Argumentation about de Broglie-Bohm pilot wave theory

Guest blog by Ilja Schmelzer, a right-wing anarchist and independent scientist

A nice summary of standard arguments against de Broglie-Bohm theory can be found at R. F. Streater's "Lost Causes in Theoretical Physics" website. Ulrich Mohrhoff [broken link, sorry] also combines the presentation of his position with an interesting rejection of pilot wave theory. These arguments I consider in a different file. Here, I consider the arguments proposed in several articles of Luboš Motl's blog "The reference frame": David Bohm born 90 years ago and Bohmists & segregation of primitive and contextual observables, Anti-quantum zeal and in off-topic responses of "Nonsense of the day: click the ball to change its color". Below, we refer to Luboš Motl simply as lumo (his nick in his blog).




Another argument (also with lumo's participation), related to Lorentz-invariance, I have considered at another place.

If you know other interesting pages critical of de Broglie - Bohm pilot wave theory, Nelsonian stochastics, non-local hidden variable theories in general, as well as ether theories, please tell me about them.




The most important thing: Measurement theory

The most important part of physics are, of course, experiments. Moreover, this is also the point where lumo is simply wrong, so it is worth to start with it.:

... it is not true that the de Broglie-Bohm theory gives the same predictions in general. It can be arranged to do so in the case of one spinless particle. But in the real quantum theories we find relevant today, such as quantum field theory, de Broglie-Bohm theory cannot be constructed to match probabilistic QFT exactly, and one can see that its very framework contradicts observable facts.

At another place, we find some hint where his misunderstanding is located:

Your equations about \(X\) are completely irrelevant for the measurement of the spin. The problem is not when one wants to measure \(X\). Indeed, the measurement of \(X\) might occur analogously to its measurement in the spinless case. The problem occurs when one actually wants to measure the spin itself.

The projection of the spin \(j_z\) is an observable that can have two values, in the spin \(1/2\) case, either \(+1/2\) or \(-1/2\). It is a basic and completely well-established feature of QM that one of these values must be measured if we measure it.

How is your 17th century deterministic theory supposed to predict this discrete value? Like with \(X\), it must already have a classical value for this quantity. Except that in this case, it has to be discrete, so it can't be described by any continuous equation. ...

Preemptively: you might also argue that any actual measurement of the spin reduces to a measurement of \(X\). But it's not true. I can design gadgets that either absorb or not absorb the electron depending on its \(j_z\). So they measure \(j_z\) directly. deBB theories of all kinds will inevitably fail, not being able to predict that with some probability, the electron is absorbed, and with others, they're not. This has nothing to do with \(X\) or some driving ways. It is about the probability of having the spin itself.

The last paragraph gives the hint: lumo has interpreted the claim that all measurements reduce to position measurements as "all measurements of the electron reduce to position measurements of the electron". If that would be true, I would concede that lumo's polemics against pilot wave theorists are justified. This was, by the way, the state of the art before Bohm's measurement theory appeared 1952. Thus, lumo's arguments illustrate in a nice way why de Broglie had given up pilot wave theory.

Once the question has been asked how the 17th century deterministic theory manages to predict discrete values, let's explain this story. As a 17th century theory, with real aristocratic origin, it leaves the hard work to servants (quantum operators), reserving for itself the final (and most important) decisions ;-).

First, there is some interaction of the wave function of the electron with the wave function of the measurement device. (There is of course also an equation for the position of the electron \(q_{el}\) – the \(X\) in lumo's text – but it is completely irrelevant, not only at this stage, but in the whole process.) The result of the measurement is, as usual, a wave function of type\[

|\psi\rangle = \alpha_1|{\rm up}\rangle|q_1\rangle + \alpha_2|{\rm down}\rangle|q_2\rangle

\] This exploitation of standard QT is not enough – now decoherence will be exploited in an equally shameless way. We leave it to decoherence considerations to decide which observables of the measurement device become amplified or macroscopic. Assume the quantum states \(|q_1\rangle, |q_2\rangle\) are decoherence-preferred. In this case, decoherence amplifies the microscopic measurement results \(|q_1\rangle, |q_2\rangle\) into classical, macroscopically different states \(|c_1\rangle, |c_2\rangle\). After finishing this hard job, it presents the following state:\[

|\psi\rangle = \alpha_1|{\rm up}\rangle|c_1\rangle + \alpha_2|{\rm down}\rangle|c_2\rangle

\] Now, everything is prepared, it remains to make the really important decision which of the wave packets is the best one ;-). At this moment a hidden variable enters the scene. But, surprise, it is not the hidden variable of the electron \(q_{el}\) (lumo's X), but that of the classical measurement device \(q_c\).

The job of \(q_c\) is not a really hard one. After driving around (no, being driven around by quantum guides) in an almost unpredictable way, it simply takes the wave packet prepared for him by the quantum operators at the point of arrival ;-). In other words, we simply have to put the actual value of \(q_c(t)\) into the full wave function \(|\Psi\rangle\) to obtain the (unnormalized) effective wave function:\[

\psi(q_e) = \Psi(q_e, q_c(t))

\]What we need for this scheme to work as an ideal quantum measurement is not much. We need that the two states of the macroscopic device \(|c_1\rangle, |c_2\rangle\) do not (significantly) overlap as functions of the hidden variable \(q_c\). In this case, whatever the value of \(q_c\), the result \(\psi(q_e)\) will be a unique choice between two effective wave functions, namely between \(|{\rm up}\rangle\) if \(q_c\) is in the support of \(|c_1\rangle\), and \(|{\rm down}\rangle\) otherwise. And we need the quantum equilibrium assumption for \(q_c\) to obtain the probabilities for these two choices as \(|\alpha_1|^2\) resp. \(|\alpha_2|^2\).

Thus, everything works as in quantum theory – Born rule as well as state preparation by measurement (only without any ill-defined wave function collapse or subdivision of the world into a classical and quantum part, or the equally ill-defined "subdivision of the world into systems" used in many worlds or other decoherence-based approaches).

But maybe one of the two assumptions we have used used are wrong? Given Valentini's subquantum H-theorem, together with the numerical results of Valentini and Westman, which show a remarkable relaxation to equilibrium already in the two-dimensional case in a quite short period of time (arXiv:quant-ph/0403034), there is not much hope for observations of non-equilibrium in our universe.

One can, of course, also doubt that macroscopically different states do not have a significant overlap in the hidden variables. Such doubts have been, for example, expressed by Wallace and Struyve for pilot wave field theories. See my paper "Overlaps in pilot wave field theories" at arXiv:0904.0764 about the solution of this problem.

About the zeros of the wave function

There is a second point where experiment is involved, with an easy solution:

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

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

That's a nice argument, and, because of this argument, today the original form of de Broglie's "pilot wave theory" is preferred in comparison with the "Bohmian mechanics" version proposed 1952 by Bohm. In pilot wave theory, the pilot wave is really a wave, and you can apply the original argument to show that these observables are quantized. In Bohm's second order version, this is different, and the quantization of certain observables becomes, indeed, problematic. This has been another reason for me (beyond history, see arXiv:quant-ph/0609184) to prefer the name "pilot wave theory" in comparison with "Bohmian mechanics".

More generally, something very singular seems to be happening near the \(R=0\) strings in the Bohmian model of space.

The "model of space" in pilot wave theory is a trivial one, nothing strange happens there if R = 0. The singularity of the velocity at these points is harmless – a simple rotor localized in a string, moreover, there is nothing in the place where velocity becomes undefined.

So even though the Bohmian mechanics stole the Schrödinger equation from quantum mechanics, the superficially innocent step of rewriting it in the polar form was enough to destroy a key consequence of quantum mechanics - the discreteness of many physical observables.

If there would be property rights for equations or functions, one could argue as well that Schrödinger has stolen the wave function from de Broglie's pilot wave theory. Fortunately, such nonsense does not exist in science. But there is a point worth to be mentioned: Without pilot wave theory, there would be no Schrödinger picture, and we would have to use the Heisenberg formalism all the time. And if some Bohm would have found the Schrödinger equation later, it would have been named, as well, an unnecessary superconstruction and banned from physics, for almost the same reasons.

