I am having a look at different philosophical interpretations of quantum physics. This is the fourth post in the series. The first post gave a general introduction to quantum wave mechanics, and presented the Copenhagen interpretations. I have subsequently looked at spontaneous collapse models and the Everett interpretation. Today it is the turn of the pilot wave interpretation.
The pilot wave model is often discussed by philosophers of physics, although has considerably less support among physicists themselves. It was first proposed by de Broglie in the 1920s, and then revived by David Bohm in the 1950s and taken up by Bell in the 1960s. Since then it has received various refinements and discussion, but it has also gained some criticism. People have tried to address the criticism. It still remains an active area of research, with its proponents. I cannot claim to be an expert on it, but will present here the model as best I can, and the main reasons why people both like it and reject it.
The pilot wave model, like the Everett model, is deterministic, and thus allows us to continue the philosophy of classical physics into quantum physics with only minor modifications. The underlying idea is that the beables in quantum physics can be split into two different classes of objects. Firstly there are the particles, and these are the things we observe: electrons, protons, and so on. Then we have an underlying pilot wave. This is not directly observed, but its effects can be seen through its interactions with the particles. Because the pilot wave cannot be directly observed, its initial state in any experiment is unknown. This leads to the apparent indeterminacy we observe in experiment. It is not an indeterminacy fundamental to physics, either through wave function collapse (or something similar) or an indeterminate dynamics, but merely arises due to our lack of knowledge of the full physical situation. This is similar to how uncertainty enters into classical physics. When we toss a Newtonian coin, the underlying physics is deterministic. But because the system is so finely balanced, and we don't know the details of the force applied to the coin, or the air movements, or maybe even the initial state, we can only assign probabilities for each possible result.
The main criticism of the pilot wave model, and the reason it is rejected out of hand by most physicists, is that it is apparently inconsistent with relativistic quantum physics. This is a big deal. A viable philosophy of quantum physics needs to fulfil two roles: to reproduce the best physical theories to within experimental tolerance (which, at the moment is the standard model of particle physics, a relativistic quantum field theory), and secondly to leave no be philosophically complete and coherent, so leave no gaping gaps in its explanations. I don't think anyone doubts that the pilot wave model satisfies the second criteria, which puts it, on that count, better than most suggested interpretations. But if it is inconsistent with the physics then it is a dead end. The formulation of the theory by Bohm is fairly obviously inconsistent with the physics, as it is based on the non-relativistic Schroedinger equation, which is not the correct model of nature. There have been attempts to modify the Pilot wave approach to bring it back into agreement with the physics, but so far (to my knowledge) none of these have yet gone far enough to actually reproduce the standard model. Most of these attempts call back to de Broglie's original pilot wave model.
The non-relativistic formulation
The pilot wave interpretation is, at its heart, a reformulation of the Schroedinger equation to introduce a second dynamical variable. I here outline Bohm's approach, which shows both the advantages and disadvantages of the interpretation. We start with the non-relativistic wave mechanics Schroedinger equation.
The starting point of the interpretation is to parametrise the wavefunction in terms of two real fields, R and S.
In terms of this decomposition, the left hand side of the Schroedinger equation then becomes
While the right hand side becomes
Comparing real and imaginary parts of the equations then gives two equations, one describing the evolution of R in time,
The other gives the evolution of S in time,
Now we need to interpret these equations, and if we manipulate them further we see a natural picture emerge. Firstly, we write a new variable P as the square of R.
In the Copenhagen interpretation of the Schroedinger equation, this would correspond to the probability density. Here, of course, it is just an underlying real field, or potential. The time evolution equation for R is re-written in a simple form in terms of P.
This takes the form of a conservation equation, with P representing a charge and P∇S as some sort of probability current. For the equation describing the motion of S, we note that the term dependent on R can be expressed as a potential Q
This leads to an equation of motion for S which is starting to look like a classical equation of motion.
If we identify ∇S as a momentum,
the equation describing the motion of S is then written as
And this is basically Newton's equation of motion, with the particle momentum changing deterministically due under a potential. The potential is split into two parts, one arising from the external potential input into Schroedinger's equation at the very start, and the other arising from the second field, P.
So far, I haven't actually done anything. This is just a reformulation of the non-relativistic Schroedinger equation. It describes exactly the same physics. The mathematics is entirely equivalent; it is just a different way of expressing the same theory. One can still interpret P as a probability density associated with the wavefunction, introduce wavefunction collapse, and get back to the Copenhagen interpretation.
However, this reformulation opens up another possibility. What if we interpret R or P as a classical field rather than being related to a probability density? What if we sat that in addition to this field, there is also a particle -- what we observe -- with a momentum p and a location conjugate to it (following the approach used in Hamilton's reformulation of classical mechanics)? Then we have an entirely deterministic physics, which obeys equations of motion which are similar to those used in Newton's mechanics. The R field carries waves, which uniquely do not convey energy or momentum -- that is just transported by the particle. It is a little bit different from Newtonian mechanics, obviously, but sufficiently similar that the same philosophy would directly carry over. All we have done is introduce a new field which in part directs the motion of the particle, and whose changes are influenced by the motion of the particle, but classical mechanics has electric and magnetic fields without any problems, so there is nothing troublesome about this.
