Some people can't get used to the fact that classical physics in the most general sense – a class of theories that identify Nature with the objectively well-defined values of certain (classical) degrees of freedom that are observable in principle and that evolve according to some (classical) equations of motion, usually differential equations that depend on time, mostly deterministic ones – has been excluded as a possible fundamental description of Nature for almost a century.
Classical physics has been falsified and the falsification – a death for a theory – is an irreversible event. Nevertheless, those people would sleep with this zombie and do anything and everything else that is needed (but isn't sufficient) to resuscitate it. Of course, it's not possible to resuscitate it but those people just won't stop trying.
Bohmian mechanics, one of the main strategies to pretend that classical physics hasn't died and hasn't been superseded by fundamentally different quantum mechanics, was invented by Prince Louis de Broglie in 1927 who called it "the pilot-wave theory". In the late 1920s, the 1930s, and 1940s, physicists were largely competent so they didn't have any doubts that the pilot wave theory was misguided by its very own guiding wave ;-). Exactly 25 years later, the approach was revived by David Bohm who made the picture popular, largely because he was a fashionable, media-savvy commie (he's almost certainly the recipient of Wolfgang Pauli's famous criticism "not even wrong" that was ironically hijacked by aggressive Shmoitian crackpots in the recent decade). Prince Louis de Broglie liked the new life that apparently returned to the veins of his old sick theory so he didn't even care too much that his theory was going to be attributed to someone else and that the someone else was a Marxist rather than an aristocrat.
A constraint that defines Bohmian mechanics is simple: it should be a classical theory that emulates quantum mechanics as well as it can. The champions of the Bohmian theory know that getting the same predictions as quantum mechanics is the maximum goal they may dream about – they can never beat quantum mechanics – and they sort of realize that even this tie is too much to ask in general. Most of the Bohmian advocates seem to know that their theory can't be accurate, especially because of its fundamental conflict with relativity – but they don't seem to care. The fact that the Bohmian mechanics agrees with their fully discredited preconception that Nature is fundamentally classical is more important for them than the (in)accuracy of the predictions extracted from their pet theory.
It's straightforward to explain why it's possible to design a classical theory that parrots quantum mechanics when it comes to certain questions.
Bohmian mechanics is at least vaguely defensible in the non-relativistic quantum mechanical models only; in more general theories, it collapses completely. How does it rebuild non-relativistic quantum mechanics for one particle, for example?
Proper quantum mechanics of this system may be written down in Schrödinger's picture that dictates the following time evolution to the wave function:\[
i\hbar\frac{\partial}{\partial t}\psi(q,t)=-\sum_{i=1}^{N}\frac{\hbar^2}{2m_i}\nabla_i^2\psi(q,t) + V(q)\psi(q,t)
\] The way how this wave is evolved in agreement with the equation above contains all the "mathematical beef" of quantum mechanics for the given system and to get the right numbers, any classical caricature of quantum mechanics simply has to contain some objects that are pretty much equivalent to \(\psi(q,t)\). These objects are then assigned totally different, wrong interpretations in the caricatures but they must be there and they must evolve according to the same Schrödinger's equation.
Bohmian mechanics buys \(\psi(q,t)\) and incorrectly interprets it as a classical wave – a field that has objective values and is in principle measurable. Of course, we know from quantum mechanics as well as experiments that the value of the wave function simply shouldn't be and isn't measurable in a single repetition of an experiment. So the Bohmian apologists must also invent convoluted mechanisms to make the wave unmeasurable – because it is unmeasurable according to the experiments – despite the fact that the wave function is fundamentally measurable in their theory.
Bohmian Rhapsody, via Dilaton.
Is this the real life? Is this just fantasy?
Caught by the guiding wave. No escape from reality.
Open your eyes. Look up to the skies and see:
I'm just [a] state vector, I need no images.
Because I'm easy come, easy go.
A little high, little low.
Anyway the [pilot] wave blows, doesn't really matter to me, to me.
The pilot-wave theory adopts \(\psi(q,t)\) as an objective classical wave – which it gives a new name, the "guiding wave" or "pilot wave" – but in order to agree with the fact that particles may be observed at sharp locations despite the fuzziness of the wave functions associated with them, they must add some additional degrees of freedom: the actual classical position of the particle. The defining philosophy of Bohmian mechanics is that the actual, classical position of the particle is "guided" by a function of the classical field emulating the wave function so that the probability distribution for the particle's positions remains what it should be according to quantum mechanics. For example, the laws that guide the actual classical particle must be such that they repel the particle from the interference minima in a double-slit experiment:
The right end of the picture (the photographic plate) shows denser and less dense regions, the interference maxima and minima.
Can you find the appropriate rules for one non-relativistic spinless quantum particle that is able to do it in a way that imitates quantum mechanics? You bet. All the tools are available in conventional quantum mechanics for this system. Recall that in quantum mechanics, \(\rho=|\psi(q,t)|^2\) is the probability density that the particle is sitting near location \(q\) at time \(t\). But quantum mechanics also allows you to define the probability current\[
\bold j = \frac{1}{m} \mathrm{Re}\left ( \psi^*\bold{\hat{p}}\psi \right )
\] Note that it is again sesquilinear (bilinear with one star) in the wave function. We act on the wave function by the momentum operator \(\bold{\hat{p}}=-i\hbar\nabla\), multiply the result by \(\Psi^*\) just like when we calculated the probability density, take the real part, and divide it by the mass \(m\). You see that it only differs from the formula for the probability density by the extra operator \(\bold{\hat{p}}/m\), the operator of the velocity \(\bold{\hat v}\), inserted in the middle. The real part could have been added to the probability density as well because it was real to start with.
At any rate, if you define the probability density and the probability current correctly, they obey the continuity equation\[
\frac{\partial \rho}{\partial t} + \bold \nabla \cdot \bold j = 0.
\] The divergence of the probability current exactly agrees with the decrease of the probability density in the given region. It means that the probability current measures how the probability has to flow into/from a given infinitesimal volume if you want the probability density to change just like it should according to Schrödinger's equation.
