Photons are incompatible with the Bohmian picture
And so is atomic emission and the energy conservation law
Bohmian mechanics is a framework (partly well-defined in some very special situations) proposed to replace the postulates of quantum mechanics by a new classical theory that largely borrows the mathematics controlling the state vector as if it were a classical wave; and that is supplemented with extra beables (in the only semi-successful example, positions of non-relativistic particles) that behave classically and whose trajectories are affected by the pilot wave.
The theory allows you to assume that if you measure the non-relativistic particles' positions, the random outcome isn't decided at the moment of the measurement. Instead, these positions were already ready before the measurement. And the defining assumption of the Bohmian paradigm is that any measurement may be reduced to such a measurement of the positions (or other beables if you knew what they are) so that the random collapse in the standard quantum mechanical treatment of the measurements isn't ever needed. For example, it's sometimes semi-successfully claimed that the measurement of particles' spins may be reduced to the measurement of the positions of the same particles. But we will see that this is not the case for observables that commute neither with the position nor with the momenta such as energy.
In several recent articles, I discussed the incompatibility of Bohmian mechanics with some of the physical phenomena that go beyond non-relativistic QM, e.g. with loop corrections in QFT. User7348 was dissatisfied with the claim that Bohmian mechanics fails in this test so he asked for a "rebuttal" of my claims. There must be one, right? Everyone who believes that Bohmian mechanics doesn't suffer from these lethal defects is invited to write the pre-planned "rebuttal" of my proof.
Proofs can rarely be disproven and so far, no one could have offered any "rebuttal". Physicists largely agree that I am right. On this blog, aside from other interactions, I've had quite a typical exchange with one Bohmian who claimed that the loops could work. So I asked him to show me the calculation of the electron's magnetic moment (up to the one-loop level) in the Bohmian theory. The answer I got was "why would I do it if it has already been done?".
He obviously meant that it was done in proper quantum field theory governed by the standard, "Copenhagen" postulates of quantum mechanics (at most reformulated with a different "accent" but not a different "content"). And because he must believe that Bohmian mechanics has "conquered" the standard quantum mechanics and may claim credit for all of the successes of quantum mechanics (while taking no responsibility for the alleged drawbacks), he just doesn't need to write anything, he believes.
This is obviously silly. If you claim to have a theory that differs from another one (e.g. the "dominant" one among the researchers) and your theory is said to have some advantages, it follows that it's inequivalent. When it's inequivalent, it may also be inequivalent in the "undesirable" situations – it may fail to reproduce the successes of the "dominant" theory you want to replace. Clearly, this general rule – modifications that have the potential to "improve" something also have the potential to "ruin" something – holds for Bohmian mechanics, too. And be sure that the basic Bohmian assumptions ruin pretty much everything.
These Bohmian people often make claims such as "all the physics of QFT works just fine in Bohmian theory". References to incoherent preprints that make similar claims are the "evidence" you may get. All these preprints contain some "extra" (and very ugly) mathematical constructions that are absolutely different from the standard QM/QFT and it's obvious that they can't be producing the same predictions in general. But if someone claims that the Bohmian theory makes sense, shouldn't he be able to write the "proper modern [Bohmian]" textbook replacing the existing textbooks of quantum field theory? At least a few chapters, up to a calculation of some annihilation processes of QED.
Needless to say, nothing like that is possible. None of these calculations may actually be done in the Bohmian theory. At most, these people may say that "they're suddenly modest and they're satisfied with the 'Copenhagen' treatment". But they won't tell you what the list of "beables" (like Bohmian particle positions) is during the annihilation or light emission or any other basic process – and how they behave during the process. What happens to make sure that the beables detect a correct statistical distribution of these quantities? These Bohmian positions are basically "one-half of the Bohmian theory" – a part that is completely absent in proper QM/QFT – so shouldn't the Bohmians add something, basically 100%, to the Copenhagen story about all these phenomena?
These people would like to dictate the overall philosophy. They often please you with extraordinarily wonderful claims such as "observers are not needed" or "the basic framework of classical physics was more or less right and no conceptual change is needed". But when you want them to use their theory to actually calculate something, they prefer to be silent. This approach is similar to the Marxists who are very loud and aggressive when they steal the assets of an "evil capitalist" but when they're expected to produce the same values that the "evil capitalist" was able to produce, their energy suddenly fades away and you hear excuses. These Marxists complain about lots of bad circumstances such as the four main enemies of communism – spring, summer, autumn, and winter – that sometimes lead to failures, and so on. Both Bohmists and Marxists are very loud and aggressive when they scream what they hate but when they're supposed to show an alternative, all the meaningful activity goes away.
Sadly, this is not just an analogy. Most Bohmists actually were and are Marxists at the same moment (starting with David Bohm himself) so these two examples of a behavior may be just two sides of the same coin.
What's even more disturbing is that these Bohmists must actually know that the theory fails in the description of very basic things – such as the propagation (and emission) of photons. There are Bohmian papers explicitly stating that according to the common Bohmian lore, something is impossible.