We need one more thing to create a full interpretation of quantum physics. This is to explain the apparent probabilistic nature of quantum measurements. The interpretation is the same as apparent randomness in classical physics, for example when we toss a coin. The result is unpredictable because we do not know all the underlying details of the coin toss. Variables such as the initial configuration of the coin, the precise force applied to it, and so on. What we have to do is create a realistic model of these hidden variables, assigning a probability to each of them, and solve the equations of motion for each possibility. That will give us a probability distribution for the final result.
Bohm's pilot wave model interprets the apparent probabilistic nature of experiments in a similar way. There are hidden variables which we do not know -- which arise from the precise functional form of the pilot wave and the initial conditions of the particle -- which means that we cannot predict with certainty the precise final result of the measurement. Instead, we assume the equilibrium condition -- that these hidden initial conditions can be parametrised according to the probability distribution P. If this is so, then the final results of the experiment will be distributed according to the updated probability distribution -- this follows because P evolves according to Schroedinger's equation, the same as the underlying pilot wave. The equilibrium condition is not something arbitrary, but it can be shown that over time a Bohmian system would tend towards this equilibrium state. There might be some differences to standard quantum physics in the early universe as the system tends towards equilibrium, but today everything is in equilibrium and we would never be able to measure any difference between a world that matches the Pilot wave interpretation, and one where Copenhagen was in fact the correct interpretation.
So it is easy to see why this interpretation is very appealing. It avoids all the philosophical problems of the Copenhagen, the spontaneous collapse, and the many-worlds interpretations. It allows us to carry over our interpretation and philosophy of classical physics practically unchanged. And it reproduces precisely the physics on non-relativistic quantum wave mechanics.
And I quite agree, that if non-relativistic quantum wave mechanics were the correct theory of nature, the Pilot wave model would be a very serious contender for the correct philosophy of physics. That is not as strong a statement as saying that it is true -- one would have to show that there are no other contenders which duplicate the physics and lack any obvious philosophical problems or gaps -- but it would be difficult to rule it out.
However, the model described above, while it has some support among philosophers of physics, is rejected by almost all physicists, myself included. The reason for this is straight-forward: the non-relativistic Schroedinger equation is not the correct theory of quantum physics. It is not even an approximation to the correct theory (in the way that Newtonian mechanics is an approximation to relativistic mechanics, valid in the limit that the speed of light becomes infinite). It is not even the correct non-relativistic limit of relativistic quantum field theory -- that is non-relativistic field theory, which is widely used in condensed matter physics. Quantum wave mechanics is just a theory that was a useful stepping stone to help us come to an understanding of relativistic quantum field theory, which replicates some but not all of the ways in which quantum field theory differs from classical physics.
So the question is: can the equations of relativistic quantum field theory be recast in a similar way to how Bohm treated non-relativistic wave mechanics, to make the equations of motion for the various elements of a pilot wave model obvious? If it can, then great. It is back on the table. But if not, then it has to be abandoned.
The problem with relativity
Obviously, the approach described above is not going to be compatible with relativity because it is based on the non-relativistic Schroedinger equation. There is an alternative, which I ought to mention. The guiding equation for the pilot wave is derived from the appropriate Schroedinger equation for the system. The change in the particle position q is then described by the equation.
This approach is not derived from the non-relativistic Schroedinger equation, so it might be adaptable to relativistic theories. The problem for me is that its derivation is not as transparent as Bohm's, and thus it is harder to be sure that it is just a reformulation of the standard theory. The equation is derived from the standard quantum mechanical conservation of probability.
where the probability current is given by
This is a standard result which follows directly from the Schroedinger equation. The relation to the guiding equation for the particle is obvious: the rate of change of the particle location is just the probability current divided by the probability density. The claim is then that if we understand the particles as being initially distributed according to a probability ψ†ψ, then it will continue to be so according to this evolution equation. The difficulty I have with this is that it switches the interpretation of ψ from a probability amplitude to a pilot wave wavefunction. How is probability even meant to be understood for a single particle? Presumably in Bayesian terms. In this case, the probability would either have to be subjective (Bayesian proper) or conditional (the logical interpretation of probability), and in both cases there seems to be an issue in jumping between using ψ to represent an objective and unconditional pilot wave wavefunction and representing a subjective or conditional probability.
