Friday, January 23, 2009

Bohmists & segregation of primitive and contextual observables

A student named Maaneli decided to defend his favorite theory, the Bohmian model of quantum phenomena.
Update: See also Anti-quantum zeal for a sociological discussion of these issues.
This picture, originally pioneered by Louis de Broglie in the late 1920s under the name "pilot wave theory", was promoted and extended by David Bohm in the 1950s. Because Bohm was a holy Marxist while de Broglie was just a sinful aristocrat, the term "Bohmian mechanics" for de Broglie's theory has been used for decades and I will mostly follow this convention, too. ;-)

At any rate, there is no real reason to fight for priority because the theory is worthless nonsense. The framework tries to describe the quantum phenomena in a deterministic way.

What is the pilot wave theory?

In this approach, the wave function is an actual wave, a "pilot wave", analogous to the electromagnetic waves. Besides these classical degrees of freedom, there are additional classical degrees of freedom, the positions and velocities of the particles.

These positions are influenced by the "pilot wave" in such a way that the pilot wave drives the particles away from the interference minima.

More precisely, if the probability distribution "rho(x)" for the (effectively unknown to us, but known to Nature in principle) particle's position "X(t)" at "t=0" agrees with "|psi(x,t)|^2", it will agree with it at later times "t", too. Such a law for "X(t)" can be written down - as a first-order equation - while the classical "psi(t)" obeys the classically interpreted Schrödinger equation.

It is not surprising in any way that the new, Bohmian equation for "X(t)" can be written down: it is clearly always possible to rewrite the Schrödinger equation as one real equation for the squared absolute value (probability density) and one for the phase (resembling the classical Hamilton-Jacobi equation). And it is always possible to interpret the first equation as a Liouville equation and derive the equation for "X(t)" that it would follow from. There's no "sign of the heavens" here.

And this construction is actually very unnatural because it picks "X" as a preferred observable in whose basis the wave vector should be (artificially) separated into the probability densities and phases: we will talk about the unequal treatment of different observables throughout this article. And because there are way too many similar classical equations appearing in this rewriting of one-particle quantum mechanics, it is very "easy" for an inexperienced physicist to assign these quantum objects with an incorrect, classical interpretation.

EPR problems

Einstein never liked the probabilistic interpretation of quantum mechanics but de Broglie's theory was so awkward that Einstein called it an "unnecessary superconstruction". More concretely, it is inconsistent with modern physics in many ways, as we will see.

Special relativity combined with the entanglement experiments is the most obvious example. Bell's theorem proves that if a similar deterministic theory reproduces the high correlations observed in Nature (and predicted by conventional quantum mechanics), namely the correlations that violate the so-called Bell's inequalities, the objects in the theory must actually send physical superluminal signals.

But superluminal signals would look like signals sent backward in time in other inertial frames. It follows that at most one reference frame is able to give us a causal description of reality where causes precede their effects. At the fundamental level, basic rules of special relativity are inevitably violated with such a preferred inertial frame.

You might think that the experiments that have been made to check relativity simply rule out a fundamentally privileged reference frame. Well, the Bohmists still try to wave their hands and argue that they can avoid the contradictions with the verified consequences of relativity. I wonder whether they actually believe that there always exists a preferred reference frame, at least in principle, because such a belief sounds crazy to me (what is the hypothetical preferred slicing near a black hole, for example?). But can they avoid the flagrant contradictions if you allow their theory to be arbitrarily unnatural?

Generically, they certainly can't: a theory of their kind will violate the relationships based on relativity by 100% or so. With a huge amount of fine-tuning, they can surely make some statistical predictions "look" relativistic even though they are fundamentally not.

But it is possible to see that one can't get relativistic predictions of a Bohmian framework for all statistically measurable quantities at the same moment, not even in principle. If a theory violates the invariance under boosts "in principle", it is always possible to "amplify" the violation and see it macroscopically, in a statistically significant ensemble. If such a violation existed, we would have already seen it: almost certainly.

The contradiction between relativity and semi-viable Bohmian models (that violate Bell's inequalities, and they have to in order not to be ruled out by experiments) is a very profound problem of these models. It can't really be fixed. But I want to look at another reason why these models are fundamentally wrong, and it is the following:

Life before the segregation of observables

What do I mean? Orthodox quantum mechanics is ontologically politically correct with respect to all measurable quantities, known as "observables", such as the position, velocity, momentum, angular momentum, spin, isospin, electric field at point (x,y,z), the gluon field's energy density elsewhere, the amplitude of the third harmonic wave on a string, and so on.