Now it's easy to realize that if you define a classical "velocity field"\[
\bold{\hat v} = \frac{\bold{\hat j}}{\rho},
\] it will be very useful for emulating quantum mechanics. It's not hard to prove that if you define Bohmian mechanics as the "classicalized" wave function together with a classical position \(\bold{\hat q}(t)\) that evolves according to the "guiding equation"\[
\ddfrac{\bold{\hat q}}{t} = \bold{\hat v} (\bold{\hat q}(t)),
\] the trajectories of the classical particles will be repelled from the interference minima, attracted to the interference maxima, and will obey a more specific rule: If you imagine that the particles in the initial state are distributed according to the probability distribution given by \(\rho(\bold{\hat q},t)\), it will be true for the final state, too.
This trick may be generalized to the case of \(N\) non-relativistic particles. In this case, the wave function \(\psi\) becomes a classical wave that is a function of the \(3N\)-dimensional configuration space. This configuration space is larger than the ordinary space and is "multi-local" and because we have this "multi-local" old-fashioned classical field, the theory becomes explicitly non-local and a violation of the Lorentz symmetry, at least in principle, is inevitable.
I would like to emphasize that it's no surprise at all that it's possible to find the equation that evolves the probability distribution in the right way. Imagine that you start with a wave function \(\psi(\bold{\hat q})\) at some time \(t_0\). Throw a trillion of dots – particles – to the space that are distributed according to \(\rho = |\psi|^2\). Do the same thing for the final moment \(t_1\) when the wave function is different. You will have two configurations of trillions of particles. It's not shocking that you may "connect the dots" from the initial state to the final state in some way.
A way that is simple enough, one based on the probability current and described above, gives you one of the solutions. But it's not the only solution. In reality, the "initial dots" could be connected with the "final dots" in infinitely many ways (well, a "mere" trillion factorial if you only have a trillion of dots). In the continuous language, you could e.g. make the particles move along spirals inside the cylinders that surround the interference maxima. Is one way to connect the dots better than others?
Of course, it's not. All of them are equally good. Quantum mechanics commands you to learn something about the initial state – some wave function or density matrix that encode the initial probability distribution – and it allows you to predict the probabilities for the final state. But it doesn't tell you which of the initial particles is connected with which final particle, i.e. how to connect the dots. It doesn't inform you about any preferred classical trajectory that connects them (and Feynman's approach orders you to sum over all trajectories). If you could actually "measure" this permutation that determines how the dots are connected, quantum mechanics would be shown incomplete.
However, it's totally obvious that there's no way to measure the trajectories or permutations inside. The particles just don't have well-defined, in principle measurable trajectories between the measurements for the usual Heisenberg uncertainty principle-based reasons. If you tried to measure the trajectory before the final measurement, you would change the experiment and destroy or damage the final interference pattern. So all the precise lines on the "caricature of the double-slit experiment"
are pure fantasy. They're just crutches for the people who need some specific picture of the intermediate states to be drawn. But the specific picture we drew is in no way better than infinitely many other pictures we could draw that would predict the same interference pattern, the same probability distributions for the final state. Everything we added because we wanted the physical system to have objective properties prior to the measurement – because we're bigots who can't accept the fact that classical physics has died – is unphysical. The added value is purely negative. Everything we added to get from proper quantum mechanics to Bohmian mechanics is rubbish. And many things we're forced to lose when we switch from quantum mechanics to Bohmian mechanics are essential.
Because the wave function has a probabilistic interpretation in proper quantum mechanics (it is a ready-to-cook meal from which one may quickly prepare various probability distributions by a calculation), it doesn't matter that it spreads. The spreading of the wave function doesn't make the world more fuzzy. It only makes our knowledge about the world more uncertain. But once we learn the answer to a question – e.g. about the position of a particle – the world fully regains its sharp character it boasted at the beginning. If you only know that the probability of 1,2,3,4,5,6 are 1/6 for some dice in Las Vegas, it doesn't mean that the dice became structureless balls or that the digits written on their sides have become fuzzy or mixed or smeared. It just means that we have one equally sharp cubic die but we just don't know its orientation in space. The uncertainty coming from quantum wave functions are analogous – they only differ from the "classical uncertainty" by their inevitability.
That's not the case of Bohmian mechanics. The wave function is interpreted as a classical field of a sort and it is objectively spreading. So something objective is being diluted all over the Universe. That's terrible because this objectively makes the Universe increasingly more fuzzy and bizarre. The useless parts of the guiding wave – the "classicalized" wave function – should be killed in some way because they became useless. But Bohmian mechanics doesn't imply anything of the sort. If you want to clean the garbage of the no-longer-needed branches of the wave function, you will have to add another independent contrived mechanism. Such a mechanism will be a new source of a violation of the Lorentz invariance.
(You also need a special mechanism that prepares the guiding wave in a certain initial state and one more mechanism that distributes the "actual particle" inside the appropriate distribution with the right odds because these two things don't follow from Bohmian mechanics as we have defined it above, either. Most of these things are ignored by the Bohmists. Note that with the right probabilistic interpretation – quantum mechanics directly connects the knowledge about the past with the knowledge about the future, without any new crutches in between – we don't need to invent any new mechanisms.)
I think that a sane, critically thinking person must be able to realize what he is doing if he is doing such things. He is drawing a ludicrous caricature of Nature – a physical system that is actually governed by the laws of proper quantum mechanics – that reproduces some properties of the correct, quantum theory. The project of drawing the caricature is motivated by the desire to defend a philosophical dogma that the world is fundamentally classical even though it is clearly not. If he has at least some conscience, he must feel analogously as if he were counterfeiting a $100 banknote. He must know that what he is producing isn't the "real thing"; it is just a forgery that can bring him greater personal benefits than the actual banknotes but that's where the advantages stop.
But every change from the proper quantum mechanics to the pilot-wave theory is clearly wrong – the "added value" is unquestionably negative. Because the Bohmists don't like the probabilistic character of the wave function, they turn it into a classical wave – the guiding wave. But a classical wave that spreads objectively makes the world ever more fuzzy. So one has to introduce new tricks to have a chance that this increasing fuzziness doesn't spoil the world. All these tricks – tricks that can't really ever be defined in such a way to imitate quantum mechanics completely accurately – have to be considered and added just in order to mask the fact that the wave function is simply not a classical field.
It's fair to say that the claim by quantum mechanics that the wave function is not an objectively real wave or field that can be in principle measured is something that we have proven by direct experiments. Attempts to pretend that the wave function is a classical wave are just attempts to mask the truth. I am confident that every Bohmist must ultimately realize it is so and he must be dishonest if he claims that his efforts are more justifiable than the efforts of creationists who are trying to obscure the explicit evidence in favor of evolution: they are exactly equally unjustifiable.
