Tuesday, May 17, 2016

Bohmists' inequivalence & dishonesty

Off-topic, annoying: The Central Committee of Both F*cked-up and Non-F*cked-Up Feminist Whores at LIGO has publicly attacked and bullied a senior LIGO memberBarry Barish who rescued the experiment from an epoch of mismanagement and is therefore the most well-deserved Nobel prize candidate among currently active members and who has also led the ILC design group – because the bitches found an image in an after-dinner speech at Pheno16 "offensive". Why is it so hard for you to f*ck off, ladies? And dear reader, can you find what offended them (PDF/PPTX)? I can't. Maybe the penis-like shape of LIGO on slide 8? It's an unedited photograph! Or a similar potential ATLAS+sex theme on the introductory slide? If LIGO had the shape of a vagina, see my undergraduate hostel collectively decorated by all the students inside, 50% of the tubes would be wasted and the acute angles would reduce the sensitivity, too.
I've seen many new bizarre responses by the advocates of Bohmian mechanics. Maybe my responses have contributed to their activity in an example of a vicious loop. ;-)

First, let me mention an old website I found, bohmianmechanics.org. It was created by James Taylor who got a PhD at Rutgers in 2003 (two years after me at the same school – he used the same macros). His adviser was Sheldon Goldstein, a noted Bohmist, and the thesis was dedicated to the ideology of Bohmism.

The actual main guru of Bohmism. (Note that Chen and Kleinert say that Madelung basically discovered the pilot wave theory a year before Louis de Broglie.)

Thankfully, Taylor didn't continue as a physicist and the website has no visitors (just like bohmian-mechanics.net) and that's good news. One of the pages in the table-of-contents of the website is Advice for debaters in support of Bohmian mechanics (that's how the page is referred to at the main page of the website).

Wow, this sounds just like the 135 million pages ;-) written by Al Gore's disciples (and courses taught by them) about how to manipulate your parents concerning the climate, how to deceive the Nobel peace prize committee, how to talk to a conservative, how to fool a denier, convince a neighbor, steal real estate in Tennessee and California, and how to escape police and laugh at everyone. ;-)

I am sorry but this not how an honest scientist behaves. On his "Advice for Bohmians" page, Taylor sensibly says that Bohmists won't convince anyone etc. But he also recommends them:
Assure them that there are mathematical results for existence, uniqueness, agreement, extensions to whatever phenomena, etc.
This hugely differs from the advice that an honest scientist such as your humble correspondent would give to everyone – for discussions about foundations of quantum mechanics, string theory, climate, or any other scientific topic:
Tell the truth whether it supports one viewpoint or another. Try to be comprehensible and assure that the listener gets as accurate a picture of the available evidence, facts, and explanations as his or her abilities and knowledge allow.
Unfortunately, Taylor's "how to brainwash" page isn't the only sign showing that the Bohmists are a deceitful, ideologically driven movement.

Ilja Schmelzer – who wrote a guest blog in 2013 – had to be banned a week ago or so. It just happens that approximately at the same moment, lots of responses to myself appeared on his forum, along with new users such as secur.

In one comment, secur tells us that he's the same person as the commenter "algore" who appeared on this blog. A nice admission. Even more interestingly, I think that there exists nontrivial evidence that secur=algore and Ilja Schmelzer are actually the same person, too. Try to read all exchanges between these two men on Ilja's forum. The degree of agreement between them is amazing. It's just like Amy's virtual friend in The Big Bang Theory. They're constantly on the same frequency, know the same things, use the same not-quite-generic acronyms such as dBB, overuse the colon (;) relatively to almost everyone else, and so on. There may be other users on Ilja's forum who are Ilja's sock puppets, too.

Well, it's funny.

One of the other possible copies of Ilja's – and at least, a Bohmist – is user7348 who has posted lots of assorted pro-Bohmian stuff on the Physics Stack Exchange recently. (I also ran into another obnoxious Bohmist crackpot named Timaeus over there.) Whoever user7348 is, he clearly plays a propaganda game, asks bogus question whether Bohm's theory suffers from one problem or another, and waits for someone to write a bogus answer – the people who answer may be the same person. But even if they're not, many of the answers are just wrong.

One omnipresent statement made by the Bohmists is that Bohmian mechanics is equivalent to quantum mechanics. That's obviously completely wrong. Equivalent theories must give the same predictions. That's pretty much possible only if they involve an "equally large set of degrees of freedom" that may be mapped to one another, that have the same physical meaning, and nothing is added or replaced on either side.

Bohmian mechanics may reproduce the whole wave function and Schrödinger's equation for its evolution. But Bohmian mechanics also adds additional "beables", such as the classical particle positions, and uses these new classical degrees of freedom to decide what will happen in an experiment. At this moment, the theory is already obviously inequivalent to quantum mechanics where all the predictions are made from the wave function – and never from some additional classical degrees of freedom.

The claim that such a modified theory is equivalent to QM is as ludicrous as the claim that communism is equivalent to the free-market economy. People have the same ingenious ideas just like in capitalism – they just need a stamp from the communist party – so the communism is equivalent to capitalism, isn't it? Well, it's not. The communists screw everything good about capitalism, and analogously the beables and the wrong interpretation of the wave function screw everything good about quantum mechanics.

It's remarkable how dense the Bohmists may be when it comes to the simple point that one either has an equivalent theory – so that he can't claim any improvement at all – or he changes something about the ways how things are predicted (or what can be predicted), and this general change means that the theory will make different or new predictions that are likely to disagree with the facts (given the fact that quantum mechanics agrees with all the facts).

It may be seen in millions of ways – conceptual as well as detailed "minor" examples – that Bohmian theory simply cannot be equivalent to quantum mechanics. As "algore" told us, Bohm & Hiley's 1993 book (which both secur=algore and Ilja Schmelzer seem to know very well – I've never seen the book and I think that almost no physicist has) made it clear that the authors realized that there can't be any valid Bohmian treatment of bosonic fields and other things, due to their stress tensor's even number of indices etc. Bohm & Hiley realized that a disagreement with relativity/locality is unavoidable and that this disagreement is a lethal problem.

Bohmian mechanics incorporates both the wave function (rebranded as the pilot wave) and the actual positions. This immediately raises the problem that in general, we don't know which of these two "copies of the information" is observed, which of them affects other things. In almost every particular situation, you may see that every answer is a problem.

For example, take the simplest "relative success story" Bohmian theory for one electron and try to ask whether the electron emits electromagnetic radiation. The radiation is composed of photons which should also be associated with some "real trajectories" because the location of photons may be measured, too. OK, these photons are emitted whenever a charge is accelerating.

An obvious question arises: Should the "presence of acceleration" be decided according to the wave function (pilot wave), or according to the Bohmian trajectory? And analogously, should the Bohmian photons with trajectories be created only near the Bohmian electrons, or near all the points where the wave function (pilot wave) for the electrons is nonzero?

Both answers lead to immediate disagreements with the observations. "Sometimes" (well, almost always), two attempts are simply not enough to guess the right answer. If the photons may arise "anywhere" where the pilot wave is supported, one predicts the existence of the synchrotron radiation at points where no charged particle is actually found. On the other hand, if the synchrotron radiation (photons) are created near the Bohmian electron's trajectory, then even a free but self-interfering electron will have to emit the radiation because the generic Bohmian trajectory is curved and accelerating in that case.

This is one of the point of the 2014 paper by Pisin Chen and Hagen Kleinert
Deficiencies of Bohmian Trajectories in View of Basic Quantum Principles
which also argues, following a paper by Kurt Jung – see also the famous 1992 ESSW paper claiming that the Bohmian trajectories are "surreal" (full PDF), that the Bohmian trajectories give a wrong prediction even for the simple double slit experiment. The ratio of intensities at the central and the following peak is about 1:1 according to QM (and experiments) but 3:1 according to Bohmian mechanics.

I have some doubts about this technical claim because to some extent, the probability distributions for non-relativistic particles are unquestionably evolving correctly in Bohmian mechanics. But I have no doubt about many other criticisms that appear in these papers.

By the way, it seems clear to me what happens when people – including Hagen Kleinert, a collaborator of Feynman in his last years and a co-author of an ingenious path-integral solution to the hydrogen atom – send a paper criticizing Bohmian mechanics to a journal. The editor almost certainly sends such a paper to some Bohmist referees. And what a surprise, Bohmists are dishonest Marxist aßholes who will prevent the publication even if they know that the paper is correct. The very existence of this would-be "subfield" of physics is a problem that should have been prevented.

Bohmian mechanics unquestionably has a severe problem to describe the Maxwellian pilot wave for a photon, emission of light (see also the synchrotron radiation dilemma above), anticommuting fields, and many other things from quantum field theory. Even in non-relativistic quantum mechanical setups, Bohmian mechanics behaves extremely bizarrely whenever the wave function is real (which it is for almost all ground states, among other \(m=0\) states): the velocity of the Bohmian trajectories is strictly zero (because the Bohmian velocity is proportional to the gradient of the quantum complex phase). The particle just sits at one place. This is also problem for molecules that just sit there, at a random but frozen place.

Also, Bohmian mechanics produces insane, asymptotically infinite, velocities of particles near the loci (strings) where \(\psi(x,y,z)=0\) as it hysterically struggles to repel the Bohmian particles from the interference minima. The superposition postulate (or linearity) of quantum mechanics guarantees that nothing special is happening near such points.

Just like it was impossible to meaningfully answer the question whether the synchrotron radiation is emitted according to the locus of the pilot wave or the Bohmian trajectories, it's impossible to define a meaningful formula for the energy. If the energy only depended on the pilot wave or only on the Bohmian trajectory, it could be easily proven that the conservation law is violated in some situations. A compromise probably implies that the law is violated in all situations. And so on.

To discuss another lethal problem in detail, take a very simple topic, the question about the heat capacity in Bohmian theory which was inspired by my blog post ruling out all realist "interpretations" by seeing the low heat capacity of atoms.

The argument was already known to Dirac – and presented at the beginning of his textbook on quantum mechanics – and its essence is ultimately very simple, so if you still haven't understood why it safely kills all Bohmian-like theories, I ask you for some patience.

Quantum mechanics predicts and experiments confirm that the entropy of one atom is always comparable to \(k\), i.e. the Boltzmann constant, near room temperatures etc. This entropy may be measured from the heat capacity of the materials. In the units \(k=1\), the statement is that the information carried by one average atom at the room temperature is comparable to one bit. It may be a few bits but it's not too much higher.

The entropy is the logarithm of the number of microstates that are macroscopically indistinguishable, \(S=k \log W\), as Boltzmann's tomb proudly says. How is it possible that for an atom in QM, \(S\sim k\)? It's because the number of microstates \(W\), if I respectfully use Boltzmann's notation, is of order one.

In fact, for \(T\to 0\) kelvins, the entropy of an atom obeys \(S\to 0\), something that was known as the "third law of thermodynamics" long before quantum mechanics was born. Why is it true that the entropy of all regular materials goes to zero when the temperature goes to zero kelvins? It's because at zero kelvins, one doesn't have enough energy to excite any degree of freedom, and within the interval of energies comparable to \(kT\), the ground state of the atom (or a crystal etc.) is unique!

It's unique, so Boltzmann's number \(W=1\) and its logarithm is zero, giving \(S=0\) for \(T=0\).

For somewhat higher, e.g. room, temperatures, it becomes possible to excite some electrons into higher orbitals (it's easier to excite them in molecular orbitals etc., atomic orbitals need much higher energies i.e. much higher temperatures). But we still have \(W\sim \O(1)\) and therefore \(S\sim \O(k)\) for an atom.

How is it possible that we have a unique ground state and \(W=1\) at very low temperatures? It's because the ground state wave function is counted as \(W=1\) state. But cannot you have "nearby" wave functions? Cannot the ground state wave function be deformed "infinitesimally"?

Yes, you can consider \(\ket{\psi_0}+\epsilon\ket\phi\), but that's enough to increase \(W\) e.g. to \(W=2\), let alone higher. Why? Because in quantum mechanics, non-orthogonal states are simply not mutually exclusive. Instead, the probability that a normalized vector \(\ket\alpha\) is "totally physically equivalent" to \(\ket\beta\) is given by the Born rule, \[

P = \abs{\bra \alpha \beta \rangle}^2

\] If you need to increase the number of states from \(W=1\) to \(W=2\) in quantum mechanics, you need to find a new state that is orthogonal to the ground state, and is therefore very different (and not just infinitesimally different) from the ground state.

This is the miraculous trick by which quantum mechanics achieves something that is totally impossible in any classical theory: It just freezes all the degrees of freedom at the low temperatures – and still freezes almost all of them at room temperatures. The energy spectrum is discrete. And what makes this statement important is that the states that are not energy eigenstates, e.g. infinitesimal deformations of the ground state, simply cannot be counted as new, mutually exclusive states that increase the value of \(W\) because quantum mechanics boldly says that they're "almost certainly" identical to the ground state!

The indistinguishability of these nearby wave functions is a manifestation of the uncertainty principle. If your wave function is too close to the ground state, you can't be any certain that it is a different wave function because all of its physical properties are the same within the error margins.

The situation is very different in any classical theory – a synonym of a "realist theory" or a "theory without observers". Bohmian mechanics is one of them. In these theories, all the configurations are in principle totally distinguishable: they are mutually exclusive. So the ground state wave function \(\ket{\psi_0}\) is reinterpreted as a classical "pilot wave".

Its infinitesimal deformations \(\ket{\psi_0}+\epsilon\ket\phi\) are distinguishable, different, mutually exclusive states, so they must be counted as new states that increase the value of \(W\), the number of classically indistinguishable microstates. Clearly, there is a continuous space so you get \(W\to \infty\) even if you restrict yourself to the states that are close to the ground state. So the entropy is \(S/k\to \infty\), too.

This problem has always existed in classical physics. The entropy was only well-defined up to a universal additive shift \(\Delta S\) in classical physics. The third law of thermodynamics didn't really say that \(S\to 0\) for \(T\to 0\). It said that \(S\to S_0\) for \(T\to 0\) where the constant entropy \(S_0\) was undetermined.

All these problems of classical thermodynamics boiled down to the fact that the number of states in the phase space was really infinite (as the number of points in any continuum), so its logarithm \(S/k\) was infinite, too. This infinite additive constant had to be removed from \(S\) – basically by assuming that the precision with which all the phase space variables may be measured was limited – and that's how a finite \(S\) could have been restored.

Quantum mechanics solved the problem by making \(W\) literally finite (at least for any collection of particles in a box and at a finite temperature). But \(W\) is no longer the total number of wave functions – which is still infinite. Instead, \(W\) is the dimension of the Hilbert space of states whose macroscopic appearance is the desired one. Only because we talk about the dimension – i.e. because we don't consider nearby ket vectors to be different i.e. because we only count the basis vectors in a basis – we may make \(W\) finite for an atom. In other words:
The uncertainty principle logically identifying nearby states as "in principle indistinguishable" (no repetition of the situation is allowed) is a necessary condition for the finite (and low) entropy or heat capacity per one atom!
Bohmian mechanics denies the uncertainty principle – it says that the nearby forms of the "pilot wave" are in principle distinguishable – so it doesn't reproduce this clever trick of quantum mechanics. Consequently, Bohmian mechanics implies \(W\to \infty\) and therefore \(S/k\to \infty\), too (for one atom). Even with some regularizations and truncations you could propose as a fix, we will unavoidably have \(S\gg k\).

I wouldn't allow a student who can't get these elementary things to pass a course that includes quantum statistical physics. This is no research-level physics. It's about basics of statistical physics.

By the way, the Bohmian mechanics has other problems besides \(S\gg k\). One of them is that the entropy isn't extensive. If you have \(N\) atoms, the entropy should scale as \(N\). If you add \(M+N\) atoms, the entropies should add as \(S(M)+S(N)\), too. Mutual interactions between the subsystems will be neglected.

In viable theories of classical physics, this result arises because the phase space of a composite system is the Cartesian product \(PS=PS_A\times PS_B\). Its volume is therefore \(V(PS)=V(PS_A)V(PS_B)\). This multiplicative behavior is a simple generalization of the formula for the area of a rectangle. Note that Boltzmann's \(W\) obeys \(W\sim PS\) and therefore \(\log W\) is additive (the logarithm of the product is the sum of logarithms).

In quantum mechanics, the same logic holds but now \(W={\rm dim}\HH\). You only count the basis vectors of the Hilbert space. A composite system has the Hilbert space described by the tensor product of the subsystems' Hilbert spaces and its dimension obeys \(d=d_A d_B\). That translates to \(\log d = \log d_A + \log d_B\) and consequently, the entropy exhibits the same additive behavior.