All of these observables are represented by Hermitean, linear operators. All of these operators have some eigenstates and eigenvectors. The wave functions can be decomposed into superpositions of eigenvectors associated with different eigenvalues. The squared absolute values of the coefficients give us the probabilities that the corresponding eigenvalue will be measured if the measurement is made right now. I have essentially sketched the postulates of quantum mechanics - the principles that every quantum theory shares, regardless of its detailed dynamical rules (such as the Hamiltonian).

You may dislike the probabilistic character of quantum mechanics but I guess that you must agree that the "philosophical democracy" between all observables is pleasing and natural.

Segregation begins

But things must work differently in Bohmian mechanics. Recall that the wave function always becomes a real wave in this framework. However, one must also supplement it with some additional classical observables - the positions of particles. What about the spin? Does it have some classical value, besides the individual components of the wave function (which may be e.g. a spinor)?

The answer is obviously a resounding No. The Bohmists assume that "X(t)" objectively exists. But if the "j_z" existed at the same moment, and had one of the allowed values "+1/2" or "-1/2", it would inevitably mean that "z" is a preferred axis and the rotational symmetry is heavily broken (much like the Lorentz invariance whose breaking by the Bohmian picture was sketched earlier). Moreover, the equation controlling the discrete function of time, "j_z(t)", couldn't be continuous.

You could also think that the "real particle" has pre-determined binary polarizations of the spin with respect to any axis in space. Well, that's no good, either. Such a picture could never predict simple facts, for example the fact that if a particle is prepared in the "spin-up" state with respect to one axis, the probability of "spin up" with respect to another axis must be "cos(theta/2)^2" where "theta" is the angle between the two axes.

Denying the existence of spin

So what do the Bohmists do? They simply deny that the spin exists in the same way as the positions or velocities do. While the wave functions - now real waves - still depend on the spin (because the wave functions must clearly be spinorial for an electron), there is no additional degree of freedom associated with them.

That's a tragic tumor, a seed of self-destruction, inserted into the very heart of the pilot wave theory but the Bohmists must obviously try to transform this tumor into a virtue. While Bell would say that the spin and similar observables are "contextual", Shelly Goldstein et al. write:
... referred to as contextuality, but we believe that this terminology, while quite appropriate, somehow fails to convey with sufficient force the rather definitive character of what it entails: Properties which are merely contextual are not properties at all; they do not exist, and their failure to do so is in the strongest sense possible.
How nicely radical these people are in writing the very opposite of the actual truth :-) - which says that the ontological reality and the degree of existence of the spin is, in the strongest possible sense, equivalent to the ontological reality of the position or any other observable. And even the previous sentence fails to convey the key point that those who fail to recognize the equal philosophical status of all observables shouldn't have passed their introductory quantum mechanics courses. :-)

In order to celebrate the Martin Luther King Jr Day, I will dedicate the rest of the text to a fight against the segregation of observables. :-) So my statement is very modest - that observables can't be segregated into the "real" primitive ones and the "fictitious" contextual ones - a fact that trivially rules out all theories (such as the Bohmian ones) that are forced to do so. There are many arguments to see why I am right:

Effective theories automatically remove segregation

Imagine a world where not all observables (which are linked to Hermitean operators in quantum mechanics) are "fundamental" at a hypothetically deeper (Bohmian) level. Only a subset of the "primitive" ones can be associated with real "properties" (additional classical degrees of freedom that accompany the wave function).

The position and the velocity are "real properties" and so is the orbital angular momentum - which is just the cross product of the position and the momentum. However, the spin can't be a "real property", as the Bohmists (and I) explained: it must be just "contextual", otherwise the Bohmian theory collapses immediately.

However, the spin and the orbital angular momentum are just two terms that add up to the total angular momentum. And the separation into the two terms is partly unphysical. For example, it depends on the resolution with which we describe the phenomena, or the characteristic scale of our effective theory, if you wish. What does it mean?

If you describe the atoms in a molecule as a chemist, you may want to describe the atoms as "indivisible" objects whose internal angular momentum is simply classified as a discrete spin. In such an approach, the whole angular momentum is "contextual". However, a physicist may know that there are electrons inside the atom. She can study their motion and identify the orbital angular momentum as an important part of the total angular momentum. Suddenly, most of the angular momentum becomes "primitive".

One can go even deeper and study shorter distances. The spin of elementary particles in the stringy tower of states can be derived from (and reduced to) internal vibrations and oscillations of a string. If you believe that the positions of all "string bits" along the string are "primitive" properties, i.e. observables equipped with additional classical degrees of freedom, you may argue that the spin is "primitive", too.