Photons' wave functions, beables, and atomic emission: impossible in the Bohmian picture
I wanted to start with the emission of atoms resulting from an atomic transition. It can't work and all Bohmian attempts are guaranteed to destroy the discreteness of the spectrum and lead to violations of the energy conservation law. But there is a simpler thing that doesn't work: the mundane propagation of free photons!
Just like electrons, photons may be used to perform the double slit experiment. Photons create individual dots on the photographic plate. By its definition, the Bohmian mechanics must have a result for the measurement of the "photon position" that is ready before the measurement. Except that there can't be any equations – at least not local or otherwise natural equations – that could govern the motion of such "real Bohmian photons". Why?
It's simple. Take a 2001 paper by Ghose+3, Bohmian trajectories for photons, that starts:
To incorporate photons that "know where they should create dots", you simply need such a 4-current \((\rho, \vec j)\) as well. However, it simply cannot exist in Maxwell's theory, at least not in any local or otherwise natural form.
The problem is that the natural bilinear object in Maxwell's theory isn't a current (a 4-component vector) but the stress-energy tensor \(T_{\mu\nu}\). The integral of \(T_{00}\) over the 3-dimensional space isn't one. Instead, it's the total energy – something that is proportional to the frequency via \(E=\hbar\omega\).
In proper quantum mechanics, the wave function of one photon basically behaves as the Maxwell's electromagnetic field – which is however reinterpreted probabilistically. Now, the detection of a photon is always sensitive to some frequencies and the Maxwell field (or photon's wave function) secretly knows about the representation of different frequencies, too.
However, if you assume that the Bohmian photon has a well-defined position, it simply cannot have a well-defined energy at the same moment. Also, the required 4-current can't be written as any function of the fields \(A_\mu\) and their derivatives. You simply can't build any conserved current (with one index) out of electromagnetic fields.
It's easy to show the complications explicitly. The quantized electromagnetic field may be built as a Fock space. In a box, for every allowed \(\vec k\), there exist two annihilation operators \(a_\lambda(\vec k)\) and two Hermitian conjugate creation operators \(a^\dagger_\lambda(\vec k)\) – two because there are two polarizations for each direction.
There is a nice formula for the total energy\[
H = \sum_{\vec k, \lambda} \hbar c |\vec k|\cdot a^\dagger_\lambda(\vec k) a_\lambda(\vec k)
\] in the position space, namely the integral of \(\rho_E = (E^2+B^2)/2\). However, there is no natural formula for the number of photons. The number of photons would differ from \(H\) by the omission of the factor \(|\vec k|\). While the required factor \(1/|\vec k|\) may look simple and natural in the momentum space, it is absolutely unnatural and nonlocal in the position space – it is some kind of \(1/\sqrt{\nabla^2}\) and believe me that it's not nice to compute the square root of some inverse Laplace operators. (Dirac found an ingenious method how to define a nice "square root" of the box operator – the Dirac operator – but that's exactly what you can't do for bosons.)
So no local or natural formula for the number of photons \(N\) may exist in Maxwell's theory. Consequently, you can't find any local or natural 4-current which is needed in Bohm's theory. And that means that there can't be any natural equations telling you how the photon should respond to the Maxwell pilot wave. It just doesn't make any sense.
Above, I assumed that you would try to define the number of photons by applying the "inverse square root of the Laplacian" – a highly non-local operator – on the formula for the total energy. Ghose et al. chose a different approach, one with some extra bizarre matrices \(\beta_\mu\), but it's ultimately equally ugly. They may draw some trajectories that respect some probability densities but there is absolutely nothing natural about the mathematical manipulations needed to do so – nothing natural about the new proposed laws of physics governing photons.
And you may check that none of the 68 followups of the paper by Ghose et al. actually uses this super-ugly, super-unnatural set of equations with extra new vectors. It's ultimately as ugly as the nonlocal square roots of the Laplacians – and probably equivalent at some level. All the papers only use the bizarre formulae by Ghose et al. as a political argument of a sort. They don't really want to do any science with such things. I think that they don't really believe that these super-ugly, arbitrary equations are correct or useful in any scientific sense. They must know that they have at most fitted a result calculated by a different theory, rather than derived a result by using their own theory. What they're doing is just an ideologically-driven propaganda. All these "solutions" are "superfluous ideological superstructures", as Werner Heisenberg accurately called the original Bohmian theory.
(If the weird Ghose et al. formalism to produce the Bohmian photon trajectories were right, people could immediately find additional evidence, consistency checks, agreement with hints found in different ways, and/or generalizations of this formalism to other fields and particles of the Standard Model such as the Yang-Mills fields or other phenomena – just like in the cute and dynamic evolution of the successful scientific theories we actually know from the physics textbooks. But the number of these "signs" that the Bohmian ideas are on the right track is strictly zero and the number of hints that they're wrong is huge. Similar advocates of these bad ideas generally don't care about an arbitrary finite number of "bad signs" because they are infinitely prejudiced.)