Moreover, it's sometimes being said or thought that the perfect emulation of quantum mechanics can be done. Because the invalidated dogma that Nature is fundamentally classical is holy for these bigots, they think that it should be done, too. But the truth is that it can't be done for a general physical system and for a general choice of observables we may measure in actual experiments described by general enough quantum theories.
Try to add the spin to a particle. If the logic of Bohmian mechanics – the wave function "is" a classical field and we should also add some classical values of a maximum set of commuting observables – were universally valid, it's clear that aside from the spinor-valued wave function \((c_{\rm up},c_{\rm down})\), we should also assume that Nature "objectively knows" about the classical bit of information that tells you whether the spin is "actually" up or down.
However, even the Bohmists realize that if every electron "objectively knew" whether its spin is up or down with respect to the \(z\)-axis, then the laws of physics would break the rotational symmetry because the \(z\)-axis would play a privileged role. Roughly speaking, the ferromagnets would always be oriented vertically, to mention an example. If the \(z\)-component of the classical angular momentum is quantized, it's totally obvious that the other components can't be quantized. A nonzero vector can't have integer (or half-integer) coordinates in each (rotated) coordinate system.
Because they sort of realize that the rotational symmetry holds exactly and the hypothesis that the classical value exists with respect to one axis would break the symmetry kind of maximally, they decide that the Bohmian rules must be "skipped" in the case of the spin – they just manually omit some degrees of freedom that should be there according to the general prescription of Bohmian mechanics and hope that the spin measurements are ultimately reduced to position measurements so that it doesn't hurt if some degrees of freedom are not doubled in the usual Bohmian way.
The reason why the case of the spin is obvious even to them is the fact that different components of the spin are non-commuting observables none of which is more "natural" than others. After all, they are exactly equally natural because they are related by the rotational symmetry.
While the spin is an obvious problem, the pathological character of Bohmian mechanics is much more general. Every (qubit-like) discrete information in quantum mechanics – information labeling a finite-dimensional Hilbert space – is incompatible with the Bohmian philosophy. Recall that Bohmian mechanics added "classical trajectories" \(\bold{\hat q}(t)\) and these coordinates were functions of time that evolved according to some differential equations. But that was only possible because the spectrum of the coordinates was continuous. If you think about observables with a discrete spectrum, it just doesn't work because they would have to "jump to a different, sharply separated discrete eigenvalue" at some points and there can't be any deterministic laws that would govern such jumps.
Quantum mechanics tells you that a quantum computer composed of a very large number of qubits may perfectly emulate any quantum system. But that's not the case in Bohmian mechanics. An arbitrarily large quantum computer is composed of qubits, e.g. many electron spins, and because the spin isn't accompanied by a classical bit, Bohmian mechanics is forced to say that an arbitrarily large quantum computer only contains the "classicalized" wave function but no additional classical information analogous to the classical trajectories. So for a quantum computer, the whole "redundant superstructure" (which is how Albert Einstein called these extra coordinates – he was a foe of the pilot-wave theory, despite his being a disbeliever in quantum mechanics) has to be omitted. This is quite an inconsistency in the Bohmian treatment of different quantum systems. The actual reason behind the inconsistency is clear, of course: some physical systems may be caricatured by the pilot-wave trick, others can't. But in Nature, there actually isn't any qualitative difference (in principle observable difference) between these two classes of situations.
I said that Bohmian mechanics doesn't allow you to consistently treat the particles' spin or any other discrete degrees of freedom, for that matter. But the inadequacy of Bohmian mechanics is much worse than that. It really doesn't allow you to correctly deal with most observables in general quantum systems, not even with observables with a continuous spectrum. I have discussed similar problems in Bohmists and the segregation of primitive and contextual observables four years ago.
The problem is that Bohmian mechanics forces you to choose some observables that "really exist" – are encoded in the objective extra coordinates that are supplemented to the "classicalized" wave function. However, quantum mechanics implies that other observables just can't have a well-defined value at the same moment – because they don't commute with the first ones, stupid. That also means that Bohmian mechanics can't have any answers to questions about the value of these observables.
The Bohmian trajectories in the picture above pretend that a particle has an objective position and an objective velocity. But what about the orbital angular momentum \(\bold{\hat L} = \bold{\hat q}\times \bold{\hat p}\)? A basic result of quantum mechanics is that the spectrum of \(\bold{\hat L}_z\) is discrete; the eigenvalues are integer multiples of \(\hbar\). Already this elementary fact in quantum mechanics – even non-relativistic quantum mechanics – is completely inaccessible to Bohmian mechanics. The cross product of the classical position and the classical momentum of the "added Bohmian trajectories" isn't quantized at all. It has really nothing to do with the angular momentum that can be measured.
And be sure that the measurement of the angular momentum is often – e.g. for electrons in atoms – much more natural and "fundamental" than the measurement of the particles' positions or momenta. It's because its eigenstates are much closer to the energy eigenstates and those are the most natural basis of a Hilbert space because they describe stationary – and therefore lasting – states. But such a direct measurement of the discrete orbital angular momentum can't be done in Bohmian mechanics. Instead, Bohmian mechanics tells you that you have to continue the evolution of the wave function according to the laws stolen from proper quantum mechanics up to the moment when you can actually convert the original measurement to a measurement of a location, and hope that Bohmian mechanics knows how to emulate the measurements of positions. It isn't quite the case, either, but even if it were the case, Bohmian mechanics is just bringing an amazing degree of inconsistency into the way how different observables – different functions of the phase space – are treated. A sensible theory should treat all functions of the coordinates and momenta i.e. all functions in the phase space equally, following unified rules. Quantum mechanics obeys this criterion, Bohmian mechanics doesn't. We could say that just like the solipsists say that their own mind is the only physical system that may be claimed to be self-aware, Bohmian mechanics remains silent and reproducing the (accurately emulated) quantum evolution up to the moment when macroscopic positions are apparently being measured (those are the "conscious events" that are supposed to replace quantum mechanics with something else). But in the real world, there's nothing special about the minds of the solipsists (except that they belong to the set of crazy people) and there's also nothing special about the positions of macroscopic objects in comparison with many other observables we may define.