About relativistic symmetry and the preferred frame

Last but not least, there are some claims that pilot wave theories will be unable to recover QFT predictions in the relativistic domain. Unfortunately for his argumentation, the equivalence theorem remains to be a theorem even in the relativistic domain – nothing used in it has any connection to the particular choice of spacetime symmetry. Thus, if the quantum theory has relativistic symmetry for it's observable predictions, the same holds for the observable predictions of pilot wave theory.

More concretely, it is inconsistent with modern physics in many ways, as we will see.

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

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

I was already afraid that lumo does not even understand that in a preferred frame everything is fine with causality. The introduction was, at least, the highly dramatic one which is typical for such crank cases.

I like the formulation "at most". Sounds as if we would really like to have more reference frames and are, now, very disturbed that at most one preferred frame is available ;-).

You might think that the experiments that have been made to check relativity simply rule out a fundamentally privileged reference frame. Well, the Bohmists still try to wave their hands and argue that they can avoid the contradictions with the verified consequences of relativity.

Who is hand waving here? Lumo might, of course, think that experiments rule out a hidden preferred frame. But it's his job, in this case, to point out which observations rule out such a preferred frame. As long as he fails to do it, I don't even have contradictions with any verified consequence of relativity to wave my hands.

I wonder whether they actually believe that there always exists a preferred reference frame, at least in principle, because such a belief sounds crazy to me (what is the hypothetical preferred slicing near a black hole, for example?).

I'm happy to answer this question: The preferred coordinates are harmonic. Given, additionally, the global CMBR frame, with time after big bang as the time coordinate, this prescription is already unique. For a corresponding theory of gravity, mathematically almost exactly GR on flat background in harmonic gauge, physically with preferred frame and ether interpretation, see my generalization of the Lorentz ether to gravity.

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

I would be interested to learn more about this mystical way to amplify high energy violations of Lorentz symmetry into the low energy domain, without access to the necessary high energies. As far, it is lumo who is waving his hands.

I know that there are some nice observations, which use the extremely large distances light has to travel for some astronomical observations, to obtain boundaries for a frequency dependence of the velocity of light. Some of the boundaries obtained in this and different ways suggest even that these Lorentz-violating effects are absent for distances below Planck length. But Planck length is merely the distance where quantum gravity becomes important. The fundamental distance where our continuous field theories start to fail may be different.

In proper quantum mechanics, locality holds. If one considers a Hamiltonian that respects the Lorentz symmetry - such as a Hamiltonian of a relativistic quantum field theory - the Lorentz symmetry is simply exact and it guarantees that signals never propagate faster than light.

In proper quantum mechanics, one can define the operators that generate the Poincaré group and rigorously derive their expected commutators. Also, it is exactly true that operators in space-like-separated regions exactly commute with each other. This fact is sufficient to show that the outcome of a measurement in spacetime point B is never correlated with a decision made at a space-like-separated spacetime point A.

These facts allow us to say that quantum field theory respects relativity and locality. The actual measurements can never reveal a correlation that would contradict these principles. And it is the actual measurements that decide whether a statement in physics is true or not. Bohmian mechanics is different because these principles are directly violated. You may try to construct your mechanistic model in such a way that it will approximately look like a local relativistic theory but it won't be one. Consequently, you won't be able to use these principles to constrain the possible form of your theory. Moreover, tension with tests of Lorentz invariance may arise at some moment.

First, there is no reason not to use some symmetry principles for one part of the theory which do not hold for another part of it. For example, the symplectic structure in the classical Hamilton formalism has another symmetry group – the group of all canonical transformations – than the whole theory including the Hamiltonian.

Then, to postulate a fundamental Poincare symmetry is, of course, a technically easy way if one wants to obtain a theory with Poincare symmetry. But what is the purpose of a postulated global Poincare symmetry in a situation where the observable symmetry is different, depends on the physics, as in general relativity? Whatever the representation of the \(g_{\mu\nu}(x)\) on the Minkowski background – it will (except for simple conformally trivial cases) have a different light cone almost everywhere. If the Minkowski background lightcone is the smaller one, one has somewhere to violate the background Poincare symmetry. It may be always the other way. But in this case, the axioms of the theory give only restrictions for the background Minkowski light cone, not for the physical light cone. Thus, tensions with the physical Lorentz invariance may arise in the same way, because the theory only looks like one which, in the particular point \(x\), has the Lorentz invariance for the metric \(g_{\mu\nu}(x)\). But really it is a theory with Lorentz invariance for a different metric \(\eta_{\mu\nu}\), with a larger light cone, thus, allows for superluminal information transfer relative to \(g_{\mu\nu}(x)\).

String theory, as far as I understand, obtains gravity as a spin two field on Minkowski background. This requires, as far as I understand, that this problem is solved in string theory. Fine. Means, it is a solvable one.

The contradiction between relativity and semi-viable Bohmian models (that violate Bell's inequalities, and they have to in order not to be ruled out by experiments) is a very profound problem of these models. It can't really be fixed.

Again, nice formulation. Sounds like poor Bohmians have tried hard not to violate Bell's inequalities and finally given up. "Semi-viable" is also a nice word. But the "very profound problem" remains hidden. (A nice place for problems in a hidden variable theory.;-))

Instead, I prefer to follow the weak suggestions one can obtain based on mathematical equivalence proofs. When I construct a pilot wave theory based on a relativistic QFT, it seems really hard to avoid the consequences of this theorem to violate Lorentz invariance. At least, I don't know how to manage this. We obtain a pilot wave theory which does not violate observable relativistic symmetries. Simply because there is an equivalence proof for observables.

Today, we have some more concrete reasons to know that the hidden-variable theories are misguided. Via Bell's theorem, hidden-variable theories would have to be dramatically non-local and the apparent occurrence of nearly exact locality and Lorentz invariance in the world we observe would have to be explained as an infinite collection of shocking coincidences.

I'm impressed by the verbal power of "dramatically nonlocal", even more by the "infinite collection of shocking coincidences". Sounds really impressive. But I would not name a nonlocality, which, because of an equivalence theorem, cannot be used even for information transfer, and can be observed only indirectly, via violations of Bell's inequality, a dramatical one. Instead, it seems to me the most non-dramatical one. As well, I would distinguish the simple and straightforward consequences of an equivalence theorem from an "infinite collection of shocking coincidences". Instead, I would be more surprised if an quantum equilibrium large distance low energy limit would not change anything in the symmetry group of a theory.

Last but not least, the Lorentz group is simply the invariance group of a quite prosaic wave equation, an equation we find almost everywhere in nature. And such, a wave equation (or it's linearization) usually defines also an effective (and in general curved) Lorentz metric, so that the wave equation becomes the harmonic equation of this Lorentz metric. As a consequence, for everything which follows such a wave equation we obtain local Lorentz symmetry. (See arXiv:0711.4416, arXiv:gr-qc/0505065 for overviews.)
To assume that a symmetry, which so often and for very different materials appears as an effective symmetry in condensed matter theory, is fundamental, is a hypothesis which seems quite unnatural for me.

... and the ether ...

The similarity with the luminiferous aether seems manifest. ...

I just don't think that this is a rationally sustainable belief. It's just another repetition of the old story of the luminiferous aether.

About the similarity with the aether I fully agree with lumo ;-)))). But what is irrational in the belief that there is an ether? I would like to hear some details. I would be really interesting to hear which of the beliefs expressed in my ether model for particle physics are not rationally sustainable.

Now, it seems we have finished the claims of empirical inadequacy. It's time to consider the metaphysical arguments.

About signs of the heavens

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

I think there are "signs of the heavens" here. First, the guiding equation for the velocity is a nice, simple, and local (in configuration space) equation. The derivation mentioned by lumo could as well lead to a dirty nonlocal one.

Then, the equation for the phase resembles the classical Hamilton-Jacobi equation, and for constant density becomes simply identical with it. Now, the same guiding equation is, as well, part of the classical Hamilton-Jacobi theory – a theory which was in no way related to the conservation law of the first derivation.

