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Bohmian mechanics is incompatible with loop corrections

...meaning the Feynman diagrams...

Florin Moldoveanu wrote his review of Hardy's paradox. Like the GHZM experiment, it's another "mostly qubits-based" quantum thought (or real) experiments that behaves in a way predicted by any local realist theory to be impossible.

Florin's presentation is a bit shallower than my blog post on Hardy's paradox or his treatment of the GHZM case but I am willing to believe that he understands how it works.

The original paper by Lucien Hardy included the annihilation of an electron and a positron into two photons; and Jean Bricmont, a pro-Bohmian ideologue, recently published his book attacking the foundations of quantum mechanics. So I think it could be a good moment to write a blog post dedicated to this specific topic of "Bohmian mechanics vs effects of QFT", something I have wanted to do for some time.

For some review of Bohmian mechanics, see e.g. Bohmian mechanics is a ludicrous caricature of Nature. For a basic sketch of a key conceptual defect of Bohmian mechanics, one that will basically play role below as well, see Bohmian mechanics requires the unphysical segregation of primitive and contextual observables.

OK. What's the problem? Bohmian mechanics is an ideologically driven candidate to replace quantum mechanics (the theory discovered and with rules rather carefully defined by Heisenberg, Jordan, Born, Bohr, Dirac, and perhaps a few others in the next wave such as von Neumann and Wigner). Quantum mechanics says that facts about Nature only exist if they are actually observed by an observer, so the "state of a physical object/system" prior to the observation must be described by complex probability amplitudes that generalize the subjective ("Bayesian") probabilities believed by the observer, not facts.

Bohmian mechanics is one of the three main philosophies attempting to return physics back to the era of classical physics – physics associated with the 17th, 18th, and 19th century where the observer played no important role. In Bohmian mechanics, originally invented by Louis de Broglie in 1927 (but named after Bohm who resuscitated in the 1950s – the theory is currently not being named after de Broglie because de Broglie was an aristocrat while Bohm and almost all the fans of this theory were Bolsheviks) postulates that the elementary building blocks of Nature don't have particle-like and wave-like properties depending on the method of observation.

Instead, Bohmian mechanics says that there objectively exists a particle with a classical trajectory; and there also exists an independent set of degrees of freedom, a classical wave called the guiding wave or pilot wave. The latter behaves just like the wave function in quantum mechanics but has a different interpretation: it is analogous to a classical field. The guiding wave directs the motion of the "actual particles" that exist on top of the waves so that the probability distributions for the "real particles" remain as predicted by quantum mechanics. In other words, the guiding wave repels the particles from all interference minima by the right "force".

In this way, assuming a random and properly distributed initial position of the actual particle, the experiment will produce the right interference patterns in double slit experiments and similar experiments. Bohmian mechanics may be extended to \(N\) non-relativistic spinless particles: there are \(N\) actual position vectors of the particles, and a guiding wave in \(3N\) dimensions mathematically coinciding with the wave function but interpreted as a classical (multi-local) field.

Bohmian mechanics can't describe the spin – there is no way to add any "actual" information about the spin because any classical information about \(j_z\) etc. would break the rotational symmetry. Bohmian mechanics is in severe conflict with relativity because the guiding wave is a multi-local, i.e. non-local, object, and it has to be able to exert superluminal influences in order to emulate quantum mechanics.

The correct way to argue is that the generic theory in the Bohmian class contains infinitely many Lorentz-violating effects and they have no reason to vanish. So the probability that all of them cancel and produce the prediction of Lorentz-invariant phenomena – which are observed – is \(1/\infty^\infty\). It is zero for all actual purposes. The theory is obviously ruled out. It's been born dead because relativity had been known since 1905 and Bohmian mechanics is fundamentally contradicting relativity. (Moreover, there really can exist no Lorentz-invariant point in the Bohmian parameter spaces at all. The probability of relativistic predictions isn't just tiny, it is strictly zero.)