In quantum mechanics, you may directly construct operators for the angular momenta and ask about their possible values, eigenvalues, and about the predicted probabilities that the measured value will be one or the other. It doesn't matter whether the angular momenta belong to large or small or conscious or unconscious objects. Quantum mechanics allows you to deal with all observables equally. In Bohmian mechanics, those things matter. Effectively, any measurement has to be continued up to the moment when it imprints itself into a position of a macroscopic object which Bohmian mechanics claims to reproduce correctly.
A totally new minefield for Bohmian mechanics is relativity. The minimum consistent relativistic theories of quantum particles are quantum field theories (QFTs). They include the spin; I have already discussed the Bohmian problems with the spin. But there are infinitely many similar problems. For example, you may choose many different bases of the QFT Hilbert space. They may be eigenstates of the occupation number operators; eigenstates of field operator distributions \(\hat \phi(\bold{x})\), and so on. It is not clear at all which of these observables are added as the "extra classical trajectories" to Bohmian mechanics. In fact, it is totally obvious that none of the choices will behave correctly in all the experiments that may test a quantum field theory. Also, you can't add many of them or all of them (e.g. both positions and particles and classical values of the fields) because it would be clearly undetermined which of these "added", mutually conflicting classical degrees of freedom defines the "actual reality" that decides about a measurement.
Sometimes, the value of the field at a given point may be measured, especially when the frequencies are low. So it would seem like you need to add a "preferred classical field configuration" to the Bohmian version of a QFT. However, especially for high frequencies, the quantum field manifests itself as a collection of particles so you may want to add the trajectories of the particles instead. Moreover, even if you represent a QFT as a system describing many particles, your Bohmian theory won't be able to deal with the basic and most universal processes that must exist in a QFT or any other relativistic quantum theory such as the pair creation of a particle and an antiparticle and their destruction.
If individual particles evolve according to the "guiding wave" equations we discussed at the beginning, it's simply infinitely unlikely (the probability refers to the selection of the initial positions from the distribution) that they will ever collide with one another. Two random lines in a 3D space simply don't intersect one another. But if they don't directly collide, it means that they can't annihilate! To allow the particles to annihilate (and be pair-created) with the (experimentally proven) nonzero probability, you would need to introduce a totally non-local extra dynamics that sometimes allows the particles to jump to a completely different place; or you would have to allow the annihilation of particle pairs that don't coincide in space. Any such an extra mechanism would force you to change the original laws of physics in a way that would almost certainly contradict some other experiments because the unmodified quantum laws simply work and it was a healthy strategy for you to emulate them "perfectly" at the very beginning. Such modifications would especially contradict some experimental tests of relativity because these modifications are so horribly nonlocal.
So you have no chance to construct an operational Bohmian caricature of a quantum field theory. Needless to say, the problems become even more extreme once you switch to quantum gravity i.e. string theory because many more observables have a discrete spectrum, there are many more ways to choose the bases, the nonzero commutators of various observables are more important than ever before, and Bohmian mechanics just can't prosper in such general quantum situations. On one hand, quantum gravity i.e. string theory is just another quantum theory. On the other hand, it is "more quantum" than all the previous quantum theories simply because the quantum phenomena affect many more questions that could have been thought of in the classical way if you worked with simpler quantum mechanical theories (for example, the spacetime topology – especially the number of Einstein-Rosen bridges in the spacetime – can't even be assigned a linear operator in a quantum gravity theory, as Maldacena and Susskind argued).
The non-local fields, collapses, non-local jumps needed for particle annihilations, and other things represent an inevitable source of non-locality that can, in principle, send superluminal signals and that consequently contradicts the Lorentz symmetry of the special theory of relativity. There's no way out here. If you attempt to emulate a quantum field theory in this Bohmian way, you introduce lots of ludicrous gears and wheels Рmuch like in the case of the luminiferous aether, they are gears and wheels that don't exist according to pretty much direct observations Рand they must be finely adjusted to reproduce what quantum mechanics predicts (sometimes) without any adjustments whatsoever. Every new Bohmian gear or wheel you encounter generally breaks the Lorentz symmetry and makes the (wrong) prediction of a Lorentz violation and you will need to fine-tune infinitely many properties of these gears and wheels to restore the Lorentz invariance and other desirable properties of a physical theory (even a simple and fundamental thing such as the linearity of Schr̦dinger's equation is really totally unexplained in Bohmian mechanics and requires infinitely many adjustments to hold Рwhile it may be derived from logical consistency in quantum mechanics). It's infinitely unlikely that they take the right values "naturally" so the theory is at least infinitely contrived. More likely, there's no way to adjust the gears and wheels to obtain relativistically invariant predictions at all.
I would say that we pretty much directly experimentally observe the fact that the observations obey the Lorentz symmetry; the wave function isn't an observable wave; and lots of other, totally universal and fundamental facts about the symmetries and the interpretation of the basic objects we use in physics. Bohmian mechanics is really trying to deny all these basic principles – it is trying to deny facts that may be pretty much directly extracted from experiments. It is in conflict with the most universal empirical data about the reality collected in the 20th and 21st century. It wants to rape Nature.
A pilot-wave-like theory has to be extracted from a very large class of similar classical theories but infinitely many adjustments have to be made – a very special subclass has to be chosen – for the Bohmian theory to reproduce at least some predictions of quantum mechanics (to produce predictions that are at least approximately local, relativistic, rotationally invariant, unitary, linear etc.). But even if one succeeds and the Bohmian theory does reproduce the quantum predictions, we can't really say that it has made the correct predictions because it was sometimes infinitely fudged or adjusted to produce the predetermined goal. On the other hand, quantum mechanics in general and specific quantum mechanical theories in particular genuinely do predict certain facts, including some very general facts about Nature. If you search for theories within the rigid quantum mechanical framework, while obeying the general postulates, you may make many correct predictions or conclusions pretty much without any additional assumptions.
If you ask any of the hundreds of questions (Is the wave function in principle observable? Are observables with discrete spectra fundamentally less than real than those with continuous spectra? Is there a way to send superluminal signals, at least in principle? And so on) in which proper quantum mechanics differs from Bohmian mechanics, the empirical evidence heavily favors quantum mechanics and Bohmian mechanics can only survive if you adjust tons of parameters to unnatural values (from the viewpoint of Bohmian-like theories) and hope that it's enough (which it's usually not).
In 2013, even more so than in 1927, the pilot-wave theory is as indefensible as a flat Earth theory, geocentrism, the phlogiston, the luminiferous aether, or creationism. In all these cases, people are led to defend such a thing because some irrational dogmas are more important for them than any amount of evidence. That's what we usually refer to as bigotry.