Now, Hamilton-Jacobi theory is really beautiful mathematics, it has all properties of "signs of the heavens", even if taken only alone. See arXiv:quant-ph/0210140 for an introduction. That one and the same simple law for velocity gives, on one hand, Hamilton-Jacobi theory in the classical limit, and, on the other hand, a Liouville equation, is, at least for me, a sufficiently strong hint from the mathematical heaven. In many worlds I have not seen any comparable signs of beauty.

And there is, of course, the really beautiful derivation of the whole quantum measurement formalism.

How to distinguish useful improvements from unnecessary superconstructions

The mechanistic models add a new layer of quantities, concepts, and assumptions.

Indeed, every new, more fundamental theory adds a new layer of quantities, concepts, and assumptions. So what?

[Einstein] called the picture an unnecessary superconstruction.

Appeal to authority does not count. And there is no reason to expect that the father of relativity would like a theory which violates his child. But how to distinguish unnecessary superconstructions from interesting more fundamental theories? Above add something to the old theory. But useful more fundamental theories allow to explain something else from the old theory: Some postulates of the old theory can be derived now. So, one has to compare what one has to add with what can be derived now.

This relation is quite nice for pilot wave theory: The new layer is, essentially, the configuration together with a single additional equation – the guiding equation for the configuration. What can be derived from this equation is, instead, the whole measurement theory of quantum mechanics, including the Born rule and the state preparation by measurement. Compared with the Copenhagen interpretation, the additional layer also replaces the "classical part" of this interpretation and removes the collapse from the theory.

These last two points have been a major motivation of other reinterpretations as well. In particular, for many worlds it seems to be the only aim. The interpretation I prefer to name "inconsistent histories" is focussed on this aim too. Thus, two things which have been obtained in pilot wave theory first, have been widely recognized today as important contributions to the foundations of quantum theory. One can object that pilot wave theory does not get rid of the classical part, but even extends it into the quantum domain. This depends on what one considers as problematic with the classical part: If the problem is the imprecision of this notion, the absence of well-defined rules for this part, then it is clearly solved in pilot wave theory. Anyway, pilot wave theory was the first interpretation with completely unitary dynamics for the wave function, without a collapse.

One can perhaps create classical mechanistic models that mimic the internal workings of quantum mechanics in many situations. For example, one can write a computer simulation. But you can't say that the details of such a program or Bohmian picture is justified as soon as you confirm the predictions of conventional quantum mechanics.

There is no necessity to justify every detail. The important point of the pilot wave interpretation is that to explain the observable facts there is no necessity to reject classical logic, realism, or to introduce many worlds, inconsistent histories, correlations without correlata or other quantum strangeness and mysticism. We have at least one simple, realistic, even deterministic, explanation of all observable facts. That's enough to reject quantum mystery. Why should we justify every detail of some particular realistic model? There may be several realistic models compatible with observation. I would expect this anyway, given large distance universality.

The mechanistic models add a new layer of quantities, concepts, and assumptions. They are not unique and they are not inevitable. The similarity with the luminiferous aether seems manifest. If they only reproduce the statistical predictions of quantum mechanics, you could never know which mechanistic model is the right one: it could be a computer simulation written by Oracle for Windows Vista, after all.

But what's the problem with this? Is Nature obliged to work with theories which can be inevitably reconstructed by internal creatures? You could never know? Big problem. Anyway, our theories are only guesses about Nature, and we can never know if they are really true. If you doubt, I recommend to read Popper. (I ignore here, for simplicity, the modern ways to recognize the truth of theories, like counting the number of papers written about them, or getting inspirations about the language in which God wrote the world.)

Moreover, science has developed lot's of criteria which allow to compare theories which do not make different predictions: Internal consistency, simplicity, explanatory power, symmetry, mathematical beauty. Lumo uses such arguments himself, thus, he is aware of their power. They are usually sufficient to rule out most of the competing models. And if there remain a few different theories, all in agreement with observation, this is not problematic at all – it is even useful: It allows to see the difference between the empirically established parts of these theories – these parts will be shared by all viable theories – and the remaining, metaphysical parts, which may be very different in the different theories. Thus, they serve as a useful tool to show the boundaries of what science can tell at a given moment.

For example, today the existence of pilot wave theory shows that almost all of the quantum strangeness, in particular the rejection of realism, "quantum logic", and the esoterics of many worlds, are in no way forced on us by any empirical evidence, but purely metaphysical choices of some particular interpretations.

What are the fundamental beables?

I could make things even harder for the Bohmian framework by looking into quantum field theory. What are the real, "primitive" properties in that case?

In the simplest case of a scalar field, the natural candidate for the "primitive property" or the "beable" is simply the field \(\phi(x)\). This is a very old idea, proposed already by Bohm. But the effective fields of the standard model are also bad candidates for really fundamental beables. They are, last but not least, only effective fields, not fundamental fields. In my opinion, one needs a more fundamental theory to find the true beables.

My proposal for such more fundamental beables can be found in my paper about the cell lattice model arXiv:0908.0591. Even if pilot wave theory is not mentioned at all in this paper, it is quite obvious that the canonical quantization proposal for fermion fields I have made there allows to apply the standard formalism of pilot wave theory to obtain a pilot wave version of this theory.

Problems with spin and with particle ontology in quantum field theories

A large part of lumo's arguments is directed against two particular versions of pilot wave theory – strangely, I don't like them too. The first one is the idea to describe particles with spin using only wave functions of particles with spin, but leaving the configuration without spin. In this case, the wave function is no longer a complex function on configuration space, but a function with values in some higher-dimensional Hilbert space. But, as a consequence, the very nice pilot wave way to obtain the classical limit via the Hamilton-Jacobi theory no longer works, and one would have to use the dirty old way based on wave packets to obtain some classical limit.

There are other examples of such pilot wave theories. First, this trick was used by Bell, who has proposed a pilot-wave-like field theory with beables for fermions, but not for bosons. Now, one can argue that this is already sufficient, and leave the bosons without beables. The reverse situation was a theory from Struyve and Westman for the electromagnetic field. Again, it has been argued that this is sufficient. And, for the purpose to obtain a realistic theory which is able to recover QFT predictions, it is. But I think that such pilot wave theories are sufficient only for one purpose: To be used as a quick and dirty existence proof for realistic theories in situations where some parts of the theory cause problems. For this purpose, they are indeed sufficient, if the part of the theory represented in the beables is large enough to distinguish all macroscopic states – a quite weak requirement. If one doubts that a theory without fermions, or without bosons, is sufficient for this, one should think about renormalization: If we use these incomplete theories to describe one type of the bare fields (for some energy), then all types of the dressed fields already depend on this single type.

The second type of theories I don't want to defend are theories with particle ontology in the domain of field theory. One reason is that semiclassical gravity shows nicely that fields are more fundamental, and the pilot wave beables have to be, of course, fundamental. Then, to handle variable particle numbers is a dirty job. There should be something more beautiful. Particles which pretend for a status of beables should be at least conserved.

Therefore, the parts of the argumentation where lumo attacks particle theories I can leave unanswered. Let's note only that a short look at the particle-based approach to field theory in arXiv:quant-ph/0303156 suggests that lumo's arguments don't hit this target as well. This version introduces stochastic jumps into the theory (showing, by the way, that pilot wave theorists are not preoccupied with determinism). But I can leave the comparison to the reader.

About the "segregation" among observables

Because experiments eventually measure some well-defined quantities, the likes of Bohm think that there must exist preferred observables - and operators - that also exist classically. They are classical to start with, they think. Positions of objects are an important example.

But the quantum mechanical founding fathers have known from the very beginning that this was a misconception. All Hermitean operators acting on a Hilbert space may be identified with some real classical observables and none of them is preferred.

I think it is a misconception to interpret pilot wave theory as preferring some observables. It is not an accident that Bell has even proposed another word, beables, for the configuration space variables in pilot wave theory. In particular, measurements of the beables play no special role at all, nor in the classical limit, nor everywhere else in pilot wave theory. To derive the measurement theory, we don't need them (this would be circular anyway). What we need are the actual values of the beables, not some results of observations. Indeed, let's assume for simplicity we consist of atoms, which are the beables of some simplified pilot wave theory. Then, a theory about our observations does not need anything about our observations of atoms – if we "observe" them at all, then only in a quite indirect way, and most people do not observe atoms at all. Therefore, observations of atoms cannot play any role in an explanation of our everyday observations. Of course, in these explanations atoms have to play a role, at least indirectly – as constituent parts of our brain cells. But these atoms inside our brain cells are nothing we observe, if we observe something in everyday life. Thus, we use only the atoms themself, not the observations of atoms, in such explanations of our observations.