An obvious additional problem is that the photons' trajectories have to be superluminal at many points. The waves generally move by the speed of light but to preserve the interference patterns, the pilot wave has to push the Bohmian photons back and forth (towards the interference maxima), so this gives them some extra motion relatively to the averaged motion of the wave, and the average speed of these particles on these "not quite straight" lines obviously has to be higher than the speed of light. If you want to keep in touch with light but you need detours on your trip, you sometimes need to move faster than the light, right? There's no way to "solve" this problem. The superluminal motion of these particles looks like the motion "backwards in time" in other relativistic inertial systems. Too bad.
So there's really no usable way to incorporate photons to the Bohmian theory.
Bohmian theory is incompatible with light emission by atoms
The absence of a 4-current in Maxwell's theory – that prevents you from defining any natural "pilot wave force" on the Bohmian photon – is far from being the only problem you would encounter if you tried to incorporate photons. Try to find the Bohmian description of a very simple process – the emission of light by atoms. Take a hydrogen atom. It should have some discrete spectrum, OK? The energy of the photons may be converted to their position using a prism (colors are separated on the rainbow).
Let's assume that the Bohmian theory correctly describes the fermions (electron, proton) that the atom is composed of. Such a positive appraisal isn't right even in this limited setup of a non-relativistic description of fermions but let's be overly generous. However, this theory doesn't automatically emit light. You need to couple the atom to the electromagnetic field. The frequency \(\omega\) modes of the electromagnetic field will be sensitive to the modes of the atom in which the energy jumps by \(\hbar\omega\). For those, the constructive interference may add up and convince the atom to create a photon.
The Bohmians want to argue that before the photons are detected, they already objectively have the soon-to-be-observed positions. If that's the case, the photon must be actually born at some moment, in the vicinity of the atom. (We neglect the previously discussed problems with the free photon which is already too general and can't be incorporated to the Bohmian theory.) OK, when is the Bohmian photon born and what are the rules that achieve this job?
Proper quantum mechanics tells you that you mustn't try to answer this question because it's meaningless. Only the observed outcomes may be treated as "facts" and the photons will only be detected a long time after they're created by the atoms. Before they're detected, the probability amplitudes are interfering with each other.
Much like in the discussion of the loop corrections, the whole potential for the photons to appear in one direction of the prism or another is due to the interference effects. To have a chance to reproduce those, the Bohmian-like theory would have to contain the whole "wave functional" (state vector of the quantum field theory) as a classical field. Great, Bohmists know how to steal but unfortunately for them, the state vector isn't the only classical degree of freedom in their proposed theory. They also need the photons' positions. How are those created, what are their initial values, what are the rules that govern them?
Any remotely viable solution will have to be sensitive to the interference so it will have to be non-local in time. You know, if parts of the state vector have the same phase at time \(t\) and the time \(t+2\pi /\omega\), there will be constructive interference that makes it likelier for a photon of energy \(\hbar\omega\) to be born. But in the Bohmian theory, you also need to create the actual photon – one that will fly in a particular direction of the prism. But to decide whether this "actual classical particle" should be first created at the time \(t\), you simply need to compare the wave function at the moment \(t\) with the wave function at other moments of time. Otherwise your theory can't know anything about the interference and the right energy (and therefore direction in the prism) of the photon that should be created.
Even if you designed some laws – nonlocal in space and time – that create the actual photon, you won't know how this photon could be "piloted" by the Maxwellian pilot wave. At any rate, the Bohmian photon's average speed will exceed the speed of light. And to make things even worse, you're guaranteed to violate the energy conservation law.
You know, the funny thing is that in quantum mechanics, if you detect the photon whose energy is equal to the energy of an atomic transition between two levels \(E_i,E_f\),\[
\hbar\omega = E_i - E_f,
\] you are performing a measurement of the atom's energy and the detection of the photon of frequency \(\omega\) automatically implies that the atom's final state has to be \(E_f\). The final wave function for the system atom+photon (before the measurement of the photon, however) is entangled and guarantees the perfect correlation between the photon's energy and the atom's final energy. This perfect correlation has to hold because it encodes the energy conservation law. And be sure that you may experimentally verify that the energy is conserved in this sense, in these simple emission processes.
No Bohmian theory is capable of guaranteeing this perfect correlation. In other words, all theories based on the Bohmian paradigm are guaranteed to violate the energy conservation law during the atomic emission of light. Why?
Well, it's simple. In quantum mechanics, the energy conservation follows from the collapse of the wave function and by the very definition of the Bohmian mechanics, Bohmian mechanics avoids the collapse at the moment of the measurement. In any Bohmian picture, the measured values must be already prepared a femtosecond before the measurement.