Bricmont is aware of the contradiction between relativity and Bohmian mechanics. But he doesn't use it to falsify the theory, Bohmian mechanics, as a participant in the scientific method would (theories are abandoned when they contradict the empirical data!). Instead, he calls the contradiction "the greatest problem of physics today". Well, it's not a problem for physics as a discipline of science. It's only a problem for wrong theories. Theories which are OK exist. Quantum field theory and string theory – and when I say these names, it means the union of the general "Copenhagen" postulates of quantum mechanics plus some particular choices of the Hamiltonian or S-matrix – are not excluded because they don't suffer from this problem at all. Within the quantum formalism, the Lorentz covariance of these theories is easy to be demonstrated. It exactly holds. Does he fail to understand that fact? Or the importance of the fact for the "competition" between the physicists' state-of-the-art theories and his "theories"?

But the conflict with relativity is extremely far from being the only lethal defect of Bohmian mechanics.

Bohmian mechanics cannot give a consistent explanation how the correctly distributed initial state is prepared; and what kind of a janitor cleans the "garbage" guiding wave away when a particle is absorbed. Without such a "janitor", the system would get contaminated by a growing number of "dead souls", guiding waves that were needed but are no longer needed etc. Like all realist "interpretations", Bohmian mechanics contradicts the observed low heat capacities of atoms thanks to its huge, infinite, number of classical degrees of freedom.

However, I still believe that all these problems, while enough to kill the philosophy in a second, aren't the main characteristic lethal bugs of Bohmian mechanics. There is something sick that is most intrinsically connected with the very assumptions of Bohmian mechanics. While Bohmian mechanics manages to emulate the wave-like behavior of particles in the double slit experiment and similar setups (the information about the relative phases is stored by the pilot wave), it still contradicts the superposition postulate e.g. the feature of quantum mechanics that whenever there are 2 or several possible states in which a physical system may find itself, we must always describe these states by probability amplitudes and the relative phase matters and can have observable consequences.

Einstein famously asked whether the Moon was there when no one was watching. The correct answer is that

in principle, even the question about the existence of the Moon refuses to be given an "objective" answer. Different answers (Yes/No) have to be represented by basis vectors in the Hilbert space and an arbitrary complex superposition of these basis vectors is always allowed!
The case of the Moon is (deliberately) misleading because all observers on the Earth have measured the answer to the question whether the Moon exists, in one way or another, and got the answer Yes. So their wave functions are collapsed and only live in the Yes subspace of the Hilbert space. The amplitudes for all the No states are zero – the only situation in which there is no information about the relative phase. However, for much smaller objects than the Moon (because they're not being observed all the time), the existence of superpositions of "exists" and "doesn't exist" becomes damn practical and damn easy to experimentally verify!

I've mentioned that Bohmian mechanics has a serious problem with at least two fundamental features of quantum field theory: the spin of particles; and the Lorentz covariance. But it actually has a serious problem with all conceptually new effects by which quantum field theory differs from non-relativistic quantum mechanics. It isn't compatible with the particle creation and annihilation. The calculations of renormalization can't be embedded into Bohmian mechanics in any way.

And in fact, the very simple insight that there are loop (Feynman) diagrams in quantum field theory contradicts Bohmian mechanics at the very fundamental level.

Take the anomalous magnetic moment of the electron. The electron exists in the initial state and the final state; that produces the two fermionic external lines. And there is one external photon line because the electron is interacting with some external magnetic field and we want to know how strong the change of the electron's velocity is. The main classical term is given by the Feynman diagram including the simple cubic vertex only.

The pictures above give us the leading quantum corrections – one-loop diagrams that modify the classical prediction for the magnetic moment by relatively small but safely measurable amounts. It's really the first diagram (a) that gives us the leading term, \(\alpha/2\pi\), to \((g-2)/2\) (the subtraction is meant to cancel the classical term). The other diagrams are cancelled in the physical result because they correct the 3 external particles (their propagators) individually, not their interaction.

This \(\alpha/2\pi\) term was first calculated by Julian Schwinger in 1948 – using a calculational framework that looked less elegant than the Feynman diagrams we like to use these days. It's a whopping 68 years ago. Communists just conquered Czechoslovakia in 1948. Both my parents were born on that year, too. They were conceived during democracy and born into totalitarianism. Believe me, it's a long time ago when loop corrections started to be computed.