Thus, as observables the beables play no special role – in particular, the theory of their measurements can be derived in the same way, without danger of circularity. In particular, their measurements have to be described by self-adjoint operators or POVMs as those of every other observable too. In this sense, there are no preferred observables in pilot wave theory.

And this construction is actually very unnatural because it picks \(X\) as a preferred observable in whose basis the wave vector should be (artificially) separated into the probability densities and phases

Configurations (I prefer "q" instead of "X", because "X" is associated with usual space, while "q" is associated with configuration space) play indeed a special role. But this is the same special role they play in the Lagrange formalism as well as in Hamilton-Jacobi theory. Above are very beautiful, useful approaches. I don't remember to have heard any objections that the Lagrange formalism is unnatural, because it picks "q" as a preferred observable. Instead, the Lagrange formalism is an extremely important tool in modern physics, in quantum field theory as well as in general relativity. Moreover, this "segregation" is a very natural one: If nothing changes, the configuration remains the same, while the velocities have to be zero. Instead, I have found the symmetry between such different things as position and momentum in the Hamilton equations (and, similarly, in the canonical approach to quantum theory) always strange and unnatural, (even if, because of its symmetry, beautiful).

So why lumo does not fight against segregation in the Lagrange formalism? The segregation is the same, the poor momentum variables are degraded to the role of "derivatives". (Or maybe he does? I have not checked. Anyway, the important role of the Lagrange formalism in modern science, which is based on exactly the same "segregation", is a fact which shows that there is nothing wrong with this particular segregation.)

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

... I guess that you must agree that the "philosophical democracy" between all observables is pleasing and natural.

I see no reason at all to find such a "democracy" pleasing. You can observe a honest guy telling us the truth. As well you can observe how a liar is telling us lies. Above are observable. There may be even more symmetry between them. They may even make the same claims: "I have seen that he has stolen the money". That means, without segregation among observables, without destroying observable symmetry, we have to give them equal status. I don't plan to follow this idea, and will always prefer a segregation between truth and lies, even if this destroys some observable symmetries.

The segregation between contextual and non-contextual observables is less important, but is part of our everyday life as well. You can ask somebody about things he has not decided yet. He will think about them, possibly argue with you, and, maybe, give you an answer. This answer does not exist before you have started to argue with him, it is, therefore, contextual. Arguing with somebody else, he could have made a different decision. (Last but not least, this is one purpose of communication – to modify our decisions, if we hear good arguments to do this.) In other words, this answer will be contextual. But in a different situation, he has already decided about this question, and the answer was already part of reality of his thoughts when you have asked him. In this case, the answer is not contextual. Above answers we observe as results of complex verbal interactions, and they are, in this sense, on equal foot. Nonetheless, a realistic theory about his thoughts has to segregate between them. Without segregation, he should be or almighty, able to think and decide about all imaginable questions before you ask him, or completely dependent, deciding about nothing before you ask him.

In all these cases, the same "formalism" is used to obtain the results – communication in human language. Thus, that the same formalism – that of self-adjoint operators, or, more general, of POVM's – is used to describe the results of interactions in quantum theory is in no way an argument against this particular segregation.

Clearly, some quantities in the real world look more classical than others. But what are the rules of the game that separates them? The Bohmists assume that everything that "smells" like \(X\) or \(P\) is classical while other things are not. ...

Clearly, they want some quantities that often behave classically in classical limits.

Clearly not. The "segregation" in pilot wave theory is between configuration and momentum variables, and it is in no way related with one of them being "more classical". In classical situations, above behave classically, and the same segregation exists in classical theory too, in the Lagrange formalism as well as in Hamilton-Jacobi theory. There is no place in pilot wave theory where one has to care that something in the behaviour of the configuration is "classical": In the classical limit, it follows automatically, from the classical Hamilton-Jacobi equation, that everything behaves classically. For other questions this is simply irrelevant.

It is the many worlds community which is focussed around the classical limit. That's reasonable – they have a very hard job to construct something which at least sounds plausible (at least if one uses words like "contains" for a linear relation between some points in a Hilbert space, talks about "evolution" of branches without defining any evolution law, and applies decoherence techniques without explaining how to obtain the decomposition into systems one needs to apply them).

In order to simplify their imagination, the Bohmists imagined the existence of additional classical objects – the classical positions.

Simplification has, it seems, been removed from the aims of science. Ockham's razor is out, simple theories have to be rejected. The higher the dimension, the better.

But the objects are in no way additional. They have been part of the Copenhagen interpretation: Its classical part contains, in particular, all the measurement results. And Schrödinger's cat proves that a unitary wave function alone is not sufficient, that we need something else. Or some non-unitary collapse, or some particular configuration as in pilot wave theory. Something – be it the collapsed wave function, or some different entity – has to describe the reality we see: or the dead, or the living cat. Many worlds claims something different, but introduces, for this purpose, the "branches" – some sort of collapsed wave functions without collapse, or configurations without a guiding equation, which is claimed to be "contained" in the wave function. (How a decomposition of some vector into a linear combination of others defines a containment relation remains unclear. A concept where a function like \(\psi(q) = 42\) "contains" all possible universes has it's appropriate place in the Hitchhiker's Guide to the Galaxy, not in scientific journals.) The approach named "consistent histories" leaves us with many inconsistent histories, subdivided into families.

Theories with physical collapse need dirty and artificial non-unitary modifications of the Schrödinger equation. The branches of many worlds are, it seems, left today without any equations at all. (A very scientific approach, indeed. Time to rename it into "many words"). Only pilot wave theory gives us a nice, simple, and beautiful equation for this "additional" entity. Moreover, it allows, just for nothing, to derive the whole measurement formalism of quantum theory.

Imagination is completely irrelevant for these questions. I see, of course, no reason to object if a theory allows to simplify our imaginations too. Instead, I would count it as one additional advantage of a theory. But I recognize that this attitude is not shared by other scientists. And there are, indeed, good reasons to prefer theories which are complex and mystical. Imagine you are in a company of nice girls (or boys, whatever you prefer), and they ask you what you are doing. Isn't it much more impressive if you can tell them about curved spacetimes, large dimensions, a strange new quantum realism, or even quantum logic, many worlds and other strange quantum things? Compare this with the poor 17th century scientist, the fighter against any form of mystery, the classical loser in every popular mystery film. The choice is quite obvious.

About history

Louis de Broglie wrote these equations for the position of one particle, David Bohm generalized them to N particles.

Not correct, the configuration space version of pilot wave theory was presented by de Broglie already at the Solvay conference. See de Broglie, L., in “Electrons et Photons: Rapports et Discussions du Cinquieme Conseil de Physique”, ed. J. Bordet, Gauthier-Villars, Paris, 105 (1928), English translation: G. Bacciagaluppi and A. Valentini, Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference”, Cambridge University Press, and arXiv:quant-ph/0609184 (2006)

I think that in analogous cases, we wouldn't be using the name of the "updater" for the final discovery.

After having read something about the history of this theory (I do not care that much about history), I use "pilot wave theory" instead of "Bohmian mechanics". But Bohm has a point too: de Broglie has broken his theory as not viable, being unable to develop the general measurement theory. This has been done by Bohm. Therefore, if I use names, I use now the combination "de Broglie-Bohm".

Of course that I have always known that Bell constructed his inequalities because he wanted to prove exactly the opposite than what he proved at the end. He was unhappy until the end of his life. Bad luck. Nature doesn't care if some people can't abandon their prejudices.