But in quantum mechanics, by measuring the position of the photon in the prism, you measure the energy, and because the photon's energy is entangled with the atom's energy, a particular result for the photon's direction (i.e. energy) collapses the atom's wave function to an energy eigenstate! Now:
The absence of the collapse in Bohmian mechanics means that the atom can simply never collapse to an energy eigenstate, even though the photon that has known about the atom's energy has gone through the prism and was detected. The wave functions of the Bohmian mechanics never really collapse. They're only meant to direct particles in some way but if you measure something else than particles' positions – i.e. atom's energy (which you can by looking at the emitted photon), this Bohmian picture simply breaks down completely.
The atom continues to be described by a pilot wave that is a general superposition that still contains nonzero probabilities for all previously allowed energy eigenstates of the atom. And the positions of the electron and proton aren't affected by the detection of the photon – even though they should really be correlated with the photon's energy. So if you measure the energy of the atom once more, Bohmian mechanics predicts that you may get wrong values of the energy and the energy conservation law is violated.
The experiments show that this never happens.
It's possible to say that Bohmian mechanics removes the "collapse" from quantum mechanics and replaces it with the "extra Bohmian positions". In this way, the collapse to the eigenstate of all particles' positions "works" (and is done in advance because the soon-to-be-measured positions are already decided before the measurement). But in quantum mechanics, one also needs infinitely many different types of collapses to different states than position eigenstates – and Bohmian mechanics is guaranteed to fail in all these situations. Its inability to collapse the "pilot wave" for an atom into an energy eigenstate was just a major example; it's obvious that Bohmian mechanics fails in the prediction of every single measurement of any property of the atom except for the locations of the fermions themselves.
One could reinterpret this "Copenhagen-Bohm match" above as a direct experimental proof of the (Copenhagen) postulates of quantum mechanics. We can really experimentally prove that when a photon emitted by an atom is detected, the atom's wave function has to collapse to an energy eigenstate – and the distributions for the positions of the nucleus and the electron have to be modified accordingly. This fact may be experimentally checked. It works. Bohmian mechanics says that the collapse doesn't take place so it's simply ruled out by this simple experiment. Almost any experiment rules it out.
You might be inventing ways to "fix the Bohmian theory" so that the atom does collapse to the energy eigenstate once the emitted photon is measured; and the Bohmian electrons' and nuclei's positions are abruptly modified once the photon is detected, too. But such a picture will obviously depend on the reference frame and break the Lorentz symmetry heavily, much like many other phenomena you need. All these fixes will be immensely ugly.
More importantly, they will be ad hoc. You will need a new fix for pretty much every qualitatively new phenomenon or experiment someone will invent. At the end, if you work on these "fixes" for weeks or months and you have some basic integrity, you should be able to realize that "your celebrated theory" isn't really producing any results. It is not a fixed theory at all. You are just constantly fixing and adapting "the theory" to fit the results calculable using a different, superior theory – (Copenhagen) quantum mechanics. Other people know that there exists exactly one fix of Bohmian mechanics that isn't embarrassing or dishonest. It's called (Copenhagen) quantum mechanics.
One may say that the process of "applying ad hoc fixes" to Bohmian mechanics is exactly equivalent to the publication of a new book of the Bible that "explains" every new feature of fossils or DNA discovered by the evolutionary biologists. A new fossil is found and exhibits certain patterns. A creationist may write a book in which God has a discussion with Moses and from the discussion, it follows that the dinosaur bones have to have one property or another. So the Biblical theory of Creation explains all the data – but only after the Biblical theory of Creation is adapted every time a new fact emerges.
The case of Bohmian mechanics is isomorphic.
You know, the problem is that such theories are never used to do any actual science. These theories (and their shameless apologists) are 100% pure parasites that want to share the credit with the actual theories that may be used to explain or predict the phenomena in Nature – in these two cases, with quantum mechanics and Darwin's evolutionary biology. And it's simply bad for someone to support or praise similar parasitic, unscientific theories. They don't deserve any credit because they're not used to explain or understand or predict anything. And these parasitic unscientific theories and their advocates don't have any scientific capital to decide about the right answers to the big questions – they can't even deal with the smallest ones. They're only proposed to muddy the waters and fool the people into thinking that certain "ideologically inconvenient" paradigm shifts (the shift from creationism to evolution, or the shift from classical to observer-dependent quantum physics) aren't really needed. Except that they definitely are needed. They are totally essential.
This was a sketch explaining why I think that Bohmian mechanics belongs to the pseudointellectual cesspool, not to serious scientific research, and that it must be treated on par with creationism.
And so is atomic emission and the energy conservation law
Bohmian mechanics is a framework (partly well-defined in some very special situations) proposed to replace the postulates of quantum mechanics by a new classical theory that largely borrows the mathematics controlling the state vector as if it were a classical wave; and that is supplemented with extra beables (in the only semi-successful example, positions of non-relativistic particles) that behave classically and whose trajectories are affected by the pilot wave.
The theory allows you to assume that if you measure the non-relativistic particles' positions, the random outcome isn't decided at the moment of the measurement. Instead, these positions were already ready before the measurement. And the defining assumption of the Bohmian paradigm is that any measurement may be reduced to such a measurement of the positions (or other beables if you knew what they are) so that the random collapse in the standard quantum mechanical treatment of the measurements isn't ever needed. For example, it's sometimes semi-successfully claimed that the measurement of particles' spins may be reduced to the measurement of the positions of the same particles. But we will see that this is not the case for observables that commute neither with the position nor with the momenta such as energy.