The reason I am saying it is that the Bohmists in 2016 believe that they may still totally ignore all these insights. They fool themselves into thinking that it is not a problem that their philosophy is totally incapable of explaining these calculations by Schwinger and millions of similar calculations that all of particle physics is about today.

And this industry of loop corrections started by Schwinger has grown a lot by 2016, indeed. Corrections up to five-loop diagrams have been computed, producing a prediction for the magnetic moment that has the accuracy of 15 significant figures, and my ex-colleague Jerry Gabrielse and co-authors has experimentally verified that all these digits are nontrivially right. The electron's magnetic moment remains the most accurately verified prediction in natural (let alone social or otherwise unnatural) sciences.

Now, let's see what the calculations of the loop diagrams actually mean e.g. for Bohmian mechanics. Imagine that we compute the Feynman diagram in the position space. We know what the diagrams are doing. Feynman was really summing over all possible histories of the particles. Each possible history contributes a complex probability amplitude to the total one. The initial particles are propagating from one spacetime point to another (with amplitudes given by the propagators) and they are allowed to merge and split (at the cubic vertices). When all these histories are summed over, we get the probability amplitude for the overall evolution from the initial state to the final state.

This sum-over-histories may also be shown to be equivalent to some intermediate evolution of the wave function for the whole set of quantum fields. The fact that the histories are numerous and they differ from each other at some particular intermediate moment, e.g. \(t=0\), as well really means that if you evolve the wave function by the accurate Schrödinger's equation, the intermediate state at \(t=0\) will be a superposition of very different states.

Now, you may ask. In the middle of the history (the experiment that measured the sensitivity of the electron's motion to the magnetic field, i.e. the magnetic moment), at \(t=0\), how many electrons, positrons, and photons were present in the space?

The answer is the same as the answer to any similar question about the state of the system that is actually not observed. The answer is:
We didn't actually measure the number of photons, electrons, and positrons at \(t=0\), so the question expecting a clearcut classical answer is just meaningless. Only quantities that are actually measured have well-defined values (one of the eigenvalues that just managed to be measured). At \(t=0\), we have to describe the state of the quantum fields as a general complex superposition of states with different values of \(N_{e^+}\), \(N_{e^-}\), and \(N_\gamma\).
There just can't be a clear answer. Superpositions of states are always allowed, even if the states differ by the number of photons or electron-positron pairs. The relative phase matters and influences some in principle measurable interference patterns.

You may see that
  • this fact – the allowed superpositions of "anything" – is absolutely critical for the theory's predictions to depend on the sum of Feynman diagrams with different numbers of loops
  • the superpositions of states with different numbers of particles are fundamentally prohibited by Bohmian mechanics (if the number of particles is allowed to go up or down at all, the change of the number must be "objective", observer-independent)
Again, the first point is true because the superpositions (of states with extra particle-antiparticle pairs and with extra photons with states without them) are everywhere in the sums of Feynman diagrams. All of the Feynman diagrammatic industry is about the superpositions of complex amplitudes – and the superpositions may involve arbitrary states with the same values of conserved quantities. The character of the states entering the superposition is virtually unconstrained at the intermediate times; only the initial and final states are fixed by the question we are asking.

The second point is true because the whole point of Bohmian mechanics is to restore realism at the fundamental level. When you restore realism, some of the measurable properties or their functions simply have to be objectively true. In the standard Bohmian non-relativistic mechanics for \(N\) particles, the number of particles is an "objective fact". So there can never exist states of the form\[

\ket\psi = \frac{ 3 \ket{e^+ e^-} + 4i \ket{ e^+e^+ e^-e^- } }{5}

\] The configuration of the Bohmian degrees of freedom always explicitly says whether the number of electrons and positrons is \(1+1\) or \(2+2\) (or something else). But in quantum field theory, complex superpositions of both states (the "Schrödinger's cat states", if you wish) must always be allowed. And the relative phase matters for some doable experiments. This is fundamentally incompatible with the basic form of Bohmian mechanics, the only one that is really discussed in many Bohmian papers, namely the non-relativistic theory of \(N\) spinless particles.