This sounds like lumo thinks that Bell has tried to prove, with his inequalities, that quantum mechanics is wrong. This does not sound very plausible. It is quite clear that he liked Bohmian mechanics, that he has seen it's nonlocality as an argument against it, and tried to remove this argument, by showing that this nonlocality is a necessary property of all hidden variable theories. About his bets before the experiments have been performed, there is the following quote: "In view of the general success of quantum mechanics, it is very hard for me to doubt the outcome of such experiments. However, I would prefer these experiments, in which the crucial concepts are very directly tested, to have been done and the results on record. Moreover, there is always the slim chance of an unexpected result, which would shake the world." (Freire, arXiv:quant-ph/0508180, p.20)

[arguing against "I've read that the Broglie-Bohm theory makes the same predictions that the normal quantum randomness theory makes but the latter was chosen because it was conceived first.":]

Concerning the first point, people can have various theories in the first run. But once they have all possible alternative theories, they can compare them.

Second, it is not true that the probabilistic interpretation was conceived "first". Quite on the contrary. Technically, it's true that de Broglie wrote his pilot wave theory in 1927, one year after Max Born proposed the probabilistic interpretation, but the very idea that the wave connected with the particle was "real" was studied for many years that preceded it. Both de Broglie (1924) and Schrödinger (1925) explicitly believed that the wave was real which is incorrect.

Given that de Broglie has given up pilot wave theory shortly after 1927, unable to find a viable measurement theory for other observables than position, one can say that pilot wave theory appeared in a viable form only 1952, with Bohm's measurement theory. At that time, the Copenhagen interpretation was already well-established (even if the label "Copenhagen interpretation" was coined only later). So there was an advantage of historical accident for the standard interpretation.

In 1952, Bohm wrote down a very straightforward multi-particle generalization of de Broglie's equations and added a very controversial version of "measurement theory". Is it a substantial improvement you expect from 25 years of progress?

Depends on how many people have worked on it during this time. In this case, most of these 25 years nobody has worked on it. In particular, de Broglie himself had broken it, because he was unable to find the "very controversial" measurement theory found later by Bohm. Bohm, who was 1927 only 10 years old, had not worked most of this time in this domain too. Thus, very few man-years have been sufficient to transform a theory broken by it's creator as not viable into a viable theory. I would name this a sufficiently efficient and substantial improvement.

The next important defender of this theory – again almost alone for a long time – was Bell. The results of his work in the foundations of quantum theory are also well-known. Despite their foundational character, they have caused a large experimental activity. Thus, also a quite efficient relation between man-years and results.

(Given that lumo has not understood the main point of Bohm's measurement theory, we can ignore the characterization of this theory as "very controversial").

About decoherence and the classical limit

Moreover, the question which of them will emerge as natural quantities in a classical limit cannot be answered a priori. Which observables like to behave classically? Well, it is those whose eigenstates decohere from each other.

The role of decoherence in the classical limit is largely overexaggerated, see the Hyperion discussion about this (Ballentine, Classicality without Decoherence: A Reply to Schlosshauer, Found Phys (2008) 38: 916-922, DOI 10.1007/s10701-008-9242-0, Schlosshauer, Classicality, the ensemble interpretation, and decoherence: Resolving the Hyperion dispute, Found Phys (2008) 38: 796-803, DOI 10.1007/s10701-008-9237-x, arXiv:quant-ph/0605249, Wiebe and Ballentine, Phys. Rev. A 72:022109, 2005, also arXiv:quant-ph/0503170).

Essentially, you can measure every operator, together with every other, if the accuracy of the common measurement is below the boundaries of the uncertainty relations. And in the classical \(\hbar \to 0\) limit they all like to behave classically.

Everything in this real world is quantum while the classical intuition can only be an approximation, and it is a good approximation only if decoherence is fast enough i.e. if the interference between the different eigenstates is eliminated. If it is so, the quantum probabilities may be imagined to be ordinary classical probabilities and Bell's inequalities are restored.

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

A nice description of the decoherence paradigm. The little dirty secret of decoherence is that it depends on some decomposition of the world into systems. Such a decomposition can be found, without problems, if we have some classical context as in the Copenhagen interpretation, or some well-defined configuration of the universe as in pilot wave theory, by considering an environment of the actual state of the universe. But without such a background structure you have nothing to start these decoherence considerations. The different systems we see around us – cats, for example – cannot be used for this purpose, at least not if we want to avoid circular reasoning. arXiv:0901.3262 The Hamilton operator, taken alone, is not enough to derive a decoherence-preferred basis uniquely.

Mechanistic models of state-of-the-art quantum theories are not available: it is partly because it's not really possible and it's not natural but it is also partly because the champions of Bohmian mechanics are simply not good enough physicists to be able to study state-of-the-art quantum theories. They're typically people with philosophical preconceptions who simply believe that the world has to respect their rules of "realism" or even "determinism".

I have a quite nice "mechanistic model" for the standard model of particle physics. One which essentially allows to compute the SM gauge group (as a maximal group which fulfills a few simple "mechanistic" axioms). How many more years (and how many more man-years) string theory needs to reach something comparable?

The idea of "philosophical preconceptions" is quite funny. My concept is quite pragmatical: If there is a simple way to do the things, use it. Simplicity is a good thing, independent of the age or the popularity of the particular concept. About determinism I don't care even today, in particular I have certain sympathies for Nelson's stochastics. And I have as well looked at non-realistic interpretations of quantum theory, like the concept I prefer to name "inconsistent histories". But I think there should be really good evidence to justify the rejection of such simple, general, fundamental and beautiful principles like realism. But pilot wave theory would be preferable even without it, simply for the beauty of the guiding equation.

Last but not least, some funny but unimportant polemics

The attempts to return physics to the 17th century deterministic picture of the Universe are archaic traces of bigotry of some people who will simply never be persuaded by any overwhelming evidence – both of experimental and theoretical character – if the evidence contradicts their predetermined beliefs how the world should work.

Well formulated. I like such polemics. Especially replacing the standard 19th century in such flames by 17th century is nice. But there is room for enhancement. In philosophy of science, I follow Popper, who likes to identify the origin of some of his ideas in Ancient Greece. I also prefer the economic system based on ideas of Adam Smith in comparison with much more modern ones developed by Lenin and Mao, so one can identify this sympathy for old ideas as deeply rooted in my personality. Indeed, I think there is nothing wrong with old ideas.

To describe pilot wavers as "predetermined" sounds really nice, but is, unfortunately, wrong. There are, of course, people who follow predetermined ideas. But these are the ideas they have learned in their youth. Where are the proponents of pilot wave ideas supposed to have learned it? What I was teached was quantum theory and Marxism-Leninism, not pilot wave theory and Adam Smith. And I remember, in particular, some uncritical fascination learning von Neumann's proof of impossibility of a classical picture. I have had nor a prejudice for 17th century determinism, nor any of the "bourgeois prejudices" the communists have liked to argue against.

It was not predetermination, but the power of arguments (in particular, of Bell's "speakable and unspeakable in quantum mechanics"), which has persuaded me to switch to pilot wave theory. And an important part of this argumentative power was the simple proof of equivalence between pilot wave theory and quantum theory. There simply is no experimental evidence against pilot wave theory.

And, indeed, the "experimental evidence" presented by lumo was (in his polarizer argument, and similar ones about spins) based on the common error not to take into account the measurement device, or (in his quantization argument) not applicable to de Broglie's version of pilot wave theory. About the theoretical evidence judge yourself.

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

It is always fun to compare the "very facts" of such claims with reality. The one-particle spinless case has never been in the focus of my interest, except if this appears sufficient to show some serious problems of other interpretations ( arXiv:0901.3262, arXiv:0903.4657). The results of my work with spins, fields, and quarks I have already mentioned. And even renormalization is on my todo list, even if some other problems have, yet, higher priority for me.

I'm not sure that naming strings and dualities "cutting-edge physics" is justified. This is clearly a domain of research I leave to lumo – it may have a value as a nice exercise in mathematics, which is an important part of human culture, even if it has nothing to do with physics. Of course, one never knows – results of pure mathematicians, who have been proud of doing things which will never find an application, are applied today in cryptography. It would be a really nice joke if some result found by lumo would find a physical application in some hidden variable ether theory ;-).

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Pacific waters as an excuse for the warming hiatus

Most of the mainstream media offered us a bizarre "story" in the recent two or three days. The absence of global warming in recent years – well, it's really 17 years now – has been "explained" by the Pacific waters. Problem solved, the belief in the global warming ideology may continue unchallenged, we're de facto told.