Edit: In view of the debate in the comments below, I think it useful to show how de Broglie's equations preserve the frequency distribution of the particles. Here I use |Ψ|2 to represent the total probability distribution, and ψi to represent the wavefunction for each individual particle. The expected number of particles in a given region Ω is given by
The rate of change of this over time is then given by
Substituting in the non-relativistic Schroedinger equation then gives
This simplifies to
Substituting in the equation of motion for the particle then gives
Which, via stokes theorem, where A represents the area bounding the surface, leads to
The left hand side of this equation can then be interpreted as the rate of change of the total numner of particles in the region. The right hand side is interpreted as the flux of particles entering or leaving the region. The equality between the two means that the number of particles entering or leaving the region in a given moment of time is equal to the change in the number of particles in the region, i.e. that the frequency distribution of the particles remains proportional to the wavefunction over time.
The key equation in all this is equation (26). To derive equation (26), we needed to use the non-relativistic Schroedinger equation. So clearly the Pilot wave theory for these equations is only valid for that equation. There are, however, equivalent expressions for relativistic quantum mechanics, albeit that they get a bit more complicated (for example, the expression for the probability density and probability current changes). The equations of motion will be different, but you can still go through the same procedure and reach the same conclusion.
It becomes a bit trickier in quantum field theory, because the Schroedinger equation works differently. In all forms of quantum wave mechanics, the Schroedinger equation is a linear equation where some power of the time differential of the wavefunction is proportional to some power of the space differential of the wavefunction plus some stuff. The proof above relies on the equation being in that form. In quantum field theory, that is not the case. The space differential does not act directly on the Fock state, but on the creation and annihilation operators. For that reason, we generally do not discuss probability density or probability currents. So it is not clear to me that there is some equivalent to equation (26) in quantum field theory, with the right properties to allow you to derive the appropriate conclusion.
One needs to formally show the equivalence (in terms of experimental predictions) between the pilot wave model and the instrumentalist Copenhagen interpretation of quantum physics. Bohm achieved that for the non- relativistic Schroedinger equation, as described above. What is needed is to create an analogous model for relativistic quantum field theory.
The problem with combining the Bohm model with relativistic quantum theory is probably easiest to describe if we start by slightly reformulating the Schroedinger equation.
In the limit that the small time difference δ t this reduces to the standard formulation of the equation listed above. One can still perform the same decomposition as above, and work through the same calculation, neglect terms of order (δ t)^2 and above, and one would end in the same place as the standard Bohm decomposition.
So why re-write the equation in this way? There are two reasons. Firstly, it is a natural way of formulating the Taylor series expansion in time.
This is equivalent to the standard differential form of the Schroedinger equation as long as the wave-function is continuous between time t and time t + δ t.
Secondly, the time evolution operator in quantum field theory cannot be expressed in terms of a straight-forward differential equation, as in the standard formulation of the Schroedinger equation. But we can write it in this exponential form.
So the time evolution operator becomes
For the simplest quantum field theory that contributes to the standard model, Quantum Electro Dynamics, the Hamiltonian operator is given by
This needs some explanation. Firstly, ψ is no longer a single particle wavefunction, but a Fock state. This basically counts how many particles are in each possible state. A represents the Photon. It is a four vector, with its time component, A0 representing the electric potential, and the three spatial components the magnetic potential. The operator with a hat over it,
combines the photon creation and annihilation operators. These represent the creation or destruction of a photon. Finally, we have the electron creation operator
and the electron annihilation operator
These represent the creation or annihilation of an electron. So if you apply a creation operator for an electron at a particular location and spin state to a Fock state without that electron, then it will output a Fock state with all the particles in the initial state plus that electron. If you apply an annihilation operator to a Fock state with that electron, then the output will be an otherwise identical Fock state but without that electron. Applying the annihilation operator to a Fock state without that electron gives zero, as does applying a Fermion creation operator to a state with that electron.
Finally, the γ in the equation above are matrices which describe interactions between different spin states. m is the electron mass. e is the electron change.
This is used to calculate amplitudes by which we can calculate probabilities for a particular outcomes given a particular initial state. For example, we might start with a state with an electron at a given location, and we want to calculate the amplitude that we will measure, after a set period of time, two electrons and a positron at their own locations. Usually we create the initial state by applying creation operators to a vacuum state with no particles (so, in this example, we would just apply a single electron creation operator). We then apply the time evolution operator to give the Fock state at the next instant of time. Then again to get the next instant of time. And so on until we reach the time we need. This will give a Fock state which is a superposition of various different possible outcomes. Then to calculate the amplitude for the particular outcome we are interested in, we apply annihilation operators for the particles in the end state (which will exclude any terms in the superposition which don't contain those particles), and contract against the vacuum state (which will exclude any terms in the superposition which contain additional particles), and this, after normalisation, gives us an amplitude for the outcome. This amplitude can then be converted into a probability, which is compared against experiment.
This discussion is a bit too simplistic (which I am sure will cause many readers to scream in terror at what the non-simple discussion might look like), as I have excluded a discussion of renormalisation, or a detailed discussion of spin and polarisation. I have also excluded the non-Abelian and Higgs fields, so this is just quantum electrodynamics rather than the full standard model. But it is good enough for my purposes here.
