with this stuff in the 2011 blog post.

But now, in his heavily misguided "Quantum Theory Without Observers" [1997-98], Sheldon Goldstein claims that this Hardy's paradox implies that the consistent histories approach to quantum mechanics is inconsistent. Let me quote Goldstein as reproduced on page 220 of Bricmont's book:
It is important to appreciate that, for orthodox quantum theory (and, in fact, even for Bohmian mechanics), the four statements above, if used properly, are not inconsistent, because they then would refer merely to the outcomes of four different experiments, so that the probabilities would refer, in effect, to four different ensembles. However, the whole point of DH is that such statements refer directly, not to what would happen were certain experimental procedures to be performed, but to the probabilities of occurrence of the histories themselves, regardless of whether any such experiments are performed.
Goldstein sort of realizes that Heisenberg and pals would point out that \(A\) and \(A'\) cannot be measured in the same experiment (without changing one another), and similarly for \(B\) and \(B'\), so it's not a problem that quantum mechanics implies that the first prediction may "sometimes" occur, while it would be classically impossible.

But Goldstein claimed that in the decoherent (or consistent) histories, the contradiction does arise because the statements about the variables \(A,A',B,B'\) have some probabilities to be true simultaneously.

Needless to say, Goldstein – and similarly Bricmont, Moldoveanu, and others who uncritically parrot him – is laughably wrong. It's the very point of the adjective decoherent or consistent in the phrase "decoherent histories" or "consistent histories" that these three or four propositions cannot be discussed simultaneously. Just check any basic introduction to consistent or decoherent histories, e.g. the introduction at Wikipedia, to see that. One may say that the point of consistent or decoherent histories is exactly the opposite than the "main point" assigned to this approach by Goldstein!

Consistent histories allow you to calculate the probabilities of some statements (about observables at many moments). These statements are branded as "histories". But once you calculate the probabilities of different histories in a set, this set of histories must be consistent. The consistency of two histories is nothing else than their orthogonality (well, a specific sesquilinear orthogonality of their "class operators"). And when you discuss histories that depend on the values of quantities \(A\) and \(A'\), the consistency or orthogonality is simply equivalent to the vanishing commutator of \(A\) and \(A'\).

So for two observables that you want to measure at the same moment – and split the histories according to their values – the required "consistency" of the histories means nothing else than the vanishing of the commutator of these two observables. So the "consistency" simply says exactly the same thing as quantum mechanics always does: noncommuting variables cannot be assumed to have some classical values at the same moment.

The consistent (=decoherent) histories approach to quantum mechanics just repackages the absolutely standard axioms and novelties of quantum mechanics into slightly different packages labeled with slightly different words. But the total "beef" contained in these packages is exactly the same as it is in standard Copenhagen quantum mechanics. You just can't measure \(A\) and \(A'\) at the same moment – and analogously, you can't measure \(x\) and \(p\) at the same moment. It's not just you who can't measure it. No observer can. And an observer is needed to make the values physically meaningful. So the values cannot simultaneously exist in the classical sense.

You know, I learned Consistent Histories from Roland Omnes who specifically admits that this approach to quantum mechanics is neo-Copenhagen – he realizes that the interpretation isn't really changing things or discovering any truly new things beyond Copenhagen. It's just a way to reshuffle all the Copenhagen axioms and optimize them for predictions of properties of the physical system at many moments of time.

In private discussions, I've noticed that Jim Hartle and Murray Gell-Mann were more obscure about the question whether they believe that that their interpretation is just a different repackaging of Copenhagen – and Griffiths, another "consistent histories" guy, is obscure, too. I think that none of them has ever said that "their interpretation has a qualitatively different substance than Copenhagen". But other people love to suggest so. In particular, you could have seen Goldstein who basically believes that "consistent histories" is another attempt to fully restore "realism" i.e. classical physics.

Sorry, it just can't be one.

In Bohr's Copenhagen jargon, we may say that \(A\) and \(A'\) are complementary observables so they can't be assumed to have classical values simultaneously. In Heisenberg's or Pauli's or Dirac's Copenhagen jargon, we notice that \(A\) and \(A'\) are non-commuting operators which have no simultaneous eigenstates so they can't be identified with \(c\)-number values at the same moment. So far so good, it's not hard to see that all the founders of quantum mechanics were really saying the same thing even if the languages were a bit different.

Consistent histories want to change the language in a deeper way. We can't talk about propositions that depend on \(A\) and \(A'\) because the splitting of the evolution to such histories would produce a set of histories that aren't consistent with each other, and that's prohibited. Now, the previous sentence sounds different – and even when translated to equations, the equation looks different – but the message of the sentence is still equivalent to the previous paragraph. One just can't measure complementary or non-commuting etc. observables in the same repetition of an experiment.