PDO: warm and cool phase

The claims are based on the paper by Kosaka and Xie in Nature,

Recent global-warming hiatus tied to equatorial Pacific surface cooling,

which is bad enough but I will mostly focus on the journalists' added spin which is even worse. The Guardian's Fiona Harvey will be used as my sample but the comments below are applicable almost universally.




The first paragraph says that it's the Pacific Ocean that "slowed" the global warming and the second paragraph tells us why this research is supposed to be important:

Their work is a big step forward in helping to solve the greatest puzzle of current climate change research – why global average surface temperatures, while still on an upward trend, have risen more slowly in the past 10 to fifteen years than previously.

I had to laugh. The most important insight of the "modern" climate science research is that the world is warming and the greatest puzzle is why it's not true according to observations. Incidentally, it is not true that one may still find a "warming trend" in the last 10-15 years. One only finds noise and in various subintervals, the cooling and warming trends are about equally represented, as shown many times on this blog and elsewhere.

If you expect that half a degree or a degree Celsius of warming per decade is now inevitable, e.g. because you were brainwashed by the AGW charlatans or because your brain has been damaged in a similar sad accident, then the absence of the warming may look like a "great puzzle". But what about the approach not to be brainwashed by unscientific superstitions? Isn't it a bit more rational approach? In that case, there's no puzzle. The temperatures are doing what they have always been doing: fluctuate by one or two tenths of a degree each decade in pretty random directions; after all, we know it from the historical data from decades when the man-made CO2 concentration was vastly lower than today that the temperatures were doing nothing less than that. CO2 may add a contribution but it's just small enough so that the CO2-induced change remains more or less negligible relatively to the natural variability for 20 years if not much longer than that.

Isn't it a more convincing explanation than the program to identify a huge puzzle and then to "solve" it by naming a culprit?




But I had to laugh throughout Harvey's article, it's just so utterly stupid and irrational. You would say that some analysis of the role of the Pacific Ocean will be presented as the research of a factor that influences the climate as a whole. Instead, we're repeatedly told that what the paper studies is the ocean's influence on the pause in global warming. This completely silly formulation – sometimes with the word "hiatus" replacing "pause" – is repeated many times in the article. It's a classic example of a tail wagging the dog. Climatology – and this paper – studies or should study the climate (which is pretty much equally likely to change in both directions, as empirically shown all the time), not "global warming" or even "hiatuses in this non-existent global warming". But a lie repeated 100 times becomes the truth according to the key beliefs of Fiona Harvey.

However, what I found most comical is the suggestion – permeating Harvey's text and pretty much all similar articles – that the "bad guy" has been found so everything is fine now. It's the evil Pacific Ocean. I can't believe my eyes because a reader must have IQ below 70 to buy this cheap trick. Is it likely that such humans or, more generally, primates have learned to read the Guardian? It is sort of hard to believe.

In reality, these AGW scammers have claimed that the global mean temperature is

1. important,
2. predictable.

It's neither important nor predictable but they have simply made a prediction 20 years ago (and similarly in more recent years) and the prediction has totally failed. The warming in the 20 years wasn't 0.6 °C as predicted but pretty much zero. The error is 100 percent. It is a complete failure. The theory claiming that the global mean temperature is predictable in the 20-year horizon that makes these particular predictions must be thrown to the trash bin immediately if you're doing science. If those who have made the predictions haven't contributed anything substantially better, they must be thrown to the trash bin, too. If they don't fit the trash bin, they must be at least humanely shot into their skulls. It's that simple. It's called science, stupid.

Instead, we're sold stories about the Pacific Ocean as a "justification" of the failure. It's no justification. If you totally screw your understanding of the Pacific Ocean, it pretty much means that you screwed your understanding of the climate on most of the globe. The Pacific Ocean covers one third of the Earth's ocean. In fact, this largest ocean's area exceeds the area of all lands on Earth. It's in no way negligible.

So even if you just consider the Pacific Ocean's contribution to the global average of the temperatures, it is a huge contribution that can't be overlooked. The Pacific Ocean was this large and this important even 20 years ago – even millions of years ago. It is no "news" that the Pacific Ocean was important. The global mean temperatures were always meant to include the contribution of the Pacific temperatures with the appropriate weights. It was the global mean temperature that did include the Pacific contribution that the global warming ideology advocates claimed to predict. They just failed. One may try to isolate which parts of the Earth had larger errors than others but it's mostly silly because every sensible person agrees that regional temperature trends can't be predicted for 20 years in advance with currently available tools. That's why you can't really say that "just the Pacific" predictions failed. Moreover, the predictions of every region are based on the same ideas so if one region empirically shows you that you misunderstand how it works, you probably misunderstand how every region works.

Now, despite the huge size, the actual temperature of the Pacific Ocean temperature isn't that important for almost any humans. But almost every elementary school alumnus knew about this fact 20 years ago, too. Nevertheless, the Pacific temperatures have been included in the "important" quantity called the global warming temperature – mainly because the contemporary climatologists are obliged to be obsessed with the greenhouse effect that operates more or less equally on every square mile of the globe. The obsessive focus on the (for the actual climate change not too important) global mean temperature that uniformly depends on each square mile of the globe has always been a key meme of the global warming ideology. You can't just throw it away and claim that nothing about the ideology has to be changed. If you change this thing, you are really proving that it has been rubbish from the beginning.

One may always cherry-pick 1/3 or even 2/3 of the globe where the disagreement with the models is smaller than it is in the rest. But if you claim that a quantity is the most important and you completely fail in predicting it, you have just failed. In scientific disciplines that may be counted as hard science and that have corresponding standards, a hypothesis is really eliminated once any clear disagreement of its predictions with the reality is found. One wrong prediction is enough to falsify an idea. If you want to be satisfied with the agreement between the rough observations of 1/2 of the numbers (that you may cherry-pick a posteriori) and your predictions, then any hypothesis will end up as "viable".

Moreover, the Pacific Ocean influences the globe by more than just this large area that "directly" influences the global temperature averages. The Pacific Ocean is the cradle of the El Niño/La Niña ENSO oscillations which have been known for quite some time to be the dominant contribution to the interannual temperature variability. If the global mean temperature changes significantly from one year to the next, chances are very high that it has something to do with the El Niños or La Niñas.

The Pacific Ocean is important at the decadal scale, too. The Pacific Decadal Oscillation describes one of the most important regional degrees of freedom – if not the most important one (AMO is a possible competitor) – that changes at the timescale comparable to tens of years. In fact, both El Niños and PDOs are mentioned by many of the articles. But they're not able or willing to deduce the most obvious consequence of all these insights, and it is the following:

The more you use natural variations such as El Niño, PDO etc. to explain what's actually going on and what's being observed, the more important the natural drivers become, the more irrelevant the CO2 gets, the more the skeptics who claim that the climate change is mostly natural are shown to be right (many skeptics have talked about the important influence of the ocean cycles and patterns for many decades and most skeptics today are well aware about the tight PDO-global-temperature correlation in the last 100 years), and the more discredited the AGW doctrine becomes along with its defenders.

Is that really so difficult for Fiona Harvey to understand this trivial point? Do we have to read all the garbage about CO2 that manifestly has nothing to do with any of the important observations or insights and not even with these not-so-important observations of the recent events in the Pacific Ocean?

Even average U.S. and EU politicians are beginning to understand what the hiatus means (why not journalists in the Guardian?) and they want a credible explanation in the IPCC AR5 summary that will be out in one month from now.

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One can't background-independently localize field operators in QG

...because the "basis" of coherent states is overcomplete...

Let me begin with something simple. John Preskill asked you "What's inside a black hole?" and offered you four options:

1. An unlimited amount of stuff.
2. Nothing at all.
3. A huge but finite amount of stuff, which is also outside the black hole.
4. None of the above.