And that's the memo.
Classical physics has been falsified and the falsification – a death for a theory – is an irreversible event. Nevertheless, those people would sleep with this zombie and do anything and everything else that is needed (but isn't sufficient) to resuscitate it. Of course, it's not possible to resuscitate it but those people just won't stop trying.
Bohmian mechanics, one of the main strategies to pretend that classical physics hasn't died and hasn't been superseded by fundamentally different quantum mechanics, was invented by Prince Louis de Broglie in 1927 who called it "the pilot-wave theory". In the late 1920s, the 1930s, and 1940s, physicists were largely competent so they didn't have any doubts that the pilot wave theory was misguided by its very own guiding wave ;-). Exactly 25 years later, the approach was revived by David Bohm who made the picture popular, largely because he was a fashionable, media-savvy commie (he's almost certainly the recipient of Wolfgang Pauli's famous criticism "not even wrong" that was ironically hijacked by aggressive Shmoitian crackpots in the recent decade). Prince Louis de Broglie liked the new life that apparently returned to the veins of his old sick theory so he didn't even care too much that his theory was going to be attributed to someone else and that the someone else was a Marxist rather than an aristocrat.
A constraint that defines Bohmian mechanics is simple: it should be a classical theory that emulates quantum mechanics as well as it can. The champions of the Bohmian theory know that getting the same predictions as quantum mechanics is the maximum goal they may dream about – they can never beat quantum mechanics – and they sort of realize that even this tie is too much to ask in general. Most of the Bohmian advocates seem to know that their theory can't be accurate, especially because of its fundamental conflict with relativity – but they don't seem to care. The fact that the Bohmian mechanics agrees with their fully discredited preconception that Nature is fundamentally classical is more important for them than the (in)accuracy of the predictions extracted from their pet theory.
It's straightforward to explain why it's possible to design a classical theory that parrots quantum mechanics when it comes to certain questions.
Bohmian mechanics is at least vaguely defensible in the non-relativistic quantum mechanical models only; in more general theories, it collapses completely. How does it rebuild non-relativistic quantum mechanics for one particle, for example?
Proper quantum mechanics of this system may be written down in Schrödinger's picture that dictates the following time evolution to the wave function:\[
i\hbar\frac{\partial}{\partial t}\psi(q,t)=-\sum_{i=1}^{N}\frac{\hbar^2}{2m_i}\nabla_i^2\psi(q,t) + V(q)\psi(q,t)
\] The way how this wave is evolved in agreement with the equation above contains all the "mathematical beef" of quantum mechanics for the given system and to get the right numbers, any classical caricature of quantum mechanics simply has to contain some objects that are pretty much equivalent to \(\psi(q,t)\). These objects are then assigned totally different, wrong interpretations in the caricatures but they must be there and they must evolve according to the same Schrödinger's equation.
Bohmian mechanics buys \(\psi(q,t)\) and incorrectly interprets it as a classical wave – a field that has objective values and is in principle measurable. Of course, we know from quantum mechanics as well as experiments that the value of the wave function simply shouldn't be and isn't measurable in a single repetition of an experiment. So the Bohmian apologists must also invent convoluted mechanisms to make the wave unmeasurable – because it is unmeasurable according to the experiments – despite the fact that the wave function is fundamentally measurable in their theory.
Bohmian Rhapsody, via Dilaton.
Is this the real life? Is this just fantasy?
Caught by the guiding wave. No escape from reality.
Open your eyes. Look up to the skies and see:
I'm just [a] state vector, I need no images.
Because I'm easy come, easy go.
A little high, little low.
Anyway the [pilot] wave blows, doesn't really matter to me, to me.
The pilot-wave theory adopts \(\psi(q,t)\) as an objective classical wave – which it gives a new name, the "guiding wave" or "pilot wave" – but in order to agree with the fact that particles may be observed at sharp locations despite the fuzziness of the wave functions associated with them, they must add some additional degrees of freedom: the actual classical position of the particle. The defining philosophy of Bohmian mechanics is that the actual, classical position of the particle is "guided" by a function of the classical field emulating the wave function so that the probability distribution for the particle's positions remains what it should be according to quantum mechanics. For example, the laws that guide the actual classical particle must be such that they repel the particle from the interference minima in a double-slit experiment:
 |
The right end of the picture (the photographic plate) shows denser and less dense regions, the interference maxima and minima.
Can you find the appropriate rules for one non-relativistic spinless quantum particle that is able to do it in a way that imitates quantum mechanics? You bet. All the tools are available in conventional quantum mechanics for this system. Recall that in quantum mechanics, \(\rho=|\psi(q,t)|^2\) is the probability density that the particle is sitting near location \(q\) at time \(t\). But quantum mechanics also allows you to define the probability current\[
\bold j = \frac{1}{m} \mathrm{Re}\left ( \psi^*\bold{\hat{p}}\psi \right )
\] Note that it is again sesquilinear (bilinear with one star) in the wave function. We act on the wave function by the momentum operator \(\bold{\hat{p}}=-i\hbar\nabla\), multiply the result by \(\Psi^*\) just like when we calculated the probability density, take the real part, and divide it by the mass \(m\). You see that it only differs from the formula for the probability density by the extra operator \(\bold{\hat{p}}/m\), the operator of the velocity \(\bold{\hat v}\), inserted in the middle. The real part could have been added to the probability density as well because it was real to start with.
At any rate, if you define the probability density and the probability current correctly, they obey the continuity equation\[
\frac{\partial \rho}{\partial t} + \bold \nabla \cdot \bold j = 0.
\] The divergence of the probability current exactly agrees with the decrease of the probability density in the given region. It means that the probability current measures how the probability has to flow into/from a given infinitesimal volume if you want the probability density to change just like it should according to Schrödinger's equation.
Now it's easy to realize that if you define a classical "velocity field"\[
\bold{\hat v} = \frac{\bold{\hat j}}{\rho},
\] it will be very useful for emulating quantum mechanics. It's not hard to prove that if you define Bohmian mechanics as the "classicalized" wave function together with a classical position \(\bold{\hat q}(t)\) that evolves according to the "guiding equation"\[
\ddfrac{\bold{\hat q}}{t} = \bold{\hat v} (\bold{\hat q}(t)),
\] the trajectories of the classical particles will be repelled from the interference minima, attracted to the interference maxima, and will obey a more specific rule: If you imagine that the particles in the initial state are distributed according to the probability distribution given by \(\rho(\bold{\hat q},t)\), it will be true for the final state, too.