One may say that Gabrielse's successful measurement of the precise electron's magnetic moment experimentally proves that
the five electron-positron pairs and/or extra photons aren't objectively there when no one is looking.
They need to be described by the superpositions. It's not quite the same statement as one about the Moon (which is larger than 10 or so leptons) but it is qualitatively the same. The measurement of the magnetic moment simply shows that any correct theory must allow you to mix the states into quantum superpositions even if they have different numbers of particles in them. By confirming the prediction based on the sum of multi-loop Feynman diagrams, the interference between the histories with these intermediate states is being almost directly measured!

A Bohmian apologist could propose that this problem may be fixed by choosing a different set of "beables". In the normal Bohmian theory, the guiding wave and the positions of the particles "objectively exist", they are "beables". The word "beable" was chosen by Bell who was driven by his hatred towards proper quantum mechanics and whose goal was to mock the dependence of quantum mechanics on the observers, and therefore the word "observables" as well. Things shouldn't be just observed, they should be, all these anti-quantum zealots always claimed in contradiction with the results of the physics research.

But whatever your "beables" will be, you will unavoidably ban most superpositions. To claim that some properties of the physical system are objectively real, independently of an observer, is so fundamentally different from the basic postulates of quantum mechanics, namely that all states may be mixed into complex superpositions and observations are needed to have any "classical facts", that you may always easily see a huge contradiction between Bohmian mechanics and quantum mechanics – or, almost equivalently (because quantum mechanics and QFT really agree with everything we have tried), between Bohmian mechanics and the experiments.

Imagine that the basic "beables" of a Bohmian theory include classical values of the electromagnetic field. There is some classical electromagnetic field \(F_{\mu\nu}(\vec x)\), after all, and its values are driven by some guiding wave which is now a functional on the space of all possible functions \(F_{\mu\nu}(\vec x)\) – well, probably just the canonical coordinates \(\vec A(\vec x)\), i.e. the pilot wave is a functional \(\Psi[\vec A(\vec x)]\).

To some extent, it's excellent. In this setup, the Bohmist may "simulate" all the relative phases between the different "continuous functional basis vectors" of a quantum field theory. But the description with these "beables" is too wave-like and it will be impossible to explain how photons may ever be detected at particular points. Why?

You must understand that the main "beables" in the original Bohmian theory – the particles' positions – were chosen exactly because Bohmists realized that those are the things that "must be possible to be measured", and they don't like the superpositions of different outcomes (they don't like when a particle "spreads" to many places, as the dissolving wave function indicates using their sloppy interpretation). That's why they included the configuration space of particles' positions – because that's a frequent measurement that may obviously be done – as extra classical degrees of freedom in their theory. The particles are "preemptively ready" for the most typical kind of a measurement, the measurement of their positions. And the Bohmists were implicitly assuming that every measurement is ultimately being reduced to a measurement of particles' positions.

In this way, Bohmists believed that they had solved the (non-existent) "measurement problem": no collapse is needed because the soon-to-be-measured location of the particle is already known before the measurement – it's the classical location of the "particle-like" part of the Bohmian classical degrees of freedom.

But the new Bohmian theory is only ready for a measurement of the value of the fields \(F_{\mu\nu}(\vec x)\). It's no good because the measurement of the locations of quanta can in no way be reduced to a measurement of \(F_{\mu\nu}(\vec x)\). If you want to guarantee that the theory will observe a photon at one particular place, you will need a collapse, anyway. You will need to borrow parts of the "standard quantum mechanics" because no value of the location \(\vec x\) of the photon is "ready" before the measurement. So the very reason why Bohmian mechanics was constructed in the first place breaks down. The new Bohmian theory won't be capable of producing sharp results for the most usual experiments. Before you measure the location of a high-energy photon, no "beable" seems to know what result you should get. It's clear that the "right result" can be neither a function of the guiding functional, nor the classical values of the fields \(F_{\mu\nu}(\vec x)\). So the Bohmian theory just can't possibly be able to predict what happens in the position measurement.

If you included both the particle positions and the fields' values among your beables, you will face even worse problems. The corresponding operators in quantum mechanics don't commute with each other, so it's always a problem if you claim that a theory determines both at the same moment. In the most general measurements, you will have no idea whether the result should be given by one beable or another. (Imagine the similar mess in the normal 1-particle Bohmian theory if you added both a classical value for \(x\) as well as one for \(p\) which would differ – and be more well-defined than – \(dx/dt\).) And if the theory chose to say that you must pick the "most relevant beable" from this redundant set, it will contradict all the experiments such as GHZM, Hardy's, and Bell's experiments.

Moreover, there is another, technical but widespread, immediate problem with the "beables that are the classical values of the fields" in a Bohmian theory attempting to emulate quantum field theory. A problem is that
the fermionic fields can't have any classical values at all.
There can't be any (nonzero) classical values of the fermionic fields because those would be Grassmannian numbers which anticommute with each other. But no two nonzero numbers \(a,b\) obey \(ab=-ba\). So there doesn't even exist a mathematically possible configuration space for these "fermionic field beables". This strategy is completely failing for fermions!

Just to be sure, quantum mechanics has never any problem with the "non-existence of particular Grassmann numbers" because we never measure them. Everything that we can actually measure by apparatuses are functions of the fields that always include "even powers" of the Grassmann numbers (or integrals/derivatives over a number of Grassmann variables which is correctly even or odd, as needed), so these observables are basically commuting. The observables we may measure are Grassmann-even. Equivalently but not "obviously" so, you may say that quantum mechanics only predicts probabilities and those end up being integrals over the Grassmann variables (these are highly abstract, formal, "Berezin" integrals) such that any dependence on the Grassmann numbers goes away in the final result.

One may say that the Grassmann numbers are just "building blocks" that are never directly "observed" in isolation (like quarks, but at a much more fundamental level). The very point of Bohmian mechanics is that it is trying to claim that all these intermediate objects in quantum mechanics are "real". This is an indefensible assertion for dozens of reasons, many of which have been enumerated in my blog posts (and above). And one of them is that "the Grassmann numbers can't be beables at all" because there's even no mathematically possible candidate configuration space in which they could take values.

But the success of QED shows that the fermionic fields (e.g. the Dirac field creating electrons and positrons) are almost certainly extremely natural parts of successful descriptions of Nature. Bohmian theories just don't allow you to construct anything that would contain things like "Grassmann fields". Quantum field theory with Grassmann fields was already getting started around 1930 – thanks to the pioneering work by Jordan, Dirac, and others. If you think that the mathematics of the fermionic fields should be included in "some way" in a promising theory/interpretation to replace the Standard Model, you may eliminate the Bohmian paradigm because it's clearly incapable of dealing with Grassmannian fields.

So these Bohmists are totally overlooking or denying particular phenomena that have been pillars of physics for more than 85 years. They don't have a problem. Despite this complete lack of viability of their pseudoscientific theory, they claim that it's a competitor to proper, i.e. observer-based ("Copenhagen"), quantum mechanics, if not a better competitor.

The stupidity and arrogance of these Bohmian and related morons is shocking. Jean Bricmont's 2016 book is a scary example of that. As I have said, it unsurprisingly ignores all these phenomena that were the "core" of physics research from the 1930s if not the 1920s, including fermions, particle production, loop corrections, renormalization, and so on. The philosophy sold in the book is absolutely incompatible with all these things. He must know that if a reader buys this stuff, he is fooled because this stuff will be unusable as long as he will try to deal with a somewhat more modern problem – fermionic fields, spin, statistics, relativity, renormalization, ... – but he doesn't care.

That didn't prevent Bricmont from de facto claiming that almost all the top physicists of the 20th century were idiots. He hasn't even dared to consider the explanation that it's him, Bricmont, and not these top physicists, who is the intellectual midget. Bricmont says that the courses of quantum mechanics in the college have never made any sense to him and he seems proud about it.