Of course, you need to consider superspace with spinor-valued coordinates to be able to obtain half-integral spins out of the internal "orbital" motion.

Decoherence, and not racial prejudices, decides about the effective segregation in the real world

That's my second point and it is a positive one because we are actually going to talk about the emergence of the classical limit in the correct picture. Clearly, some quantities in the real world look more classical than others. But what are the rules of the game that separates them? The Bohmists assume that everything that "smells" like "X" or "P" is classical while other things are not.

Well, that's not the case. Everything in this real world is quantum while the classical intuition can only be an approximation, and it is a good approximation only if decoherence is fast enough i.e. if the interference between the different eigenstates is eliminated. If it is so, the quantum probabilities may be imagined to be ordinary classical probabilities and Bell's inequalities are restored.

So if you want to know whether a particular quantity may be imagined to be classical, you need to know how quickly its eigenvectors decohere from each other. And the answer depends on the dynamics. Decoherence is fast if the different eigenvectors are quickly able to leave their distinct fingerprints in the environment with which they must interact.

Usually, position eigenstates produce radiation etc. described by microstates that are almost exactly orthogonal to each other, after a short time, for different choices of the position. That makes the basis of position eigenstates "effectively preferred", much like the basis {dead, alive} of the Schrödinger cat.

On the other hand, the state "alive+dead" of the Schrödinger cat doesn't decohere from "alive-dead", even though they are orthogonal to one another, because these two states of the cat don't emit microstates of radiation (imprints in the environment) that would be orthogonal to one another. So these two states of the cat never become good descriptions of "0" and "1" in a classical bit.

However, it is not a universal fact that the decohering eigenvectors are eigenvectors of a quantity labeled by the letter "X". They can be eigenvectors of "Omega" or any other letter and the geometric representation of "Omega" can be very different from a position.

Quantum computer: a contextual model of the whole world

The most extreme example of this fact is a quantum computer. Note that with a big enough quantum computer, you may emulate any quantum system with a satisfactory accuracy, by brute force. That includes quantum systems which have particles with positions "X" and momenta "P". However, the quantum computer is entirely made out of spins or other discrete "qubits"!

So according to the Bohmists, there are no additional classical degrees of freedom similar to "X". They only have the wave function as the quantum mechanicians do: the only difference is that they equip it with a wrong, deterministic interpretation. Because any additional "X"-like degrees of freedom are absent, they can never make a "measurement" in their system.

If the Bohmian theory is compatible enough with the observations, it must also be compatible enough with the conventional quantum theory because the latter seems to match the experiments. If it is so, it is very likely that the quantum predictions for the quantum computers are valid. One of them says that the events inside a particular quantum computer can be a perfect model of other quantum systems: they are really indistinguishable.

There can exist brains and people constructed out of quantum observables in such as quantum computer. However, they will never be able to realize themselves or measure anything. They will stay in the linear superpositions forever. The "measurement problem" is completely unsolved in such a picture.

Of course, the correct way to solve the "problem" is to accept the probabilistic interpretation of the wave function and of the quantum mechanical postulates. If you do so, there is really no problem left. In order to simplify their imagination, the Bohmists imagined the existence of additional classical objects - the classical positions.

But only in the simplest case of spinless 1-particle quantum mechanics, such a picture could be argued to survive the basic tests. Whatever you add, it makes the framework collapse. And you can make physics of the spins so complicated that virtually all of physics becomes "contextual", i.e. composed out of "non-real properties".

In that case, your wave functions will spread indefinitely, which should force you either to accept their probabilistic character, or to try to invent new levels of a physical way to imagine what's happening during the "collapse". I assure you that if you choose the latter approach, you will be failing forever. There can't be any "deeper" unexplained mechanism behind the random outcomes of experiments. They're genuinely random and they must be random, as the free-will theorem guarantees.

Dualities

I was talking about a quantum computer but we know all kinds of dualities where one discrete quantity becomes continuous in a limit, while another quantity becomes discrete. For example, in the BMN limit of the CFT, i.e. the Penrose limit of the AdS space, the number of particle excitations in a chain (the BMN operator) becomes interpreted as a spin of a particle (graviton or a low-lying stringy excitation) in the AdS space.

The Bohmists would almost certainly choose the number of particles to be a real, "primitive" quantity, while the spin is just "contextual". That's clearly wrong because these two types of observables can really be the very same thing: the difference is often purely linguistic in character. The physics may be identical, and it is just a matter of terminology whether we call a certain quantity "the spin" or "the number of some particles".