This trick may be generalized to the case of \(N\) non-relativistic particles. In this case, the wave function \(\psi\) becomes a classical wave that is a function of the \(3N\)-dimensional configuration space. This configuration space is larger than the ordinary space and is "multi-local" and because we have this "multi-local" old-fashioned classical field, the theory becomes explicitly non-local and a violation of the Lorentz symmetry, at least in principle, is inevitable.
I would like to emphasize that it's no surprise at all that it's possible to find the equation that evolves the probability distribution in the right way. Imagine that you start with a wave function \(\psi(\bold{\hat q})\) at some time \(t_0\). Throw a trillion of dots – particles – to the space that are distributed according to \(\rho = |\psi|^2\). Do the same thing for the final moment \(t_1\) when the wave function is different. You will have two configurations of trillions of particles. It's not shocking that you may "connect the dots" from the initial state to the final state in some way.
A way that is simple enough, one based on the probability current and described above, gives you one of the solutions. But it's not the only solution. In reality, the "initial dots" could be connected with the "final dots" in infinitely many ways (well, a "mere" trillion factorial if you only have a trillion of dots). In the continuous language, you could e.g. make the particles move along spirals inside the cylinders that surround the interference maxima. Is one way to connect the dots better than others?
Of course, it's not. All of them are equally good. Quantum mechanics commands you to learn something about the initial state – some wave function or density matrix that encode the initial probability distribution – and it allows you to predict the probabilities for the final state. But it doesn't tell you which of the initial particles is connected with which final particle, i.e. how to connect the dots. It doesn't inform you about any preferred classical trajectory that connects them (and Feynman's approach orders you to sum over all trajectories). If you could actually "measure" this permutation that determines how the dots are connected, quantum mechanics would be shown incomplete.
However, it's totally obvious that there's no way to measure the trajectories or permutations inside. The particles just don't have well-defined, in principle measurable trajectories between the measurements for the usual Heisenberg uncertainty principle-based reasons. If you tried to measure the trajectory before the final measurement, you would change the experiment and destroy or damage the final interference pattern. So all the precise lines on the "caricature of the double-slit experiment"
|
are pure fantasy. They're just crutches for the people who need some specific picture of the intermediate states to be drawn. But the specific picture we drew is in no way better than infinitely many other pictures we could draw that would predict the same interference pattern, the same probability distributions for the final state. Everything we added because we wanted the physical system to have objective properties prior to the measurement – because we're bigots who can't accept the fact that classical physics has died – is unphysical. The added value is purely negative. Everything we added to get from proper quantum mechanics to Bohmian mechanics is rubbish. And many things we're forced to lose when we switch from quantum mechanics to Bohmian mechanics are essential.
Because the wave function has a probabilistic interpretation in proper quantum mechanics (it is a ready-to-cook meal from which one may quickly prepare various probability distributions by a calculation), it doesn't matter that it spreads. The spreading of the wave function doesn't make the world more fuzzy. It only makes our knowledge about the world more uncertain. But once we learn the answer to a question – e.g. about the position of a particle – the world fully regains its sharp character it boasted at the beginning. If you only know that the probability of 1,2,3,4,5,6 are 1/6 for some dice in Las Vegas, it doesn't mean that the dice became structureless balls or that the digits written on their sides have become fuzzy or mixed or smeared. It just means that we have one equally sharp cubic die but we just don't know its orientation in space. The uncertainty coming from quantum wave functions are analogous – they only differ from the "classical uncertainty" by their inevitability.
That's not the case of Bohmian mechanics. The wave function is interpreted as a classical field of a sort and it is objectively spreading. So something objective is being diluted all over the Universe. That's terrible because this objectively makes the Universe increasingly more fuzzy and bizarre. The useless parts of the guiding wave – the "classicalized" wave function – should be killed in some way because they became useless. But Bohmian mechanics doesn't imply anything of the sort. If you want to clean the garbage of the no-longer-needed branches of the wave function, you will have to add another independent contrived mechanism. Such a mechanism will be a new source of a violation of the Lorentz invariance.
(You also need a special mechanism that prepares the guiding wave in a certain initial state and one more mechanism that distributes the "actual particle" inside the appropriate distribution with the right odds because these two things don't follow from Bohmian mechanics as we have defined it above, either. Most of these things are ignored by the Bohmists. Note that with the right probabilistic interpretation – quantum mechanics directly connects the knowledge about the past with the knowledge about the future, without any new crutches in between – we don't need to invent any new mechanisms.)
I think that a sane, critically thinking person must be able to realize what he is doing if he is doing such things. He is drawing a ludicrous caricature of Nature – a physical system that is actually governed by the laws of proper quantum mechanics – that reproduces some properties of the correct, quantum theory. The project of drawing the caricature is motivated by the desire to defend a philosophical dogma that the world is fundamentally classical even though it is clearly not. If he has at least some conscience, he must feel analogously as if he were counterfeiting a $100 banknote. He must know that what he is producing isn't the "real thing"; it is just a forgery that can bring him greater personal benefits than the actual banknotes but that's where the advantages stop.
But every change from the proper quantum mechanics to the pilot-wave theory is clearly wrong – the "added value" is unquestionably negative. Because the Bohmists don't like the probabilistic character of the wave function, they turn it into a classical wave – the guiding wave. But a classical wave that spreads objectively makes the world ever more fuzzy. So one has to introduce new tricks to have a chance that this increasing fuzziness doesn't spoil the world. All these tricks – tricks that can't really ever be defined in such a way to imitate quantum mechanics completely accurately – have to be considered and added just in order to mask the fact that the wave function is simply not a classical field.
It's fair to say that the claim by quantum mechanics that the wave function is not an objectively real wave or field that can be in principle measured is something that we have proven by direct experiments. Attempts to pretend that the wave function is a classical wave are just attempts to mask the truth. I am confident that every Bohmist must ultimately realize it is so and he must be dishonest if he claims that his efforts are more justifiable than the efforts of creationists who are trying to obscure the explicit evidence in favor of evolution: they are exactly equally unjustifiable.
Moreover, it's sometimes being said or thought that the perfect emulation of quantum mechanics can be done. Because the invalidated dogma that Nature is fundamentally classical is holy for these bigots, they think that it should be done, too. But the truth is that it can't be done for a general physical system and for a general choice of observables we may measure in actual experiments described by general enough quantum theories.