Why don't we simply expand the exponentials, as is usual in wave mechanics? Because such an expansion is only valid if the Fock state is continuous over the time interval. That is not the case, because each creation or annihilation event represents a discontinuous change to the Fock state. We also consider each time slice individually because the Fermion creation and annihilation operators are non-commutative, which makes combining exponentials, as one would do for exponentials of normal commutative functions, a non-trivial task.
So the challenge to adapt the Bohm interpretation to quantum electrodynamics can be simply expressed. How does one perform a similar decomposition that will reduce the time evolution operator for QED into a form that looks a bit like classical mechanics? To my knowledge (which admittedly is not complete) this has not been successfully achieved.
So the first question is do we need to do this? In my view, yes. Obviously the solution won't be Bohm's decomposition described above. There are other options for Pilot-Wave theories. For example, de Broglie's original approach was somewhat simpler in its construction and might be more directly applicable to a QFT. But ultimately to establish the interpretation as a viable theory, you need to come up with a set of equations which describe the dynamics of the particles and the underlying Pilot waves which both resemble the sort of equations you get in Newtonian mechanics and are equivalent to the QFT evolution (the equation above would be a starting point; in practice one would also need to incorporate non-Abelian gauge theories and renormalisation). As in the case of the Bohm theory, this would just be a reformulation of the mathematics, and doesn't actually change the theory. But it would be a reformulation that suggests the philosophical interpretation. Without this being done, it is not proven that the pilot wave interpretation can be applied to the standard model. When you have alternative interpretations which are clearly applicable to the standard model (even if they have philosophical problems), then the pilot wave interpretation will be the least appealing option for physicists. Remember, a viable philosophy of quantum physics has to both make sense philosophy, and actually reduce to our current best model of physics. Physicists care more about that second condition than the first, and a failure to show that is a big red flag.
Of course, that no solution to the problem is currently known does not mean that there is no solution. Maybe it just needs more research. But there are far more challenges which have to be overcome than compared to the non-relativistic Schroedinger equation.
The first problem is that both the non-relativistic Schroedinger equation and Newton's laws (whether applied to particles or wave equations) are described by linear differential equations. Quantum Field Theory cannot be expressed in that mathematical form. This is why I personally regard Quantum Field Theory as a different theory to non-relativistic quantum wave mechanics, rather than simply an adaptation of it to relativistic fields. Yes, Quantum Field Theory contains differential operators. But they are applied to creation and annihilation operators, not the Fock state itself, which is the generalisation of the wavefunction. Clearly, if the theory is not reducible to a linear differential equation, then it is not reducible to Newtonian mechanics.
The problem comes in that the creation and annihilation operators are not expressible as differential operators in general. I should make a caveat here. One of the toy models used to introduce QFT is the simple harmonic oscillator in quantum wave mechanics. In the one dimensional case, the potential is ω2x2, with ω some constant value. The solution is a system with an infinite number of energy levels, from ℏω/2 with increments in steps of ℏω. The energy levels can be found by solving the non-relativistic Schroedinger equation, but there is an alternative approach, which is to note that there is a pair of operators, a and a†, such that a† acting on an eigenstate of energy En gives an eigenstate of energy En + 1, while a does the reverse. These operators are effectively the sum of the location operator and its differential (baring a few constants scattered around here and there). The analogy with creation and annihilation operators in quantum field theory is obvious. Can creation and annihilation operators be expressed in terms of differential operators in a similar way? If so, it would allow us to express the time evolution equation operator for quantum field theory to a straight-forward differential equation. However, this is not as easy as it would seem. Firstly, the raising and lowering operators correspond to a spin zero Boson field. Fermion fields and gauge fields would have to be represented in a different way. Secondly, the creation and annihilation operators in QFT represent a particle localised to a particular location or momentum (depending on the basis). The wavefunctions for the various excited states in the Harmonic operator are spread out over a region of space.
And how should we interpret the creation and annihilation operators in a pilot wave model? In the standard interpretations, they are associated with the creation and annihilation of a particle (or, at least, something along those lines, differing in details according to the interpretation). However, in the pilot wave model, the Schroedinger evolution describes the evolution of the underlying pilot wave. In this case presumably the theory would be interpreted with a field rather than particle ontology, and one might consider them as creation and annihilation of field excitations rather than in terms of particles. But these creation and annihilation events would act on the pilot wave, not the particle trajectories, because in de-Broglie's formulation of the pilot wave model the evolution equation is for the pilot wave and not the particle. Now we experimentally observe the creation and annihilation of particles. So the question is how a creation or annihilation of an excitation in the pilot wave is related to the creation or annihilation of a particle, whose motion is described by an equation derived from the classical Hamiltonian-Jacobi equations.
Secondly, there is the issue of how we might decompose the theory into a part that describes the particles and one that describes the pilot waves. Is that decomposition applied to the creation and annihilation operators, or to the Fock state itself? In both cases, there are problems. If we apply the decomposition to the Fock state (so write it as a product of a pilot wavefunction and a Fock state describing the particles) and leave the operators as they are, then the end result will not be a linear differential equation. Furthermore, the initial state in the prescription is usually constructed by applying creation operators to the Fock state. If we decompose the operators into (say) the product of an operator that acts on the particles and another operator that acts on the pilot waves (while expanding the Fock state to count both particles and pilot wave excitations), then that is very different to what was done by Bohm and would not be mapped to a form similar to classical mechanics.
Next we have the problem of spontaneous creation and annihilation of particles. This is not something which can happen in classical mechanics, where you have a set of simultaneous differential equations which describe a fixed number of particles. The non-relativistic Schroedinger equation likewise does not permit the particle number to change. In Quantum field theory, it obviously does. The solution to this problem would be to regard Quantum Field Theory as a theory of fields rather than a theory of particles, so you have a single differential equation describing each field. (I personally prefer a particle ontology for QFT, as that is more easily mappable to what we observe, but this is just a personal preference and might not mean anything in practice.) Particles would then correspond to localised excitations of the fields. But then you encounter a different problem. A particle creation or annihilation event would be a discontinuous change in the field. So, for example, when an electron emits a photon, positron-electron pair, before you would have no excitations in the photon field, and then you would have one. It is an instantaneous change (unless we want to change the predictions of standard model physics), and thus a discontinuous change to the field. But this cannot be the case if the field dynamics are described by a linear differential equation, as would be required if the Pilot wave model can be mapped to a version of classical physics.
There are a few proposals to overcome this problem which I am aware of. For example, this work (expanded on here) proposed introducing spontaneous jumps into Bohmian mechanics to correspond to each creation and annihilation event. The problem with this is that it loses its philosophical attractiveness. You no longer have a classical dynamics, but a classical dynamics plus spontaneous events which modify the pilot wave, which seems no better to me than a spontaneous wavefunction collapse model. The Dirac sea approach assumes a constant Fermion number (particles minus anti-particles), which is inconsistent with various weak decays. Field ontology approaches to Bohmian mechanics have difficulties defining the equilibrium measure for Fermion fields, and so far have not quantised the gauge fields. Another approach hides the problem of particle creation and destruction in some hand-waving over entanglement with the measuring device. This, in my view, just hides the problem rather than removing it.
So while there have been attempts to solve the problem of particle creation and annihilation, thus far none of them have been proved satisfactory. Work has been done here, and is continuing to be done, but all of the approaches being tried have gaps in their physics, and many of them require a modification in the philosophy, undermining the main advantage of the pilot wave interpretation.
Then there is another problem related to the interpretation of the apparent indeterminism of the experimental results. The quantum mechanical formulation of the pilot wave model depends on an equivalence between the square of the wavefunction and an epistemic probability for the original location of the particle. But there is no wavefunction in quantum field theory. What we have instead is a Fock State, but this is not directly translatable to the amplitude that a particle is in a particular location, or even the amplitude that there is a particular set of particles. To construct an amplitude, we need to take the inner product between the Fock state and another state (and then normalise). The Fock state is not a simple function of location. The initial Fock state cannot therefore be directly translated into an epistemic probability that the given particle is at a particular location. One can construct such an location dependent amplitude: perhaps by taking the inner product of the Fock state with a state representing the location of the desired particle, or by constructing a theory with a superposition of Fock states each multiplied by an appropriate amplitude. These amplitudes can then be converted to an epistemic probability for the initial location of the particle. But it is not these amplitudes which undergo Schroedinger evolution: it is the Fock state. Presumably, then, the Fock state would have to in some way represent the pilot wave. The amplitudes derived from taking the product of the Fock state with a target would correspond to the epistemic probability. But these are not the same thing, and they evolve according to different equations. Therefore it seems unlikely that we can make a mathematical equivalence between the modulus square of the pilot wave and the epistemic probability for the location of the particle, and without this the pilot wave interpretation breaks down. It is also not clear what it means to take the derivative of the Fock state, as required to construct the probability current, as, unlike the wavefunction of quantum mechanics, the Fock state is not a direct function of location.
Furthermore, amplitudes in quantum mechanics are absolute ψ(x) represents the amplitude that the particle can be found at location x. This need not be conditional on anything, and so can be mapped to the absolute value of a pilot wave. Amplitudes in quantum field theory are conditional. We calculate the amplitude for a final state conditional on a particular initial state (and the correct formulation of the Hamiltonian). Because these amplitudes are conditional on an initial state, they cannot be mapped to a non-conditional pilot wave. In the pilot wave interpretation, the pilot wave takes on a particular value at any moment of time, and this is objective and not conditional on anything. Possibly this can be avoided by the initial state of the universe as the initial state for the Fock state evolution. This would work, but only if the dynamics is deterministic, which would rule out some of the models referenced above to deal with the creation and annihilation of particles.
Possibly the most cited problem with the pilot wave interpretation, with regards to its consistency with special relativity, is its non-locality. The equation of motion for S is divided by the square of the value of R. Dividing by something which could (in principle) be zero is one problem, but the main one is that for systems with multiple particles, the divisor in this equation depends on wavefunction at the locations of all the particles in the system. The equation of motion for one particle depends on the value of the wavefunction at other locations in space. This is a feature of the pilot wave model which cannot be avoided, even if we replaced the non-relativistic formulation of Bohm with a relativistic Schroedinger equation. The response to this is that we know that quantum physics is not local in any case, due to entanglement and EPR type experiments. And, I think, that is correct -- there is non-locality in quantum physics which can't be eliminated (at least without introducing something like backwards time causation, or other ideas which are equally problematic). Does this leave the pilot wave interpretation off the hook with this objection?
The first question is why might non-locality, or simultaneous action at a distance in any valid reference frame, a problem? The issue is that it is inconsistent with Lorentz symmetry, which is the driving force behind special relativity, and embedded in the standard model of particle physics. Lorentz symmetry, among other things, forbids non-local interactions, so if there were non-local interactions that would imply that the Lagrangian would not satisfy Lorentz symmetry in one way or another. As far as we can tell, and it has been measured to an incredible precision, Lorentz symmetry is exact. It is also one of the assumptions behind the construction of the standard model, and prevents a large number of additional terms from entering the Lagrangian. One can either say that there is Lorentz symmetry, or one needs to introduce those extra terms into the Lagrangian, multiplied by a suitable constant, and then come up with an ad-hoc explanation of why those constants are too small to be measured. One of the implications of Lorentz symmetry is that interactions between particles should be local. So a non-local interaction between two particles would break a lot of established physics. And if the equation of motion for the particle is non-local, then that implies a non-local interaction.
The problem with this argument, of course, is, as noted above, that in quantum physics we know that in EPR and related experiments there are non-local correlations between certain measurement events. The natural "classical" explanation -- that these events report the results of classical parameters describing the entangled particles established when they were in contact -- leads to different predictions to what is observed (albeit that this calculation has certain other assumptions which would cause even greater problems if broken). And this is also a direct consequence of the standard model of particle physics. So the standard model enforces locality in some contexts, and enforces its violation in others. So the question becomes whether the non-locality in the guiding equation in the pilot wave model is the sort of non-locality which is allowed in the standard model, or the sort of non-locality which is forbidden.
There is an important distinction to be made between what I call substance causality and event causality. Substance causality asks which particle state did this particle state emerge from. Event causality asks why this particular event with this particular outcome occurred. In terms of the Copenhagen interpretation applied to relativistic quantum field theory, substance causality describes particle creation and annihilation, and particle propagation. These are described by the Hamiltonian or Lagrangian of the theory, and Lorentz symmetry is conserved in the Lagrangian. Event causality is in play when there is a wavefunction collapse, for example caused by a measurement event. The process of wave function collapse is described by Born's rule (albeit something of a mystery philosophically), and unrelated to the Hamiltonian evolution of the wavefunction. And, of course, wavefunction collapse is indeterminate. There is no known physical cause (in the Copenhagen interpretation, and others which admit indeterminacy) why a particular outcome should happen. All we can do is assign a probability to each outcome, and use that to predict a frequency distribution. The non-local correlations occur in the outcomes of these events. In Bell's theorem, it is not even individual events which provide evidence for non-local correlations, but a comparison between two frequency distributions. If the detectors in a spin experiment are perfectly aligned, then the spins measured between those measurements for an individual pair of entangled particles would be correlated -- but this can be explained using a classical hidden variables theory. The classical hidden variables theory cannot explain the frequency distributions that arise when the two detectors measure spin or polarisation in different bases. This suggests that the correlations between the measurement events for individual entangled particles (as opposed to comparing frequency distributions of numerous independent events) are also non-local. But in any case, the non-locality arises from wavefunction collapse rather than any particle propagation or creation or annihilation, and as such does not conflict with the standard model's use of Lorentz symmetry, or special relativity in general.
The Pilot wave interpretation, however, lacks this clear distinction between measurement outcomes and particle propagation. The non-locality arises in the equations of motion for the particle. When converted to a quantum field theory, this would imply Lorentz symmetry breaking terms in the Hamiltonian and Lagrangian of the theory. This would imply that there are creation and annihilation events at a distance (i.e. a change in the pilot wave here would lead to a particle creation event over there). The Lagrangian would not be Lorentz invariant. Since the assumption of Lorentz symmetry is crucial in the construction of the Lagrangian (and explains why numerous possible terms describing potentially observable interactions in the Lagrangian are not present in reality). Even if one could construct a Pilot wave theory which preserved apparent Lorentz symmetry for the observed particles, this would be somewhat ad-hoc and artificially imposed. It lacks the naturalness we see in the conventional construction of the Lagrangian, where the symmetry is exact from the start. One would need a Lagrangian which is not Lorentz invariant, but fine tuned so that apparent Lorentz symmetry emerges from it. If you break a symmetry somewhere in the Lagrangian, there is no reason not to break it everywhere including in the emergent theory. Even if a version of the Pilot wave model were constructed from the standard model Hamiltonian, it would just create this new fine-tuning problem.
The technical term for having a privileged reference frame is foliation. Non-relativistic physics creates a natural foliation (due to its underlying Galilean relativity), so the problem of a preferred reference frame is not an issue. In special relativity it becomes unnatural, and as stated in quantum field theory it would destroy one of the primary motivations for restricting the Lagrangian to its standard form. All attempts to create a relativistic pilot wave theory that I am aware of introduce foliation. This work by Detlef Duerr and collaborators discusses that in detail. There are two models discussed; firstly where the foliation is introduced a priori, and secondly where it emerges from the wavefunction. The conclusions of that paper are that while pilot wave theory involves a foliation which naively breaks the symmetries of special relativity, the ability to construct a foliation dynamically from the wavefunction implies that this would be true for any formulation of quantum physics. So the pilot wave model is no worse than any other interpretation. I am not, however, convinced by this argument. In the pilot wave model, the foliation is required for the underlying ontology. In other models, it is merely an artefact of one way of looking at the system, with no fundamental dynamical or ontological relevance. The success of the standard model, which is consistent with relativity, and which does not require any foliation to calculate the Lagrangian or an amplitude, shows that a manifestly relativistic quantum theory is possible. But it would not be possible to create a manifestly relativistic pilot wave model, as the construction of the theory requires a privileged frame. In my view, the necessity of foliation remains a major problem for any attempt to create a pilot wave model. Even if a pilot wave model consistent with the empirical Lorentz invariance of the standard model could be produced, it would leave open the question of why there should be that symmetry in observation when it is not present in the underlying ontology. While the pilot wave model might solve the problem of wavefunction collapse, it would open up this (in my view even worse) problem. This undermines the argument for the pilot wave model that it is a clean understanding of quantum physics, without any philosophical loose ends.
I am outlining difficulties which stand in the way of constructing a viable Pilot wave model from the standard model Hamiltonian. The next one in my list concerns gauge symmetry. The gauge transformation can be thought of as a local rotation of the phase of the particle wavefunction (and it also modifies the gauge fields). Local here means that the variable parametrising the transformation varies from one place to another; in a global symmetry the variable is constant across all space. The gauge transformation also simultaneously modifies the gauge field -- again this can be expressed in terms of a rotation. The standard model Lagrangian is invariant under gauge transformations, and, once again, in the standard construction and motivation for the standard model, this is crucial in explaining why various gauge-symmetry breaking terms are not present in the Lagrangian.
However, when we look at the decomposition used to construct Bohm's pilot wave model, we see that the particle variable S is just a phase to the wavefunction. This means that it can be eliminated through a gauge transformation. This is obviously troublesome, because this parameter represents the observed particles, and parameters which can be absorbed into gauge transformations are not observable (this explains why we only see one Higgs Boson rather than four). This again does not rule out the Pilot wave interpretation -- maybe one can use a different decomposition -- but it shows that Bohm's method does not have a direct analogue in QFT. One needs a different approach.
A similar difficulty applies to representing gauge fields in a pilot wave model. In the standard model, the representation of the gauge fields contains redundant variables due to the gauge symmetry. There are various configurations of gauge fields which are physically equivalent. This makes some calculations difficult. Integrating over these configurations in an incomplete series expansion of the theory leads to an infinite result, which is why in perturbation theory we tend to choose one particular gauge before proceeding. In practice, whichever gauge you choose will get the same results if you do it consistently. The process of gauge fixing in a non-Abelian theory (the weak interaction and QCD) does add an additional term into the path integral, which is usually interpreted in terms of pseudo-particles known as ghosts. These are just artefacts from the method of calculation, but contribute to the Feynman diagrams.
A pilot wave re-interpretation of the standard model would have to accommodate these additional redundant degrees of freedom. In classical mechanics, one does also have gauge symmetry in the electric and magnetic potentials, but here the observed qualities are the electric and magnetic fields. The potentials are not physical -- just an artefact of the calculation -- and so the redundancy caused by the gauge symmetry can be hidden away. In quantum physics, the potentials take the centre stage, and the electric and magnetic fields a more secondary role. In the pilot wave decomposition, there would (presumably) be a "particle" representing the photon, with an underlying unobservable pilot wave for the photon. This particle is in a determinate state, which would include being in a particular gauge. The only way I could see that this would work is for the pilot wave interpretation to privilege a particular gauge. In effect, there would be some natural gauge fixing before we start constructing the theory. The gauge symmetry of the apparent Lagrangian would then once again emerge from the theory. That would lead to another fine-tuning problem. If gauge symmetry is not built into reality, then why should the theory be tuned precisely so that it appears to emerges when we reconstruct the full standard model Lagrangian? And how does this natural gauge fixing avoid ghost fields?
Next, we have the problem of renormalisation. This is an essential part of the quantum field theory formula. So far I have been discussing the bare fields, as they appear in the simplest way of constructing the Lagrangian. The problem is that one can perform changes of basis, mixing the electron, quark, photon and gluon fields in the Lagrangian. The creation and annihilation operators that give us the initial and final states need to correspond to whatever basis represents the particles in reality. We have to change the basis used in the Lagrangian to match this. To fail to do so leads to inconsistencies in the calculation, which manifest themselves in infinities. In practice (in perturbation theory), the way this is done is to regulate the theory, and then modify the electron mass, normalisation, and electron-photon interaction at each term in perturbation theory so that the infinities disappear when the regulator is removed. The standard regulator used is dimensional regularisation, (i.e. represent the theory in 4-ε dimensions, which is sufficient to remove the infinities) as this is consistent with gauge symmetry.
I don't see this as a fundamental bar to deriving a pilot wave model from quantum field theory, but it adds an additional complication, which I have not seen addressed by those trying to reconstruct a field theory version of a pilot wave model. The starting action for the decomposition would not be bare action represented above, but the far more complicated action for the renormalised fields. And, indeed, the renormalised action is unknown -- one does not need to know it in the standard methods of computation, as one can renormalise after computing the Feynman diagrams in perturbation theory or after calculating the amplitudes for the bare fields in a non-perturbative lattice calculation. But, I think, when constructing a pilot wave model one would have to start with the renormalised fields, as there is no correspondence to the basis transformations needed for renormalisation in a classical system. One would have to construct something something similar to Newton's equations of motions for pilot wave "particles" that correspond to the renormalised fields. It is not obvious that doing so for the for the bare fields would allow you to construct a similar structure once you have applied the renormalisation procedure.
Other objections to the Pilot wave model
There have been some attempts to mimic some particular pilot wave model in a laboratory experiment, for example using surface waves on a fluid of oil to represent the pilot waves and an silicone droplet to represent the particle. These models do have some success in producing similar results to those seen in some quantum systems. A study by Andersen, Bohr and others is significant because it shows that the double slit of quantum physics result cannot be replicated by a pilot wave model where the particle interferes with the wave. This is because the presence of the particle on one path or another breaks the symmetry of the system and consequently of the pilot wave. I am not fully convinced that these studies rule out pilot wave models in general: there are differences between their setup and the actual quantum mechanical models. It is not clear to me that this applies to all pilot wave models (the paper states that it applies to a de-Broglie style model, but not Bohm's which I illustrated above), but it would have to be borne in mind when constructing a pilot wave model for quantum field theory. The studies need to be mentioned, but I am not sure that they say anything conclusive.
There are a lot of philosophers of physics with a fondness for the pilot wave model. And it is easy to see why: it promises to reproduce the results of quantum physics without having to make major changes to the philosophy behind Newtonian mechanics. However, it has not received wide acceptance among physicists. There were historical objections raised for this, but today the primary objection is the one I have given here: Bohm's approach was based on a reformulation of the non-relativistic Schroedinger equation. While it is possible to extend this argument to the Dirac equation, the structure of relativistic quantum field theory is very different to that of wave mechanics: it is not based around a simple linear differential equation. It seems fairly clear that a reformulation along the lines of that done by Bohm is not possible. Without such a reformulation, it is not proved that the theory is experimentally equivalent to the standard picture.
Having done research for this approach, that summary dismissal looks to be a little too quick. There is work being done on combining the pilot wave model with relativistic quantum field theory. However, this work has not progressed sufficiently to demonstrate that it is possible; and much of it undermines the philosophical simplicity which is the main appeal of the pilot wave interpretation.
And this is a big deal for the physicist. Any interpretation of quantum physics has to be shown to mathematically equivalent to the instrumental Copenhagen interpretation of the standard model, as that is what has been experimentally tested. Unlike most interpretations of quantum physics, this has not been proved for the pilot wave model. Now "not proved" is not the same as saying it is false; but there are strong reasons to suspect that it cannot be shown to be consistent with the standard model, at least not without (for example) violating Lorentz invariance at the ontological level (even if an apparent Lorentz invariance might emerge when we consider the experimental results), or introducing spontaneous jumps from one particle number sector to another which means that the dynamics is not a modified Newtonian, and we need to come up with a new philosophy after all. When there are other interpretations of quantum physics which lack these issues, then it is difficult for a physicist to get enthusiastic about this interpretation.
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