I could make things even harder for the Bohmian framework by looking into quantum field theory. What are the real, "primitive" properties in that case? Clearly, they want some quantities that often behave classically in classical limits. Except that there is no simple, universal description what they should be.

If you have low-frequency electromagnetic waves, it is the electric and magnetic vectors at each point, "E(x)" and "B(x)", that appear as natural classical quantities in a classical limit. That's why the classical electromagnetic field was the first historically understood limit of QED. On the other hand, typically for higher frequencies when "hf" is pretty high, we describe physics in terms of photons and we would prefer to imagine that its position is the real, "primitive" property that accompanies the wave function. Note that our choices would be very different in these two situations - but the two situations only differ by a boost because a boost is enough to change the frequency of an electromagnetic wave!

Absorption and "primitive" information loss

We have just explained that there are dualities (equivalences) that relate observables that are "primitive" with those that are "contextual" according to the Bohmian racist picture. ;-) However, dualities are not necessary to see the tension: evolution is good enough.

In fact, observables that look "primitive" to the Bohmists can evolve from the "contextual" ones, and vice versa. More generally, the amount or percentage of "primitive" (vs "contextual") information about the physical system is not constant in time.

There are all kinds of inconvenient processes that demonstrate this fact. For example, new coordinates "X(t)" of particles must be "invented" whenever new particles are produced at the colliders. Inconveniently enough, a collider can smash two protons or antiprotons and produce 10 of them. It often does so.

The Bohmian theories can have no self-consistent rules to "invent" the new classical values of all these new hypothetical classical hidden observables. The new particles could either be born at a random place, violating the local charge conservation, or they could be created at the same point. In both cases, you immediately face contradictions.

If you fundamentally violate the local charge conservation, you will be able to amplify this effect and obtain macroscopic violations of the charge. On the other hand, if you create all the new particles at the same point where the wave function happens to have some mysterious property, you will fundamentally violate the verified, microscopic T-reversal symmetry of the laws of particle physics because it is virtually impossible for several particles to meet at the same point which would be needed for them to annihilate (the reverse process: yes, be sure, in the real world, the number of particles can drop, too).

All these comments are huge question marks about a general problem that was understood to be a defect of the pilot wave theory from the very beginning: no one could ever produce a sensible explanation what happens with their "classical" wave function and with their classical hidden variable "X(t)" when a particle is absorbed. Although the pilot wave theory was motivated entirely by its goal to "shed light" on the measurement procedure, it has really no sensible, new picture to offer.

When a particle is created, one faces the T-reversal image of the same problems. They can't be solved.

Reduction of spin measurements to positions: not possible in general

There are many other arguments to see that the Bohmist segregated picture is indefensible. We have seen some of them but I want to sketch a similar argument from a different viewpoint.

By saying that the spin is not a real property, they also say that all the spin measurements (and the measurements of other contextual observables) must always reduce to the measurement of real, "primitive" properties such as the position and velocity: for example, the Stern-Gerlach devices change the position of a particle in the magnetic field, depending on the projection of its spin, and measure the position. Except that it's not the case in general.

In the quantum computer example, everything is made out of spins. Nevertheless, measurements must exist in that world, as long as the quantum computer emulates the environment causing the decoherence, too. Obviously, these measurements of certain functions of the spins (qubits in the quantum computer) can't be reduced to the measurement of any fundamentally "primitive" quantities because there are no fundamentally "primitive" quantities in such a quantum computer!

One doesn't have to consider a quantum computer. A simple polarizer is a gadget that can measure the x/y linear polarization of a photon - its form of an internal, discrete, spin-like degree of freedom - without changing the location of the photon (which would be pretty hard to do). You can do similar things with the electron, too. In all these cases, the Bohmian theories have no chance to explain why the photon is sometimes absorbed and sometimes it is not. The Stern-Gerlach device is not the only method to feel the spin!

Bohmian mechanics doesn't imply quantization of observables

There is one more key problem with the Bohmian interpretation which is a bit different from the problems discussed above but I will mention it, anyway: it doesn't predict the correct rules of quantization (discreteness) of various quantities. For the sake of concreteness, consider the orbital angular momentum of the Hydrogen atom.

How do we know that "m=l_z/hbar" must be an integer? Well, it is because the wave function "psi(x,y,z)" of the "m"-eigenstates depends on "phi", the longitude (one of the spherical or axial coordinates), via the factor "exp(i.m.phi)" which must be single-valued. Only in terms of the whole "psi", we have an argument.