Try to add the spin to a particle. If the logic of Bohmian mechanics – the wave function "is" a classical field and we should also add some classical values of a maximum set of commuting observables – were universally valid, it's clear that aside from the spinor-valued wave function \((c_{\rm up},c_{\rm down})\), we should also assume that Nature "objectively knows" about the classical bit of information that tells you whether the spin is "actually" up or down.
However, even the Bohmists realize that if every electron "objectively knew" whether its spin is up or down with respect to the \(z\)-axis, then the laws of physics would break the rotational symmetry because the \(z\)-axis would play a privileged role. Roughly speaking, the ferromagnets would always be oriented vertically, to mention an example. If the \(z\)-component of the classical angular momentum is quantized, it's totally obvious that the other components can't be quantized. A nonzero vector can't have integer (or half-integer) coordinates in each (rotated) coordinate system.
Because they sort of realize that the rotational symmetry holds exactly and the hypothesis that the classical value exists with respect to one axis would break the symmetry kind of maximally, they decide that the Bohmian rules must be "skipped" in the case of the spin – they just manually omit some degrees of freedom that should be there according to the general prescription of Bohmian mechanics and hope that the spin measurements are ultimately reduced to position measurements so that it doesn't hurt if some degrees of freedom are not doubled in the usual Bohmian way.
The reason why the case of the spin is obvious even to them is the fact that different components of the spin are non-commuting observables none of which is more "natural" than others. After all, they are exactly equally natural because they are related by the rotational symmetry.
While the spin is an obvious problem, the pathological character of Bohmian mechanics is much more general. Every (qubit-like) discrete information in quantum mechanics – information labeling a finite-dimensional Hilbert space – is incompatible with the Bohmian philosophy. Recall that Bohmian mechanics added "classical trajectories" \(\bold{\hat q}(t)\) and these coordinates were functions of time that evolved according to some differential equations. But that was only possible because the spectrum of the coordinates was continuous. If you think about observables with a discrete spectrum, it just doesn't work because they would have to "jump to a different, sharply separated discrete eigenvalue" at some points and there can't be any deterministic laws that would govern such jumps.
Quantum mechanics tells you that a quantum computer composed of a very large number of qubits may perfectly emulate any quantum system. But that's not the case in Bohmian mechanics. An arbitrarily large quantum computer is composed of qubits, e.g. many electron spins, and because the spin isn't accompanied by a classical bit, Bohmian mechanics is forced to say that an arbitrarily large quantum computer only contains the "classicalized" wave function but no additional classical information analogous to the classical trajectories. So for a quantum computer, the whole "redundant superstructure" (which is how Albert Einstein called these extra coordinates – he was a foe of the pilot-wave theory, despite his being a disbeliever in quantum mechanics) has to be omitted. This is quite an inconsistency in the Bohmian treatment of different quantum systems. The actual reason behind the inconsistency is clear, of course: some physical systems may be caricatured by the pilot-wave trick, others can't. But in Nature, there actually isn't any qualitative difference (in principle observable difference) between these two classes of situations.
I said that Bohmian mechanics doesn't allow you to consistently treat the particles' spin or any other discrete degrees of freedom, for that matter. But the inadequacy of Bohmian mechanics is much worse than that. It really doesn't allow you to correctly deal with most observables in general quantum systems, not even with observables with a continuous spectrum. I have discussed similar problems in Bohmists and the segregation of primitive and contextual observables four years ago.
The problem is that Bohmian mechanics forces you to choose some observables that "really exist" – are encoded in the objective extra coordinates that are supplemented to the "classicalized" wave function. However, quantum mechanics implies that other observables just can't have a well-defined value at the same moment – because they don't commute with the first ones, stupid. That also means that Bohmian mechanics can't have any answers to questions about the value of these observables.
The Bohmian trajectories in the picture above pretend that a particle has an objective position and an objective velocity. But what about the orbital angular momentum \(\bold{\hat L} = \bold{\hat q}\times \bold{\hat p}\)? A basic result of quantum mechanics is that the spectrum of \(\bold{\hat L}_z\) is discrete; the eigenvalues are integer multiples of \(\hbar\). Already this elementary fact in quantum mechanics – even non-relativistic quantum mechanics – is completely inaccessible to Bohmian mechanics. The cross product of the classical position and the classical momentum of the "added Bohmian trajectories" isn't quantized at all. It has really nothing to do with the angular momentum that can be measured.
And be sure that the measurement of the angular momentum is often – e.g. for electrons in atoms – much more natural and "fundamental" than the measurement of the particles' positions or momenta. It's because its eigenstates are much closer to the energy eigenstates and those are the most natural basis of a Hilbert space because they describe stationary – and therefore lasting – states. But such a direct measurement of the discrete orbital angular momentum can't be done in Bohmian mechanics. Instead, Bohmian mechanics tells you that you have to continue the evolution of the wave function according to the laws stolen from proper quantum mechanics up to the moment when you can actually convert the original measurement to a measurement of a location, and hope that Bohmian mechanics knows how to emulate the measurements of positions. It isn't quite the case, either, but even if it were the case, Bohmian mechanics is just bringing an amazing degree of inconsistency into the way how different observables – different functions of the phase space – are treated. A sensible theory should treat all functions of the coordinates and momenta i.e. all functions in the phase space equally, following unified rules. Quantum mechanics obeys this criterion, Bohmian mechanics doesn't. We could say that just like the solipsists say that their own mind is the only physical system that may be claimed to be self-aware, Bohmian mechanics remains silent and reproducing the (accurately emulated) quantum evolution up to the moment when macroscopic positions are apparently being measured (those are the "conscious events" that are supposed to replace quantum mechanics with something else). But in the real world, there's nothing special about the minds of the solipsists (except that they belong to the set of crazy people) and there's also nothing special about the positions of macroscopic objects in comparison with many other observables we may define.
In quantum mechanics, you may directly construct operators for the angular momenta and ask about their possible values, eigenvalues, and about the predicted probabilities that the measured value will be one or the other. It doesn't matter whether the angular momenta belong to large or small or conscious or unconscious objects. Quantum mechanics allows you to deal with all observables equally. In Bohmian mechanics, those things matter. Effectively, any measurement has to be continued up to the moment when it imprints itself into a position of a macroscopic object which Bohmian mechanics claims to reproduce correctly.