In several recent articles, I discussed the incompatibility of Bohmian mechanics with some of the physical phenomena that go beyond non-relativistic QM, e.g. with loop corrections in QFT. User7348 was dissatisfied with the claim that Bohmian mechanics fails in this test so he asked for a "rebuttal" of my claims. There must be one, right? Everyone who believes that Bohmian mechanics doesn't suffer from these lethal defects is invited to write the pre-planned "rebuttal" of my proof.
Proofs can rarely be disproven and so far, no one could have offered any "rebuttal". Physicists largely agree that I am right. On this blog, aside from other interactions, I've had quite a typical exchange with one Bohmian who claimed that the loops could work. So I asked him to show me the calculation of the electron's magnetic moment (up to the one-loop level) in the Bohmian theory. The answer I got was "why would I do it if it has already been done?".
He obviously meant that it was done in proper quantum field theory governed by the standard, "Copenhagen" postulates of quantum mechanics (at most reformulated with a different "accent" but not a different "content"). And because he must believe that Bohmian mechanics has "conquered" the standard quantum mechanics and may claim credit for all of the successes of quantum mechanics (while taking no responsibility for the alleged drawbacks), he just doesn't need to write anything, he believes.
This is obviously silly. If you claim to have a theory that differs from another one (e.g. the "dominant" one among the researchers) and your theory is said to have some advantages, it follows that it's inequivalent. When it's inequivalent, it may also be inequivalent in the "undesirable" situations – it may fail to reproduce the successes of the "dominant" theory you want to replace. Clearly, this general rule – modifications that have the potential to "improve" something also have the potential to "ruin" something – holds for Bohmian mechanics, too. And be sure that the basic Bohmian assumptions ruin pretty much everything.
These Bohmian people often make claims such as "all the physics of QFT works just fine in Bohmian theory". References to incoherent preprints that make similar claims are the "evidence" you may get. All these preprints contain some "extra" (and very ugly) mathematical constructions that are absolutely different from the standard QM/QFT and it's obvious that they can't be producing the same predictions in general. But if someone claims that the Bohmian theory makes sense, shouldn't he be able to write the "proper modern [Bohmian]" textbook replacing the existing textbooks of quantum field theory? At least a few chapters, up to a calculation of some annihilation processes of QED.
Needless to say, nothing like that is possible. None of these calculations may actually be done in the Bohmian theory. At most, these people may say that "they're suddenly modest and they're satisfied with the 'Copenhagen' treatment". But they won't tell you what the list of "beables" (like Bohmian particle positions) is during the annihilation or light emission or any other basic process – and how they behave during the process. What happens to make sure that the beables detect a correct statistical distribution of these quantities? These Bohmian positions are basically "one-half of the Bohmian theory" – a part that is completely absent in proper QM/QFT – so shouldn't the Bohmians add something, basically 100%, to the Copenhagen story about all these phenomena?
These people would like to dictate the overall philosophy. They often please you with extraordinarily wonderful claims such as "observers are not needed" or "the basic framework of classical physics was more or less right and no conceptual change is needed". But when you want them to use their theory to actually calculate something, they prefer to be silent. This approach is similar to the Marxists who are very loud and aggressive when they steal the assets of an "evil capitalist" but when they're expected to produce the same values that the "evil capitalist" was able to produce, their energy suddenly fades away and you hear excuses. These Marxists complain about lots of bad circumstances such as the four main enemies of communism – spring, summer, autumn, and winter – that sometimes lead to failures, and so on. Both Bohmists and Marxists are very loud and aggressive when they scream what they hate but when they're supposed to show an alternative, all the meaningful activity goes away.
Sadly, this is not just an analogy. Most Bohmists actually were and are Marxists at the same moment (starting with David Bohm himself) so these two examples of a behavior may be just two sides of the same coin.
What's even more disturbing is that these Bohmists must actually know that the theory fails in the description of very basic things – such as the propagation (and emission) of photons. There are Bohmian papers explicitly stating that according to the common Bohmian lore, something is impossible.
Photons' wave functions, beables, and atomic emission: impossible in the Bohmian picture
I wanted to start with the emission of atoms resulting from an atomic transition. It can't work and all Bohmian attempts are guaranteed to destroy the discreteness of the spectrum and lead to violations of the energy conservation law. But there is a simpler thing that doesn't work: the mundane propagation of free photons!
Just like electrons, photons may be used to perform the double slit experiment. Photons create individual dots on the photographic plate. By its definition, the Bohmian mechanics must have a result for the measurement of the "photon position" that is ready before the measurement. Except that there can't be any equations – at least not local or otherwise natural equations – that could govern the motion of such "real Bohmian photons". Why?
It's simple. Take a 2001 paper by Ghose+3, Bohmian trajectories for photons, that starts:
It is generally believed that only massive fermions have Bohmian trajectories but bosons do not. This is usually attributed to the impossibility of constructing a relativistic quantum mechanics of bosons with a conserved four-vector probability current density with a positive definite time component. However, it has now been shown [1] that a consistent relativistic quantum mechanics of spin 0 and spin 1 bosons can be developed using the Kemmer equation [2]...In the normal Bohmian theory (for a fermion), one uses the probability 4-current \((\rho, \vec j)\) where \(\rho=|\psi|^2\) integrates to one. The current must obey the continuity equation Рand as you know from some mathematics of the non-relativistic Schr̦dinger's equation, it does obey this equation.
To incorporate photons that "know where they should create dots", you simply need such a 4-current \((\rho, \vec j)\) as well. However, it simply cannot exist in Maxwell's theory, at least not in any local or otherwise natural form.
The problem is that the natural bilinear object in Maxwell's theory isn't a current (a 4-component vector) but the stress-energy tensor \(T_{\mu\nu}\). The integral of \(T_{00}\) over the 3-dimensional space isn't one. Instead, it's the total energy – something that is proportional to the frequency via \(E=\hbar\omega\).
In proper quantum mechanics, the wave function of one photon basically behaves as the Maxwell's electromagnetic field – which is however reinterpreted probabilistically. Now, the detection of a photon is always sensitive to some frequencies and the Maxwell field (or photon's wave function) secretly knows about the representation of different frequencies, too.
However, if you assume that the Bohmian photon has a well-defined position, it simply cannot have a well-defined energy at the same moment. Also, the required 4-current can't be written as any function of the fields \(A_\mu\) and their derivatives. You simply can't build any conserved current (with one index) out of electromagnetic fields.
It's easy to show the complications explicitly. The quantized electromagnetic field may be built as a Fock space. In a box, for every allowed \(\vec k\), there exist two annihilation operators \(a_\lambda(\vec k)\) and two Hermitian conjugate creation operators \(a^\dagger_\lambda(\vec k)\) – two because there are two polarizations for each direction.
There is a nice formula for the total energy\[
H = \sum_{\vec k, \lambda} \hbar c |\vec k|\cdot a^\dagger_\lambda(\vec k) a_\lambda(\vec k)
\] in the position space, namely the integral of \(\rho_E = (E^2+B^2)/2\). However, there is no natural formula for the number of photons. The number of photons would differ from \(H\) by the omission of the factor \(|\vec k|\). While the required factor \(1/|\vec k|\) may look simple and natural in the momentum space, it is absolutely unnatural and nonlocal in the position space – it is some kind of \(1/\sqrt{\nabla^2}\) and believe me that it's not nice to compute the square root of some inverse Laplace operators. (Dirac found an ingenious method how to define a nice "square root" of the box operator – the Dirac operator – but that's exactly what you can't do for bosons.)
So no local or natural formula for the number of photons \(N\) may exist in Maxwell's theory. Consequently, you can't find any local or natural 4-current which is needed in Bohm's theory. And that means that there can't be any natural equations telling you how the photon should respond to the Maxwell pilot wave. It just doesn't make any sense.
Above, I assumed that you would try to define the number of photons by applying the "inverse square root of the Laplacian" – a highly non-local operator – on the formula for the total energy. Ghose et al. chose a different approach, one with some extra bizarre matrices \(\beta_\mu\), but it's ultimately equally ugly. They may draw some trajectories that respect some probability densities but there is absolutely nothing natural about the mathematical manipulations needed to do so – nothing natural about the new proposed laws of physics governing photons.
And you may check that none of the 68 followups of the paper by Ghose et al. actually uses this super-ugly, super-unnatural set of equations with extra new vectors. It's ultimately as ugly as the nonlocal square roots of the Laplacians – and probably equivalent at some level. All the papers only use the bizarre formulae by Ghose et al. as a political argument of a sort. They don't really want to do any science with such things. I think that they don't really believe that these super-ugly, arbitrary equations are correct or useful in any scientific sense. They must know that they have at most fitted a result calculated by a different theory, rather than derived a result by using their own theory. What they're doing is just an ideologically-driven propaganda. All these "solutions" are "superfluous ideological superstructures", as Werner Heisenberg accurately called the original Bohmian theory.
(If the weird Ghose et al. formalism to produce the Bohmian photon trajectories were right, people could immediately find additional evidence, consistency checks, agreement with hints found in different ways, and/or generalizations of this formalism to other fields and particles of the Standard Model such as the Yang-Mills fields or other phenomena – just like in the cute and dynamic evolution of the successful scientific theories we actually know from the physics textbooks. But the number of these "signs" that the Bohmian ideas are on the right track is strictly zero and the number of hints that they're wrong is huge. Similar advocates of these bad ideas generally don't care about an arbitrary finite number of "bad signs" because they are infinitely prejudiced.)
An obvious additional problem is that the photons' trajectories have to be superluminal at many points. The waves generally move by the speed of light but to preserve the interference patterns, the pilot wave has to push the Bohmian photons back and forth (towards the interference maxima), so this gives them some extra motion relatively to the averaged motion of the wave, and the average speed of these particles on these "not quite straight" lines obviously has to be higher than the speed of light. If you want to keep in touch with light but you need detours on your trip, you sometimes need to move faster than the light, right? There's no way to "solve" this problem. The superluminal motion of these particles looks like the motion "backwards in time" in other relativistic inertial systems. Too bad.
So there's really no usable way to incorporate photons to the Bohmian theory.
Bohmian theory is incompatible with light emission by atoms
The absence of a 4-current in Maxwell's theory – that prevents you from defining any natural "pilot wave force" on the Bohmian photon – is far from being the only problem you would encounter if you tried to incorporate photons. Try to find the Bohmian description of a very simple process – the emission of light by atoms. Take a hydrogen atom. It should have some discrete spectrum, OK? The energy of the photons may be converted to their position using a prism (colors are separated on the rainbow).
Let's assume that the Bohmian theory correctly describes the fermions (electron, proton) that the atom is composed of. Such a positive appraisal isn't right even in this limited setup of a non-relativistic description of fermions but let's be overly generous. However, this theory doesn't automatically emit light. You need to couple the atom to the electromagnetic field. The frequency \(\omega\) modes of the electromagnetic field will be sensitive to the modes of the atom in which the energy jumps by \(\hbar\omega\). For those, the constructive interference may add up and convince the atom to create a photon.
The Bohmians want to argue that before the photons are detected, they already objectively have the soon-to-be-observed positions. If that's the case, the photon must be actually born at some moment, in the vicinity of the atom. (We neglect the previously discussed problems with the free photon which is already too general and can't be incorporated to the Bohmian theory.) OK, when is the Bohmian photon born and what are the rules that achieve this job?
Proper quantum mechanics tells you that you mustn't try to answer this question because it's meaningless. Only the observed outcomes may be treated as "facts" and the photons will only be detected a long time after they're created by the atoms. Before they're detected, the probability amplitudes are interfering with each other.
Much like in the discussion of the loop corrections, the whole potential for the photons to appear in one direction of the prism or another is due to the interference effects. To have a chance to reproduce those, the Bohmian-like theory would have to contain the whole "wave functional" (state vector of the quantum field theory) as a classical field. Great, Bohmists know how to steal but unfortunately for them, the state vector isn't the only classical degree of freedom in their proposed theory. They also need the photons' positions. How are those created, what are their initial values, what are the rules that govern them?
Any remotely viable solution will have to be sensitive to the interference so it will have to be non-local in time. You know, if parts of the state vector have the same phase at time \(t\) and the time \(t+2\pi /\omega\), there will be constructive interference that makes it likelier for a photon of energy \(\hbar\omega\) to be born. But in the Bohmian theory, you also need to create the actual photon – one that will fly in a particular direction of the prism. But to decide whether this "actual classical particle" should be first created at the time \(t\), you simply need to compare the wave function at the moment \(t\) with the wave function at other moments of time. Otherwise your theory can't know anything about the interference and the right energy (and therefore direction in the prism) of the photon that should be created.
Even if you designed some laws – nonlocal in space and time – that create the actual photon, you won't know how this photon could be "piloted" by the Maxwellian pilot wave. At any rate, the Bohmian photon's average speed will exceed the speed of light. And to make things even worse, you're guaranteed to violate the energy conservation law.
You know, the funny thing is that in quantum mechanics, if you detect the photon whose energy is equal to the energy of an atomic transition between two levels \(E_i,E_f\),\[
\hbar\omega = E_i - E_f,
\] you are performing a measurement of the atom's energy and the detection of the photon of frequency \(\omega\) automatically implies that the atom's final state has to be \(E_f\). The final wave function for the system atom+photon (before the measurement of the photon, however) is entangled and guarantees the perfect correlation between the photon's energy and the atom's final energy. This perfect correlation has to hold because it encodes the energy conservation law. And be sure that you may experimentally verify that the energy is conserved in this sense, in these simple emission processes.
No Bohmian theory is capable of guaranteeing this perfect correlation. In other words, all theories based on the Bohmian paradigm are guaranteed to violate the energy conservation law during the atomic emission of light. Why?
Well, it's simple. In quantum mechanics, the energy conservation follows from the collapse of the wave function and by the very definition of the Bohmian mechanics, Bohmian mechanics avoids the collapse at the moment of the measurement. In any Bohmian picture, the measured values must be already prepared a femtosecond before the measurement.
But in quantum mechanics, by measuring the position of the photon in the prism, you measure the energy, and because the photon's energy is entangled with the atom's energy, a particular result for the photon's direction (i.e. energy) collapses the atom's wave function to an energy eigenstate! Now:
The Bohmian theory never collapses the atom's wave function to an energy eigenstate. In fact, in the Bohmian theories, the "real particle" doesn't influence the pilot wave at all!This is a rather brutal feature of the Bohmian theory: the theory is very loud about the influence of the pilot wave on the particle but it basically assumes that the particle doesn't affect the pilot wave at all. You should always be suspicious about theories with similar "asymmetric" influences. They sound like a theory about God who can influence everyone else but can't be influenced. In proper physics at the fundamental level, all interactions go in both ways. You may say that some kind of Newton's third law (actions induce reactions) always applies.
The absence of the collapse in Bohmian mechanics means that the atom can simply never collapse to an energy eigenstate, even though the photon that has known about the atom's energy has gone through the prism and was detected. The wave functions of the Bohmian mechanics never really collapse. They're only meant to direct particles in some way but if you measure something else than particles' positions – i.e. atom's energy (which you can by looking at the emitted photon), this Bohmian picture simply breaks down completely.
The atom continues to be described by a pilot wave that is a general superposition that still contains nonzero probabilities for all previously allowed energy eigenstates of the atom. And the positions of the electron and proton aren't affected by the detection of the photon – even though they should really be correlated with the photon's energy. So if you measure the energy of the atom once more, Bohmian mechanics predicts that you may get wrong values of the energy and the energy conservation law is violated.
The experiments show that this never happens.
It's possible to say that Bohmian mechanics removes the "collapse" from quantum mechanics and replaces it with the "extra Bohmian positions". In this way, the collapse to the eigenstate of all particles' positions "works" (and is done in advance because the soon-to-be-measured positions are already decided before the measurement). But in quantum mechanics, one also needs infinitely many different types of collapses to different states than position eigenstates – and Bohmian mechanics is guaranteed to fail in all these situations. Its inability to collapse the "pilot wave" for an atom into an energy eigenstate was just a major example; it's obvious that Bohmian mechanics fails in the prediction of every single measurement of any property of the atom except for the locations of the fermions themselves.
One could reinterpret this "Copenhagen-Bohm match" above as a direct experimental proof of the (Copenhagen) postulates of quantum mechanics. We can really experimentally prove that when a photon emitted by an atom is detected, the atom's wave function has to collapse to an energy eigenstate – and the distributions for the positions of the nucleus and the electron have to be modified accordingly. This fact may be experimentally checked. It works. Bohmian mechanics says that the collapse doesn't take place so it's simply ruled out by this simple experiment. Almost any experiment rules it out.
You might be inventing ways to "fix the Bohmian theory" so that the atom does collapse to the energy eigenstate once the emitted photon is measured; and the Bohmian electrons' and nuclei's positions are abruptly modified once the photon is detected, too. But such a picture will obviously depend on the reference frame and break the Lorentz symmetry heavily, much like many other phenomena you need. All these fixes will be immensely ugly.
More importantly, they will be ad hoc. You will need a new fix for pretty much every qualitatively new phenomenon or experiment someone will invent. At the end, if you work on these "fixes" for weeks or months and you have some basic integrity, you should be able to realize that "your celebrated theory" isn't really producing any results. It is not a fixed theory at all. You are just constantly fixing and adapting "the theory" to fit the results calculable using a different, superior theory – (Copenhagen) quantum mechanics. Other people know that there exists exactly one fix of Bohmian mechanics that isn't embarrassing or dishonest. It's called (Copenhagen) quantum mechanics.
One may say that the process of "applying ad hoc fixes" to Bohmian mechanics is exactly equivalent to the publication of a new book of the Bible that "explains" every new feature of fossils or DNA discovered by the evolutionary biologists. A new fossil is found and exhibits certain patterns. A creationist may write a book in which God has a discussion with Moses and from the discussion, it follows that the dinosaur bones have to have one property or another. So the Biblical theory of Creation explains all the data – but only after the Biblical theory of Creation is adapted every time a new fact emerges.
The case of Bohmian mechanics is isomorphic.
You know, the problem is that such theories are never used to do any actual science. These theories (and their shameless apologists) are 100% pure parasites that want to share the credit with the actual theories that may be used to explain or predict the phenomena in Nature – in these two cases, with quantum mechanics and Darwin's evolutionary biology. And it's simply bad for someone to support or praise similar parasitic, unscientific theories. They don't deserve any credit because they're not used to explain or understand or predict anything. And these parasitic unscientific theories and their advocates don't have any scientific capital to decide about the right answers to the big questions – they can't even deal with the smallest ones. They're only proposed to muddy the waters and fool the people into thinking that certain "ideologically inconvenient" paradigm shifts (the shift from creationism to evolution, or the shift from classical to observer-dependent quantum physics) aren't really needed. Except that they definitely are needed. They are totally essential.
This was a sketch explaining why I think that Bohmian mechanics belongs to the pseudointellectual cesspool, not to serious scientific research, and that it must be treated on par with creationism.
Bohmians' self-confidence evaporates as soon as they're expected to calculate anything
![Bohmians' self-confidence evaporates as soon as they're expected to calculate anything]()
Reviewed by MCH
on
May 14, 2016
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