Well, the option (D) may have been at the beginning and an obvious suboption of (D), "The black hole interior is a region just like any other region and independent from others", should have been offered as a special choice (E). A surprising result is that (E) is almost certainly wrong. Instead, (C) is right – at least if we omit the very highly curved region near the singularity that could justify (A) in a complicated way and if we allow the definition of a black hole to cover its rare microstates – if we only allowed the most generic black hole microstates, the answer would be (B): the interior has to be empty.

Well, (B) may also be interpreted as a claim allowing a firewall, in which case it's wrong in general (the firewall isn't necessary or generic) but of course that there are rare black hole microstates that contain something that burns you near the horizon much like there are rare black hole microstates with a bunny in the interior.

This point is simple but often misunderstood. A black hole is defined by its event horizon but it doesn't follow that the interior has to be empty. There can be a bunny in it. However, among microstates of localized matter, a black hole with a bunny is an exponentially rarer class of microstates. Most of the mass \(M\pm \delta M/2\) black hole microstates look empty – that's why the entropy-increasing evolution converges towards these states as the black hole keeps on devouring the surrounding matter to clean its interior (and vicinity). But don't make a mistake about it: a bunny in a black hole (or a nonzero occupation number of freely falling field operator modes) is unlikely yet possible.

But let me switch to a more complicated question.




During Suvrat Raju's talk at a recent Fuzz-Or-Fire workshop in Santa Barbara, a core of the pro-firewall/anti-firewall conflict became rather visible. On one hand, the Papadodimas-Raju "state-dependence" of the definition of the black hole interior field operators seems unacceptable to the firewall champions although it looks pretty much inevitable to many of us.

On the other hand, this disagreement may be described as a criticism of Polchinski's and pals' alternative: They believe that the bulk field operators and especially their location in a quantum theory with a dynamical geometry may be defined by a recipe described operationally in a "background-independent way". For example, start at the AdS boundary that must be close to the empty AdS geometry, pick a direction as if it were an empty AdS, and go in this direction for a certain proper time or proper length. Then you turn in the direction of the greatest curvature (defined in some other way) and walk for 5 meters of proper distance or 2 microseconds of proper time, and so on. You get to a point and there is a scalar field at that point that you may call \(\phi(x,y,z,t)\) and ask about its eigenvalues or what it does when it acts on a state \(\ket\chi\) etc.




The classical counterpart of such a prescription sounds totally OK in classical general relativity. You may imagine that one particular spacetime geometry is the "right one" and whatever the spacetime geometry is, the operational procedure involving the proper times, proper distances, and angles may be followed and the right value of the field at the point we just found becomes a well-defined \(c\)-number (let's talk about scalar functions of local tensor fields and their derivatives only).

Joe Polchinski and others believe that the same background-independent operational definition of field operators at various points may be used in quantum gravity, too. This belief is incorrect. There are several ways to see why. They seem very different but ultimately they are rooted in the same general properties or at least "spirit" of quantum gravity that is imposed on us by consistency.

To maintain their belief that the background-independent localization of field operators (and therefore state-independence) is possible, the firewall advocates must assume that

• the metric tensor is a good and precisely and uniquely defined degree of freedom (quantum observable) at arbitrarily short distances
• every ket vector in a quantum gravity theory may be uniquely rewritten as a sum of ket vectors each of which comes with a well-defined classical geometry

Both of these assumptions are incorrect, however. In some sense, the second problem is more damning for the firewall advocates' plans than the first one.

The metric tensor isn't any good at (sub)Planckian distances

The first point has to be true because they want to determine proper distances. You need a metric tensor for that. Because the definition must work even in rather general, potentially extreme environments near collapsing and other black holes where we often need an exponential precision to locate the events (note the coordinate singularity at the horizon etc.) while we have to resist high matter densities etc., the definition of the metric tensor has to be really exact for Joe's and pals' background-independent operational definitions of the points in a general spacetime to make any sense.

However, quantum gravity doesn't allow you things like that. The metric tensor is only good and well-defined in an effective description of quantum gravity. At shorter distances, it just ceases to be a good observable. Well-defined observables in quantum gravity are different; the gauge fields in the \(\NNN=4\) Yang-Mills theory involved in the most famous example of the AdS/CFT correspondence are an example. The matrices \(X,P,\Theta\) in Matrix Theory are another example.

Even if you had something like a "closed string field theory" that would apparently contain the metric tensor "everywhere", you would have to solve the problems of the mixing of its field modes with some other modes of fields arising from heavy excited string states (with the same charges and spin). To make the procedure well-defined, you would have to overcome the problem that there are many ways (related by field definitions involving all the massive string fields) how to define the metric tensor. They may be thought of as different "renormalization schemes". You may imagine that a different "renormalization scheme" amounts to switching the metric from something like the string frame to something like the Einstein frame but the rescaling depends on the massive scalar fields \(h\) in string/M-theory rather than the dilaton \(\phi_D\). Classically, \(h\) is constant so this rescaling doesn't change much. However, quantum mechanically, \(h\) is a dynamical, fluctuating field so an \(h\)-dependent redefinition of the metric tensor does matter.

But even if your procedure directing someone to walk over some proper distances in a general spacetime etc. did specify a particular "renormalization scheme", it would still be no good because at very short, near-Planckian distances, the geometry becomes brutally fluctuating and the proper distances and times, when accurately measured over the violent landscape of the quantum foam, are probably divergent and/or ill-defined. So Joe's prescription would break down.

My point is that whatever "renormalization scheme" you pick, \(g_{\mu\nu}(x,y,z,t)\) is a fluctuating degree of freedom that has nonzero probability amplitudes to be nonzero and substantial even in the vacuum state of the spacetime. By dimensional analysis, the magnitude of the contribution \(\delta L\) of these fluctuations to a proper distance \(L\) comparable to the Planck length is comparable to the Planck length i.e. 100 percent; I believe that this dimensional analysis, assuming \(g_s=O(1)\), is OK even in string theory despite its ability to "calm down" the quantum foam. You simply shouldn't assume that the flat and peaceful spacetime offers you good expectations about the behavior of proper distances, times, and angles near/below the Planck scale. Try to follow Joe's algorithms on the quantum foam (the picture at the bottom):



It's pretty obvious that you get caught in the weird tunnels and valleys of this quantum foam whatever recipe you choose. What you actually need is a geometric prescription that is allowed to use the smooth, nearly flat spacetime similar to the upper part of the figure. But using the proper distances and proper times calculated from the dynamic metric tensor just don't give you anything like a flat space even in the vacuum-like ket vectors. The quantum foam picture at the bottom of the picture above is an eigenstate of \(g_{\mu\nu}(x,y,z,0)\) and even the Minkowski-like vacuum state in quantum gravity is a superposition of states whose geometry looks like this. You won't really get anywhere with the background-independent protocols to isolate a location in the spacetime.

Non-uniqueness of a "geometry" associated with a ket vector in QG

But it's the second complaint against Joe's paradigm, if you allow me to call it in this way, that seems more damning and conceptual. You could imagine that for some unknown reasons, string theory calms down the quantum foam so nicely that the sub-Planckian terrain may still be imagined as a smooth space rather than the quantum foam and the procedure could get through with a potentially natural choice of the "renormalization scheme".

However, the procedure will still fail due to some facts that don't depend on the short-distance, Planckian physics. What are these general problems with the background-independent approach to the location of points in a dynamically curved quantum spacetime?

For the sake of simplicity, let's assume that the procedure "go here for 5 meters, turn left etc." is only used to move through a slice of the spacetime at a fixed value of the coordinate \(t\), whatever it is. If we considered trajectories deviating from the slice, we would open yet another can of worms because the metric tensor doesn't commute with its time derivatives (the uncertainty principle!) so it's just downright impossible to imagine that these behave classically in any ket vector (this assumption is as wrong as the assumption that arbitrarily sharp trajectories in the quantum phase space make sense).