A totally new minefield for Bohmian mechanics is relativity. The minimum consistent relativistic theories of quantum particles are quantum field theories (QFTs). They include the spin; I have already discussed the Bohmian problems with the spin. But there are infinitely many similar problems. For example, you may choose many different bases of the QFT Hilbert space. They may be eigenstates of the occupation number operators; eigenstates of field operator distributions \(\hat \phi(\bold{x})\), and so on. It is not clear at all which of these observables are added as the "extra classical trajectories" to Bohmian mechanics. In fact, it is totally obvious that none of the choices will behave correctly in all the experiments that may test a quantum field theory. Also, you can't add many of them or all of them (e.g. both positions and particles and classical values of the fields) because it would be clearly undetermined which of these "added", mutually conflicting classical degrees of freedom defines the "actual reality" that decides about a measurement.
Sometimes, the value of the field at a given point may be measured, especially when the frequencies are low. So it would seem like you need to add a "preferred classical field configuration" to the Bohmian version of a QFT. However, especially for high frequencies, the quantum field manifests itself as a collection of particles so you may want to add the trajectories of the particles instead. Moreover, even if you represent a QFT as a system describing many particles, your Bohmian theory won't be able to deal with the basic and most universal processes that must exist in a QFT or any other relativistic quantum theory such as the pair creation of a particle and an antiparticle and their destruction.
If individual particles evolve according to the "guiding wave" equations we discussed at the beginning, it's simply infinitely unlikely (the probability refers to the selection of the initial positions from the distribution) that they will ever collide with one another. Two random lines in a 3D space simply don't intersect one another. But if they don't directly collide, it means that they can't annihilate! To allow the particles to annihilate (and be pair-created) with the (experimentally proven) nonzero probability, you would need to introduce a totally non-local extra dynamics that sometimes allows the particles to jump to a completely different place; or you would have to allow the annihilation of particle pairs that don't coincide in space. Any such an extra mechanism would force you to change the original laws of physics in a way that would almost certainly contradict some other experiments because the unmodified quantum laws simply work and it was a healthy strategy for you to emulate them "perfectly" at the very beginning. Such modifications would especially contradict some experimental tests of relativity because these modifications are so horribly nonlocal.
So you have no chance to construct an operational Bohmian caricature of a quantum field theory. Needless to say, the problems become even more extreme once you switch to quantum gravity i.e. string theory because many more observables have a discrete spectrum, there are many more ways to choose the bases, the nonzero commutators of various observables are more important than ever before, and Bohmian mechanics just can't prosper in such general quantum situations. On one hand, quantum gravity i.e. string theory is just another quantum theory. On the other hand, it is "more quantum" than all the previous quantum theories simply because the quantum phenomena affect many more questions that could have been thought of in the classical way if you worked with simpler quantum mechanical theories (for example, the spacetime topology – especially the number of Einstein-Rosen bridges in the spacetime – can't even be assigned a linear operator in a quantum gravity theory, as Maldacena and Susskind argued).
The non-local fields, collapses, non-local jumps needed for particle annihilations, and other things represent an inevitable source of non-locality that can, in principle, send superluminal signals and that consequently contradicts the Lorentz symmetry of the special theory of relativity. There's no way out here. If you attempt to emulate a quantum field theory in this Bohmian way, you introduce lots of ludicrous gears and wheels Рmuch like in the case of the luminiferous aether, they are gears and wheels that don't exist according to pretty much direct observations Рand they must be finely adjusted to reproduce what quantum mechanics predicts (sometimes) without any adjustments whatsoever. Every new Bohmian gear or wheel you encounter generally breaks the Lorentz symmetry and makes the (wrong) prediction of a Lorentz violation and you will need to fine-tune infinitely many properties of these gears and wheels to restore the Lorentz invariance and other desirable properties of a physical theory (even a simple and fundamental thing such as the linearity of Schr̦dinger's equation is really totally unexplained in Bohmian mechanics and requires infinitely many adjustments to hold Рwhile it may be derived from logical consistency in quantum mechanics). It's infinitely unlikely that they take the right values "naturally" so the theory is at least infinitely contrived. More likely, there's no way to adjust the gears and wheels to obtain relativistically invariant predictions at all.
I would say that we pretty much directly experimentally observe the fact that the observations obey the Lorentz symmetry; the wave function isn't an observable wave; and lots of other, totally universal and fundamental facts about the symmetries and the interpretation of the basic objects we use in physics. Bohmian mechanics is really trying to deny all these basic principles – it is trying to deny facts that may be pretty much directly extracted from experiments. It is in conflict with the most universal empirical data about the reality collected in the 20th and 21st century. It wants to rape Nature.
A pilot-wave-like theory has to be extracted from a very large class of similar classical theories but infinitely many adjustments have to be made – a very special subclass has to be chosen – for the Bohmian theory to reproduce at least some predictions of quantum mechanics (to produce predictions that are at least approximately local, relativistic, rotationally invariant, unitary, linear etc.). But even if one succeeds and the Bohmian theory does reproduce the quantum predictions, we can't really say that it has made the correct predictions because it was sometimes infinitely fudged or adjusted to produce the predetermined goal. On the other hand, quantum mechanics in general and specific quantum mechanical theories in particular genuinely do predict certain facts, including some very general facts about Nature. If you search for theories within the rigid quantum mechanical framework, while obeying the general postulates, you may make many correct predictions or conclusions pretty much without any additional assumptions.
If you ask any of the hundreds of questions (Is the wave function in principle observable? Are observables with discrete spectra fundamentally less than real than those with continuous spectra? Is there a way to send superluminal signals, at least in principle? And so on) in which proper quantum mechanics differs from Bohmian mechanics, the empirical evidence heavily favors quantum mechanics and Bohmian mechanics can only survive if you adjust tons of parameters to unnatural values (from the viewpoint of Bohmian-like theories) and hope that it's enough (which it's usually not).
In 2013, even more so than in 1927, the pilot-wave theory is as indefensible as a flat Earth theory, geocentrism, the phlogiston, the luminiferous aether, or creationism. In all these cases, people are led to defend such a thing because some irrational dogmas are more important for them than any amount of evidence. That's what we usually refer to as bigotry.
And that's the memo.
Bohmian mechanics, a ludicrous caricature of Nature
Reviewed by DAL
on
July 15, 2013
Rating:
Reviewed by DAL
on
July 15, 2013
Rating:
No comments: