## Saturday, April 07, 2018

### A nice video challenging Bohmian mechanics

After numerous wonderful videos about the Bible, Jesus, and the Big Flood that you've hopefully watched ;-), the Inspiring Philosophy YouTube channel released another video about the foundations of quantum mechanics eight hours ago (thanks to Werner Heisenberg for the URL):

Because I've praised numerous other videos on that channel in the past, it shouldn't be surprising that I liked it – and more or less agreed with everything that was said in the video. After all, a 2013 blog post by your humble correspondent is quoted at 4:33 in the video above.

There are several basic ideas that are wonderfully covered in the video – and some ideas that are missing. Among the ideas that are covered, we often hear about the disagreement between Bohmian mechanics and special relativity; and its being ad hoc, adding lots of concepts, detailed laws, and detailed observables that are unobservable and unhelpful to explain anything.

So for example, I am quoted as saying that a generic Bohmian mechanics – a realistic theory – contradicts the Lorentz symmetry of special relativity. Such a disagreement will probably be obvious but even the rough agreement with the observations requires (at least) infinitely many parameters to be adjusted. So Bohmian mechanics may describe the relativistic phenomena at most as another revival of Lorentz's aether – some environment that picked a preferred reference frame but that mechanically "caused" the length contraction and time dilation that are similar to those automatically implied by relativity.

A typical Bohmian theory attempting to emulate a relativistic quantum field theory would have infinitely many terms, all of them would have to be adjusted to produce seemingly relativistic predictions, and the exact predictions couldn't be precisely relativistic, anyway. On top of that, even the simple fact that the evolution of the wave function is linear – which, in quantum mechanics, follows from pure logic because this linearity is a complex extension of various laws of addition for probabilities – is totally unnatural in Bohmian mechanics where the wave function is claimed to be a pilot wave i.e. just another classical field. There's no reason why such classical fields should evolve in a linear way.

So all the things that are natural or guaranteed by consistency or basic logical rules in orthodox quantum mechanics become unnatural, ad hoc, and generically broken in Bohmian mechanics. All of these Bohmian additions conflict with Occam's razor and they're motivated by an ideological wishful thinking. The proponents of Bohmian mechanics simply want to preserve the objective reality in which the observer plays no fundamental role. They want to avoid the "need for the observer to define what he perceives" at any cost.

While I agree with the video, I think that it's far from presenting the full set of known arguments against Bohmian mechanics. The video says that Bohmian mechanics "only works for non-relativistic quantum mechanics". Well, I would only agree with the assertion that Bohmian mechanics solves a certain important task needed to emulate non-relativistic quantum mechanics. But it cannot fully replace quantum mechanics, not even in the non-relativistic situations.

The video says that the pilot wave evolves to the unlimited superpositions, including the possible events that don't take place. Because the "events that accidentally didn't take place" are still included in the increasingly complicated wave function, the Bohmian picture implicitly includes the "many worlds", too. There is still a part of the wave function (the Bohmian guiding wave) in which Hitler won the Second World War.

Well, is it true? It's hard to answer because no one has ever given a "complete definition of Bohmian mechanics", including how to use it, what happens after the absorption or measurement of a particle, and so on. So I think that some people do believe that the pilot wave never collapses and we live in a world co-inhabited by an increasingly complex wave function describing zombie events that could have taken place but didn't. But I also think that some Bohmists believe that some "cleaning" that collapses the pilot wave does take place in some way although they haven't ever formulated possible laws that could govern this "cleaning" (rules of the "funerals for zombies").

I am not just criticizing their ambiguity about that point – and other points. I have actually analyzed both possibilities and concluded that none of them can work. So the Bohmists are trying not to address this question and others in order to "preserve their hopes". If they actually tried to say something particular about the "cleaning" or "not cleaning" mechanisms, they would be forced to see that there's no hope for their theory.

I've discussed lots of other problems with Bohmian mechanics that aren't covered in the video because they may be more technical. One of them is that the loop corrections arising from Feynman diagrams with loops can't possibly arise from any Bohmian theory because the loop corrections represent the interference between intermediate histories that qualitatively differ from each other. They have different numbers of particles in the intermediate states, among other things. The interference between states with different numbers of particles prove that the number of particles in the intermediate state cannot be objectively real! So the very existence of the loop corrections proves that there can't exist any particular positions of the particles.

Just to be sure, there are lots of other ways to see that Bohmian mechanics has lethal problem with the particles' spin (or with the rotational invariance if you try to introduce the spin as a classical bit), and with the pair creation or annihilation of particles and antiparticles. The disagreement between the relativistic quantum phenomena (of QFT) and Bohmian mechanics is not just a formality. Bohmian mechanics really seems to prevent you from having all these important aspects of QFTs – particle creation, annihilation, antiparticles, spin, gauge invariance etc.

But as any realist theory, Bohmian mechanics also contradicts the low heat capacities of atoms. The heat capacities are numbers such as $$C_V=3k/2$$ for many atoms. Note that the heat capacity is the heat (energy) per one degree of warming (temperature). So it has the same units as Boltzmann's constant. Indeed, the heat capacity of an atom (e.g. in the gas) is a small multiple (of order one) of Boltzmann's constant. This translates to the statement that one atom carries $$N$$ bits of information where $$N$$ is of order one – and changes by $$\Delta N$$ which is also of order one if we remain at temperatures well below the ionization temperatures.

But realist theories such as Bohmian mechanics add lots of unphysical wheels and gears. So they unavoidably increase the number of bits of microscopic information carried by one atom to $$N\gg 1$$. This would unavoidably show up as much higher, and probably infinite, heat capacities. The precise position of the Bohmian particles carry $$N\gg 1$$ bits. The pilot wave carries an even higher number $$N\gg 1$$ of bits. The growth of the number of bits may even be faster than linear if you increase the number of atoms. You don't even expect the heat capacity to be additive or extensive. Well, the number of hidden bits is infinite even for simplest classical "models" but even if you tried to regulate the number and make it finite, it would almost certainly be much higher than one.

This problem is tightly linked to the "cleaning" problem above. You know, whether you enforce some cleaning of the zombie wave functions or not, your theory must be capable of describing the "gas at thermal equilibrium" because that may be observed. And in that equilibrium, all possible configurations – points in your (extensive, complicated) phase space – that have the right value of the energy and other conserved quantities must be equally likely.

When it's so, you may always count or measure the number of microstates (the volume in the phase space) that quantifies the entropy carried by the system. Whether you like it or not, the temperature will be close to the energy divided by the entropy. It means that the entropy will affect the temperature of objects – and therefore the question whether the heat will flow from the first object to the second or in the opposite direction. All these things may be easily measured, it's normal thermodynamics. And the Bohmian mechanics therefore predicts that atoms will have a much higher, probably infinite, heat capacity – a prediction that is instantly falsified by a simple experiment with any lukewarm gas. ;-)

You know, the magic new rules of quantum mechanics guarantee the tiny heat capacity in an ingenious way. An atom carries roughly 1 bit of information if it has reasonable chances to be excited from the ground state to an excited state. Why? Imagine a hydrogen atom that is either in 1s or in 2s. They're two states. The relevant accessible Hilbert space is two-dimensional. So the number of possibilities is equivalent to two tiny cells of the phase space – to $$2\times 2\pi\hbar$$.

That's true despite the fact that the wave functions of the hydrogen atom (and especially more complex atoms let alone molecules) looks very complicated and are constructed out of infinitely many mathematical variables. But the variables needed to describe a wave function aren't observable – and they aren't observables in the foundational, quantum mechanical sense. Only operators are observables. And only the observable for energy $$E$$ that distinguishes two possibilities, 1s and 2s, is relevant in our situation. If they're equally likely, the entropy is one bit. For atoms, the information is always comparable to one bit. Aside from this energy, all other degrees of freedom are completely and literally frozen. They are completely determined. Well, they're determined to be distributed according to a probabilistic distribution that only depends on $$c_{1s}$$ and $$c_{2s}$$, two complex amplitudes associated with the two energy eigenstates. These two complex amplitudes describe the pure state of the atom completely. There is no freedom for the other observables to change in any other way. Only energy may change and it only has two values it may change. One qubit describes everything and thermodynamically, because of the rules of logic, one qubit is exactly as much as one bit was in a classical theory.

That's why quantum mechanics gives you heat capacities such as $$C_V=3k/2$$ per atom and why any theory with lots of additional realist wheels, gears, and whistles isn't just conflicting with some vague philosophical Occam's razor principle. These whistles contradict the observations of the low heat capacity, too.

If you added the spin as a "beable", so that an electron would either be "objectively spinning up" or "objectively spinning down", you would pick a preferred axis, the $$z$$-axis, and the fundamental laws would break the rotational symmetry. Even the seemingly simple rotational symmetry would become as "broken in principle" as the Lorentz symmetry of special relativity that we discussed above. So Bohmists typically imagine that there's no "beable" associated with the spin.

Why is this "spin beable" a problem? It's because the "spin up" relatively to another, tilted axis is a complex superposition of "up" and "down". So according to the very basic mathematics of quantum mechanics that they want to keep, the superpositions of states are essential to get the spin along all axes, to preserve the rotational symmetry.

But superpositions are needed and experimentally provable in lots of other cases – in principle, in the case of any observable. The number of particles $$N$$ may be measured; but we also need to allow different values of $$N$$ to interfere with each other because this interference produces the loop corrections to physical quantities (Feynman diagrams with loops). Quantum mechanics allows both: $$N$$, like any observable, may be measured; but its eigenstates may also be combined into arbitrary complex superpositions, like any two state vectors.

The Bohmian paradigm doesn't allow you to do both in general. It basically demands that every observable that may be measured already has some value that is determined before the measurement, like the particle's position. But the problem is that infinitely many observables – that generically have nonzero commutators with each other – may be measured. And you simply can't have classical values for all of them prepared before the measurement. That would conflict with the uncertainty principle.

For this reason, the promotional Bohmian mechanics for non-relativistic spinless particles only seems to "work" because it basically assumes that the particle positions are the only observables that we ultimately measure. Particles land at different places of the photographic plate – they have had some position before the detection, it's being assumed. So the position becomes "special" in some way – and that's why the Bohmian particle positions are the allowed, privileged "beables" and things seem consistent.

But that's just an illusion. Instead of considering particles that may fly to very different places, consider a brain and/or a computer where the place of everything seems fixed. The brain is just sitting in the skull, the CPU is sitting inside your laptop, but things are still happening inside. Some electric impulses run through both your brain as well as your laptop. Do the particle positions help you to predict what happens in your brain or your laptop? Not really because the relevant observables aren't really positions. The relevant observables are voltages of transistors or energy states of atoms.

When an electron has a well-defined position, it can't have a well-defined energy in the atom because these two observables don't commute with each other. For this reason, Bohmian mechanics that picks the particle positions as the preferred observables fails to describe brains and computers. You would need different relevant beables, just like you needed the "real spins" in the case of the spin up and down states, but you don't have them. In lots of situations, you would need "discrete beables" but there can't be any differential equations dictating the evolution of "discrete beables" at all (because they need to discretely jump, not to differentially evolve).

These are just examples that should convince you about a much more general point: It's fundamentally wrong to pick any preferred observables that may become "beable" because, just like quantum mechanics says, there are absolutely no preferred observables. The set of possible observables on an $$N$$-dimensional Hilbert space is a real $$N^2$$-dimensional space of Hermitian matrices. Almost any pair of observables has a nonzero commutator with each other. All of them are equally good observables, all of them may be measured by some apparatus.

It's just wrong to try to pick preferred observables such as the particle positions. There are no preferred observables in Nature – according to all the observations we know. To create a new ad hoc rule "which observables in a given physical system are real" i.e. associated with "beables" is not only unnatural and conflicting with Occam's razor (and your answer must therefore be considered ad hoc whatever it is). It usually leads to unavoidable disagreements with the observations.

The right theory must be capable of dealing with all observables and predicting probabilities of their values – because those can be measured; and it must allow all the superpositions of the eigenstates, too – simply because physical systems in such superpositions may always be prepared. It's just fundamentally wrong to do what Bohmian mechanics is doing, how it's separating observables to beables and others.

One may write down – and I have partially written down – explicit examples (analogous to Bell's theorem) that show a sharp contradiction. But you should understand a much more general point, namely that such examples are absolutely everywhere and none of them is more important than others (just like Bell's theorem isn't more important than other examples of quantum surprises). We basically directly observe that there are no "preferred observables" or "preferred bases" anywhere in Nature, so Bohmian mechanics directly contradicts something that we basically directly observe.

At the end, quantum mechanics may be viewed as a "modest incremental" change of classical physics. It still deals with observables – all the things that may be in principle measured by a procedure in a single repetition of the situation – but they're not mathematically represented as mutually commuting, $$c$$-number-valued functions on the phase space; instead, they're Hermitian matrices acting on the Hilbert space. So they have nonzero commutators with each other which is why you can't imagine that they simultaneously have objective values.

But these observables evolve according to the Heisenberg (picture) equations of motion as a function of time. These equations are totally analogous to the dynamical laws in classical physics; after all, in many cases, they just differ by the addition of some hats. To understand that all of quantum mechanics and its Copenhagen principles are unavoidable, you must simply convince yourself that the observables do follow the Heisenberg (analogous to classical) equations as functions of time; they do have some algebraic relationship that turns them into matrices with nonzero commutators (the quantum evolution becomes nicer and "easier" because commutators seem simpler than Poisson brackets); and then you must think how some predictions may be extracted from the "nonzero-commutator-enhanced" classical physics. And the only way to extract prediction from this picture is to allow the observer to pick what he wants to measure – because no justifiable canonical choice may exist without the observer's choice – and accept the identification of the initial state as the eigenstate of the initial measurements; and Born's rule for the probabilities – because you may see that it makes the theory nice and the probabilistic predictions may be verified in millions of examples.

There's really no freedom left. There's no freedom to invent "interpretations". Every interpretation that tries to deny the basic logic – that was already discovered by the ingenious physicists almost a century ago – is guaranteed to be on the wrong track. The only freedom is the freedom to pick the jargon, notation, and the methods how to teach and visualize the theory so that someone learns it. But these things don't change anything about the actual inner workings of a theory and what it predicts for any well-defined question. They can't change the linearity of the evolution of the state vector; the association of the observables with Hermitian linear operators; the need for the "observable of interest" to be provided before any answers are calculated; Born's rule; and the unavoidable intrinsic randomness of the outcomes.

So your freedom is the freedom of a teacher who picks the examples; who picks the jargon (which may be more or less "spiritual" i.e. which may more or less provoke the people who have problems with the novelties of quantum mechanics); you may pick the Heisenberg, Schrödinger, Dirac, or Feynman picture which are mathematically equivalent; you may assume that the observables are $$x$$-like or binary like, projection operators for Yes/No questions; you may decide whether you want to discuss decoherence as a relevant approximation that makes a point or whether you live without it; you may decide whether the final perceived observables are states of your brain and mind or states of the apparatus that you assume to be tightly connected to your perceptions; and so on. None of these choices really makes any difference for the predictions and the set of things that are predictable by the theory. Every modification that does change something about the true predictions or even the set of things that are predictable is bound to be wrong.