He faithfully quotes lots of the great men who were explaining the departure from classical physics clearly and unambiguously. They used different words (something Bricmont would love to abuse as "disharmony" as well – but it's natural for different men to use different words) but all of them say that it's physically meaningless to ask about the state of a physical system before (or without) an actual observation etc. The authors of the quotes that Bricmont reprints include
Niels Bohr, Werner Heisenberg, Max Born, Pascual Jordan, Wolfgang Pauli, John von Neumann, Eugene Wigner, Rudolf Peierls, Bernard d'Espagnat, David Mermin, Anton Zeilinger, even Margaret Thatcher ;-), Aage Petersen, Leon Rosenfeld,
while Pauli and Dirac are also quoted in the "neutral" section. (That's also misleading because these quotes also say that it's perfectly right that the theory – quantum mechanics – says nothing about the "state of the system" prior to the observations.) Just imagine that. These people are some of the pillars of the 20th century physics. It's possible that just this small group is collectively responsible for more than 50% of the genuine progress in physics in the 20th century. And the progress was closely linked to the quotes.

But Bricmont just doesn't have a problem to say that these physicists' insights about physics just "didn't make sense" – just because Bricmont is a peabrain incapable of understanding quantum mechanics and especially its defining feature that it may only be applied relatively to an observer and pre-given classification "what is an observation" – and he chooses to discuss whether they're a "fringe group". They're surely not a fringe group. Every good physicist knows that they were right while their critics such as Einstein, Schrödinger, Bell, Bohm, ... were wrong. A physicist may remain silent because he's not interested in these philosophical debates and he knows how to actually use the laws of physics in cases he cares about but when he says something, he can't contradict the basic rules he is actually using in his work. This fact is not a matter of interpretations. It's a part of the basic knowledge of physics. You simply shouldn't be allowed to pass an undergraduate or graduate course of quantum mechanics if you're incapable of getting this elementary point – e.g. that Bohr was right and Einstein was wrong in their debates.

There are lots of 100% crackpots such as Bricmont and then there are the "confused" people like Florin Moldoveanu who sometimes write something indicating that they understand something, e.g. the workings and implications of the GHZM or Hardy's experiment, but they immediately neutralize it by saying that Bricmont's is a well-argumented defense of Bohmian mechanics. Does Moldoveanu really believe this crap or does he lick the aß of the aßes like Bricmont for financial reasons? Because he may need their recommendation letters?

Bricmont's book doesn't ever mention things from modern physics such as the spin, loop diagrams, Grassmann numbers etc., I've mentioned that – those things easily show that Bricmont and other Bohmians are full of crap. But he doesn't even attempt to discuss things that are undoubtedly relevant in these discussions, according to his "soulmates" in the "foundation community". The book only mentions the GHZM experiment in one content-free footnote; and Hardy's paradox is only sketched and "responded to" by a bizarre quote by Bohmist Sheldon Goldstein who said that the "problem is with the problem" and "there is no contradiction because there are different ensembles every time you repeat the experiment".

Oh, really? Is that supposed to be a sufficient reply preventing Bohmian mechanics from being immediately excluded thanks to Hardy's paradox, too? In Hardy's paradox, any local realist theory predicts the probability of a certain combined outcome to be \(P=0\) while experiments and quantum mechanics say \(P=1/16\). Could you please show us the calculation in Bohmian mechanics that reproduces \(P=1/16\)? This is a very simple or elementary experiment. If you're not capable of getting \(P=1/16\) in your theory claimed to be relevant for these very phenomena, and you say that it's "OK" for your theory nevertheless, you realize that it becomes obvious that you lack all the integrity needed to do science, don't you?

It is not OK. Hardy's paradox is one way among hundreds to show that Bohmian mechanics – its detailed versions but also its totally universal assumptions and philosophy – is utterly incompatible with the experimental facts in modern physics. It's extremely shameful that people like Bricmont try to obscure this basic fact. It's shameful that the likes of Moldoveanu try to defend the indefensible.

BTW I haven't heard about Bricmont as a physicist before – at most I could have thought he was an assistant of Alan Sokal's in the battles with the postmodernists. But I could have predicted that he was an extreme leftists. After the blog post above was written, I looked at Wikipedia and learned that he's active in all sorts of far left-wing causes and collaborates with Noam Chomsky, too. The correlation between the ideological garbage and scientific garbage is far from perfect but it is safely positive, too.

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