It works. However, Bohmian mechanics is a strange hybrid: it is conceptually classical but tries to recycle the mathematical objects of quantum mechanics. The Hilbert space \(\HH_A\otimes \HH_B\) (or the set of "rays" in it – this issue doesn't make much difference) must be considered a "phase space" of the Bohmian theory – the set of all in-principle distinguishable, mutually exclusive states. But the number of points in this \(\HH_A\otimes \HH_B\) tensor product space is in no way a "product" of factors from \(A\) and \(B\) – it's much higher than that because \(\HH_A\otimes \HH_B\) is a "much bigger space" than \(\HH_A\times \HH_B \equiv \HH_A\oplus \HH_B\) – so you simply couldn't get any additivity for the entropy of a system in Bohmian mechanics.

The totally basic things such as the additivity of heat capacity fail to work in Bohmian mechanics.

These totally elementary, lethal bugs are never discussed by the Bohmians because they're inconvenient. Formally, they have never gotten "this far" because they haven't described what's happening with all their pilot waves and beables when a macroscopic system tries to reach an equilibrium state etc. However, a minute of a careful reasoning is enough to see that independently of any details, it is absolutely guaranteed that no theory based on the Bohmian paradigm may ever reproduce the extensive entropy or the entropy that is comparable to \(S\sim k\) for one atom. Every intelligent student should have understood the proof above. Nothing is really missing in it.

But the dishonest ideologues' self-respect depends on their misunderstanding of similar elementary arguments so you can be sure that these ideologically driven crooks will be "confused" about these elementary matters for centuries to come – even though all these arguments were absolutely clear to Dirac (and others) before 1930.

A bonus: Ilja Schmelzer added the following answer concerning the Bohmian heat capacities:
One should distinguish here de Broglie-Bohm theory for the general situation outside the equilibrium, and that for quantum equilibrium. Entropy is defined as usual by \(H=-\int \rho \ln \rho dq\). Outside the quantum equilibrium it is useful to split it into the entropy relative to the quantum equilibrium \(H=-\int \rho \ln (\rho/|\psi|^2) dq\). This relative entropy has been used by Valentini to prove a "subquantum H-theorem" that a general initial distribution will tend toward quantum equilibrium, see for example, http://arxiv.org/abs/1103.1589 for details.

In quantum equilibrium, we have \(\rho=|\psi|^2\), so that the formula becomes \(H=-\int |\psi|^2 \ln (|\psi|^2) dq\), thus, the standard quantum-mechanical one. After this, you can apply standard quantum theory.
That's great that he wrote it in this way because everyone familiar with the basics of quantum statistical physics knows that the first formula as well as the last formula (Schmelzer's equilibrium formula for the entropy) totally disagrees with the actual entropy in quantum mechanics. Instead, the entropy is \(S=0\) for any pure state \(\ket\psi\) while for a mixed state, we must use the von Neumann entropy\[

S = -{\rm Tr}(\rho \log \rho).

\] At very low temperatures, \(\rho\approx \ket{\psi_0}\bra{\psi_0}\) which implies \(S/k \ll 1\), something that neither formula of Schmelzer's can ever give. At the same temperatures, Schmelzer's formula gives a large value of \(H\) (which is meant to be \(S\)) so the theory is clearly falsified. You may see that Schmelzer doesn't understand statistical physics (and probably quantum mechanics) at all. He confuses pure states and mixed states (mixed states are absolutely needed to meaningfully discuss any nonzero values of entropy etc.) and does many other stupid things.

But as I have shown above, all these silly things are not just artifacts of his personal misunderstanding of basic physics. Instead, there can't exist any "fix" of Bohmian mechanics that would imply \(S/k\ll 1\) for one atom at \(T\to 0\). This remarkably low entropy requires the uncertainty principle (fundamental indistinguishability of states that are too close to each other, which is why only the basis vectors are counted into \(W\)) and the Bohmian mechanics denies this principle when it declares \(\ket\psi\) to be an objective reality (distinct from other values of \(\ket\psi\)).

(I was being extremely generous. Schmelzer's formula for \(H\) wasn't even dimensionally correct because \(\psi(\vec r)\) is dimensionful and the logarithms of dimensionful arguments have ill-defined units. Moreover, Schmelzer's result should have included two terms, one from the Bohmian particle and one from the Bohmian pilot wave. But even his "maximally QM-like sketch" must make it clear to every reader who has a clue that regardless of any "details", the Bohmian result for the entropy has nothing whatsoever to do with the correct, experimental or quantum mechanical, answer.)

Again, because the low heat capacities are experimentally verified, one may say that we have an experimental proof of the uncertainty principle – an experimental falsification of all theories that contradict this principle. And Bohmian mechanics is just an example of those dead hypotheses.

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