However, when you rewrite the complex function "psi(r,theta,phi)" in the polar form, as "R.exp(iS)", the condition for the single-valuedness of "psi" becomes another condition for the single-valuedness of "S" up to integer multiples of 2.pi. If you write the exponential as "exp(iS/hbar)", the "action" called "S" here must be well-defined everywhere up to jumps that are multiples of "h = 2.pi.hbar".

That's too bad because this is an inherently quantum condition and you may only explain the allowed "monodromies" (behavior under the trip around the string) naturally in terms of "psi". In the Bohmian picture, the crucial region that decides about the single-valuedness of "S" is the "string" (co-dimension two locus) in space where "R=0" (i.e. "psi=0"). To explain what can happen with "S" when you make a circular trip around this string, you really need at least "old quantum mechanics" - Bohr's ad hoc quantization rule for the integer number of waves - or another principle that inserts the "quantization" itself.

So even though the Bohmian mechanics stole the Schrödinger equation from quantum mechanics, the superficially innocent step of rewriting it in the polar form was enough to destroy a key consequence of quantum mechanics - the discreteness of many physical observables. More generally, something very singular seems to be happening near the "R=0" strings in the Bohmian model of space.

But quantum mechanics correctly says a completely different story: there is nothing "stretched" around the "R=0" i.e. "psi=0" strings at all. They're just the regions where the particle won't be observed but nothing really goes to infinity over there. The very special role of the "R=0" as imagined by the Bohmian picture is just another consequence of its incorrect preference of the position "X" over other quantities.

For example, in the momentum "P" space, we would get completely different loci where "psi(p)=0". They wouldn't be special, either. If you take a superposition of two functions "psi(x,y,z)", the location of the string completely changes, too. The Bohmian mechanics thinks that this location is physical, singular, and important. But it is easy to see that in the actual world, as described by proper quantum mechanics, the location is physically uninteresting, smooth, regular, and unimportant.

Get real: it is just an ordinary interference minimum. Just because the electron is not located somewhere, doesn't mean that the place is filled with mysterious, ambiguous (undetermined "S"), and singular dragons: in most places of the Universe, the electron is not located, after all. That's why a careful physicist shouldn't use the "R exp(iS)" form of the wave function "psi" in cases when it routinely falls to zero - for example in cases with interference, small quantized quantities, or elsewhere, far away from the classical limit.

The separation of the phase from the absolute value is only useful in the WKB, semiclassical approximation. The more likely and important the "psi=0" value becomes (i.e. the more quantum-ish regime we consider), the more defective and misleading the polar form of "psi" becomes.

Summary

It is therefore very important that we treat all observables with the same philosophical tools, even though the mathematical details (such as the spectrum) clearly depend on the particular operator we study.

The segregation of the observables into the "primitive" and "contextual" ones is unphysical, contradicts our freedom to consider effective theories and to construct computer models, disagrees with the existence of dualities (equivalences) between the different descriptions, behaves discontinuously under time evolution, leads to fundamental microscopic violations of the time-reversal symmetry, amplifies the problems of the Bohmian theories to survive the successful tests of relativity, introduces new singular regions of space that require a special (quantum) treatment to restore the quantization rules, and is surely guaranteed to give us no correct predictions that couldn't be extracted from quantum mechanics even if you imagined that all the inconsistencies above would be miraculously cured.

While most of us have surely hated the probabilistic nature of quantum mechanics at some point - I surely did 20 years ago - it is an established fact and mature physicists should be able to see it. The attempts to return physics to the 17th century deterministic picture of the Universe are archaic traces of bigotry of some people who will simply never be persuaded by any overwhelming evidence - both of experimental and theoretical character - if the evidence contradicts their predetermined beliefs how the world should work.

If someone still believes this Bohmian nonsense, he should at least reduce his arrogant suggestions that he is ahead of the other physicists. In fact, he is still mentally living in the 17th century and is unable to grasp the most important revolution of the 20th century science. He may overwhelm others with his strong belief that all the problems with spins, fields, quarks, renormalization, dualities, and strings can be overcome in the Bohmian framework.

But the very fact that the Bohmists actually don't work on the cutting-edge physics of spins, fields, quarks, renormalization, dualities, and strings is enough to lead us to a very different conclusion: they're just playing with fundamentally wrong toy models and by keeping their focus on the 1-particle spinless case, they want to hide the fact that their obsolete theory contradicts pretty much everything we know about the real world.

And that's the memo.