A reasonable many-worldist could actually express the same idea in completely different words, too. There is a splitting of the worlds when an experimenter decides whether he measures \(A\) or \(A'\) i.e. when he chooses the axis with respect to which the spin of a particle is measured. (Well, the reasonable many worlds are splitting when an experimenter decides what to measure, not when Nature decides what outcome he should get.) In some worlds, the experimenter near the particle A measures \(A\), in other worlds, he measures \(A'\), but there are no worlds in which he measures both \(A\) and \(A'\). Such worlds don't exist and what happens in these non-existing worlds doesn't have to be discussed and doesn't need to be "consistent". It's OK for the physical theory to prohibit the discussion of such things at the same moment.

The "reasonable many-worldist's story" conveys the same point as Bohr's stories about complementarity. Many-worlds champions are almost always absolutely unreadable when it comes to fundamental "detailed" questions – e.g. whether the worlds split when an experiment decides what he wants to measure or when Nature produces an outcome or both – but if a many-worldist is happy with the "splitting during the experimenter's decision" and proud about the insight that "propositions about measurements of \(A\) and \(A'\) don't have to be consistent with each other because they occur in different 'branches' of the many worlds", she should be sensible and respectful enough to appreciate that this idea was realized by Bohr and not Everett (or DeWitt) and has been known as the Bohr complementarity before the idiotic anti-Copenhagen Marxist movement took off. You're just stealing the most precious ideas of science if you claim that this insight due to Bohr was basically done by Everett or someone else.

Obviously, there's no simple contradiction (as promoted by Goldstein and his parrots) in the consistent histories approach to quantum mechanics. The very point of the interpretation is that one isn't allowed to discuss the truth value (or probability) of various propositions that are inconsistent – result from non-commuting, non-decoherent observables – at the same moment. My more general point is that most people don't understand quantum mechanics which also means that when you express the same idea using two slightly different choices of the words, they're lost. They just can't see that the ideas expressed in two languages (perhaps even in Dirac's English and Heisenberg's German) are the same.

There's another "linguistic" difference between Copenhagen and consistent histories that the "realists" just want to misunderstand, and it's the existence of the observers. In Copenhagen, you want to emphasize that every application of a quantum mechanical theory requires an observer who chooses what he or she or it measures, perceives, and considers to be a measurement. Consistent histories seemingly avoid the words "observer", "observation", and "measurement" altogether.

Does it mean that consistent histories are returning us to the epoch in physics in which observers aren't needed at all, in which they play no role?

If you're only doing comparative literature and see that the words "observer" have been eliminated from many consistent histories papers, you could say "Yes, the consistent histories have achieved this goal to eliminate observers again". But if you actually understand the meaning of the claims about physics and the implications of all the equations and mathematical propositions, you know that the morally correct answer is "No".

Consistent histories may have eliminated the identification of an observer as a subset of the physical objects in an experiment. But the consistent histories approach hasn't eliminated the need for someone to decide what should be considered an observation i.e. what observables should be used as sources of some ordinary, "classical" information that may be discussed or predicted. Whoever decides which set of consistent histories you want to discuss is the observer. They play exactly the same role. They are needed for quantum mechanics to be applied to any actual physical situation.

The choice of the "set of the consistent histories" isn't uniquely determined in general. Someone has to decide how the "pie" is split to the different histories. And the person or entity doing this decision is the observer. OK, you may imagine that it's some external person who isn't sitting in the lab at all and just writes equations. You may say that he's not the same guy as the Copenhagen observer. So let me clarify what I said: As long as the laws of physics are being verified in any way, the person who is splitting the 100% pie into the individual "consistent histories" is the same person as the Copenhagen observer. It's the guy who actually perceives the results of the measurements that carry the information which of the consistent histories has taken place.

The human language wasn't optimized to talk about quantum mechanics effectively and accurately. So it shouldn't be surprising that the attempts to translate quantum mechanics to the human language (or, if you wish, to the language of journalists or philosophers) lead to confusions, inaccuracies, as well as redundancies. The same thing may seemingly be said in many different ways that look completely different to a linguist. But a physicist must understand the substance of the principles and claims. It basically means that he must know how to use the rules in the most general situations and see why it's always the same rules that are being applied. He doesn't need to refine the "human language" needed to talk about the physics – although I am and others are trying to refine it, anyway.

People should stop arguing – and physicists have basically stopped arguing – about the choice of the "best words" when it seems very likely that two such physicists mean the same "beef". Only philosophers argue about different words whose meaning is often undefined – and in particular, their differences are often undefined.

But there exist disagreements about physics that are sharper. For example, as I mentioned, Bohmists want to deny that the right underlying theory that most accurately describes the phenomena in the microscopic world uses linear operators and exactly linear operators and equations. And that the predictions are made in the form of probabilities that are computed from complex probability amplitudes. When the disagreement gets this deep, the proponents of the "alternative interpretation" – Bohmists, in this case – must acknowledge that they're trying to promote a totally different theory. And it's one that has absolutely no chance to compete against the correct theory, quantum mechanics (defined by the universal Copenhagen or equivalent postulates).

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