Fine. Polchinski's procedure is meant to tell you what is the action of an operator \(\phi(P)\) on a general quantum gravity ket vector \(\ket\psi\). The point \(P\) is specified by an operational, background-independent procedure of the type "go for 5 meters, turn left, do this and that". Now, Joe believes that the action\[

\phi(P)\ket\psi

\] is another well-defined ket vector. We can see it can't be the case. Why? Well, the vector \(\ket\psi\) isn't an eigenstate of the metric tensor operators \(g_{\mu\nu}(Q)\) at the relevant points \(Q\) that may appear along the trajectory. To avoid the immediate ill-definedness of a recipe based on proper distances, we must decompose \(\ket\psi\) into eigenstates of the metric tensor variables \(g_{\mu\nu}(Q)\):\[

\ket\psi = \sum_j \ket{\gamma_j}

\] Well, the sum could actually be an integral and the normal people would tend to normalize \(\ket{\gamma_j}\) to unity and write the normalization factor as a special coefficient, and so on, but the equation above is good enough. In the previous section, I discussed the problems resulting from the violent character of the geometry in the \(g_{\mu\nu}\)-eigenstate. But even if you forget about these short-distance troubles and ambiguities and you assume that the proper distances through the apparent quantum foam behave just like your long-distance intuition suggests (up to a universal renormalization coefficient for the distances), you face insurmountable problems, even at long distances. They're related to the short-distance problems discussed previously but the arguments below hopefully make their independence on the UV physics more obvious.

Imagine that we want to apply the procedure to the most peaceful yet nontrivial state we can imagine, a smooth macroscopic gravitational wave in an otherwise empty spacetime. This state containing a gravitational wave may be written as a coherent state\[

\ket\psi = \exp\left[\int d^d k\,\alpha(k) c^\dagger(k)\right] \ket 0.

\] It's the exponential of a superposition of creation operators for some graviton states. As a homework exercise ;-), add sums over the polarizations and other indices and everything else you like or need. Now, additional particles may be created on top of the state \(\ket\psi\) and I think that Polchinski would say that the right way to apply his procedure is for the distances in the states that contain a few particles on top of the curved spacetime \(\ket\psi\) to use the geometry of this curved spacetime when we try to follow the procedure to "find the location in a general spacetime".

You should already feel uncomfortable at this point because the state \(\ket\psi\) is an excitation of the Minkowski vacuum state, too. Rewrite the exponential as a Taylor expansion if you want to make the point more suggestive. Gravitons are particles, too. You might say that it couldn't be a hopeless idea to use the flat spacetime's metric when you try to locate points in the spacetime except that it would also be obvious why the relationship between the local operators on top of the excited coherent, curved space \(\ket\psi\) with the local operators on top of the Minkowski space \(\ket 0\) is extremely convoluted.

So let me assume that Polchinski et al. really want to use the curved geometry from the coherent state \(\ket\psi\) when they follow their background-independent procedure. It means that to find the action of a local operator \(\phi(P)\) on \(\ket\chi\), they need to decompose \(\ket\chi\) into "matter-like" (and therefore geometry unchanging) excitations of coherent states of the type \(\ket\psi\) above for which the metric tensor is known.

The trouble with this background-independent physics is that the "basis" of the harmonic oscillator Hilbert space consisting of the coherent states is overcomplete.

See basic introductions to coherent states if you have any doubt about the statement. So even if you restrict your calculations to ket vectors \(\ket\chi\) that only contain purely gravitational excitations, you will need "the" decomposition of such states to coherent vectors to identify \(\phi(P)\) but "the" decomposition actually isn't unique.

This is a problem that makes your background-independent procedure break down even for states \(\ket\chi\) that are as simple as a low-energy, single-graviton excitation of the Minkowski vacuum state. On one hand, you could consider this excitation to only change the background geometry infinitesimally and use the Minkowski geometry to follow the procedure. The first excited state of a harmonic oscillator is proportional to a superposition of coherent operators weighted by \(\delta'(a)\) all of which are infinitesimally close to the origin of the phase space (interpreted as a flat space in the Fock space of gravitons). On the other hand, you may rewrite this first excitation of the harmonic oscillator as some linear superposition of coherent states centered elsewhere, even very far from the center at zero (effectively a linear superposition of highly curved spacetimes). It's clear that the point \(P\) where you get by following these spacetimes will depend on the way how you decompose your states to the coherent states. This way isn't unique and the infinitely many choices differ by differences that are unbounded from above.

If the procedure doesn't work for single-graviton states, you may be sure that the problems become exponentially worse if you try to apply the procedure to a black hole spacetime with a significant density of mass, coordinate singularities, and many other things. It's completely hopeless.

Incidentally, if you tried to replace the decomposition into coherent states by a decomposition into \(g_{\mu\nu}\)-eigenstates – in the harmonic oscillator analogy, \(x\)-eigenstates – discussed at the beginning, you could cure the overcompleteness problem of the basis but you would also totally delocalize the vectors in the values of \(\partial_t g_{\mu\nu}\) which means that the time-like geodesics of the recipe would probably become infinitely singular (the coherent states naturally balance the needs of the metric in the spatial and temporal directions); you wouldn't be guaranteed that the proper distances are well-behaved and finite at short distances. At the end, any attempt to define the recipe will fail because what all of them actually contradict is the equivalence principle: they are assuming that the spacetime geometry is classical enough so that the proper length/time of some generic trajectories going in many directions may be accurately measured which isn't so.

An alternative for the background-independent operational localization protocols

Once I have shown that the background-independent way of identifying locations of operators isn't possible, it may seem polite for me to tell you what's a legitimate replacement of it. We could be saying that no calculations based on strictly local operators attached to "points" are possible in quantum gravity. Except that I think that they are possible. However, you have to assume (manually and, whenever possible, cleverly choose) a background – a particular "curved space" vacuum-like state of the quantum gravitational theory which may also be obtained as a coherent state built from other vacuum-like states – and construct many other microstates out of this vacuum-like state by the action of a "finite" (not scaling with various parameters called \(N\) that would be increasing functions of the curvature radius etc.) number of field operators where these field operators are behaving much like they are behaving in the flat space, at least locally in regions where the curvature may be neglected. Papadodimas and Raju explain these conditions more quantitatively. In some sense, I believe that the ER-EPR correspondence with its ER bridges is a special visualizable "Ansatz" for solutions of such constraints.

Here I must say that people like Lee Smolin have been saying totally idiotic things about "background independence" for years. They would even criticize string theory for being able to write the Hilbert space of quantum gravity as a de facto Fock space built upon a particular background. Remember all the silly demagogy that no backgrounds can ever be talked about because GR imposes a democracy between all of them, and all this rubbish.

Feel free to impose a ban on talking about backgrounds but then you will be unable to make any calculations that may be compared with the experiments, too. The adjective "background-independent" may be given many meanings and some of them are respectable, at least in some contexts, but be sure that if your interpretation is that "we can't use any backgrounds in calculations at all", then you are throwing the baby out with the bath water.

Because I properly learned many of the computational techniques that existentially depend on the choice of a background (in the spacetime or the world sheet) from Joe Polchinski, I wouldn't have believed 14 years ago that he would ever be saying things "remotely similar" to the Smolinian rubbish on the background independence.

If we want to organize a Hilbert space (or, more typically, its subspace) as some collection of states with a spatial interpretation (states that tell us what is being observed here or there), then we simply need to associate the microstates with a background. We also need to gauge-fix the diffeomorphism gauge symmetry or redundancy, if you wish. Only when it's done, it's possible to define how local field operators act in between the states in this subspace of the Hilbert space. It's clear that if you create too many things in your background, or if you deform the geometry by too many gravitons, to be more specific, the added gravitons or the backreaction to the added matter make the original background's geometry an unnatural (or perhaps more accurately, practically not too useful) way to measure distances and times. You should better pick a different background to parameterize the relevant portion of the Hilbert space if you consider states whose geometry is too different from the original background. But you must choose a background because trying to leave the "job to measure the geometry" on the microstates without a choice of background requires a decomposition of the gravitons' Fock space states to coherent states which isn't unique.

ER-EPR's definitions of operators are clearly background-dependent, too

The state-dependence – well, really background-dependence – of the definitions of the black hole interior (and perhaps all other) local field operators is something most tightly associated with the insights by Papadodimas and Raju. But I believe that the Maldacena-Susskind ER-EPR correspondence makes this inevitable background dependence equally if not more self-evident.

Why?

It's simple. They say that the Hilbert space of one Einstein-Rosen bridge (a pair of black holes geometrically connected by a non-traversable wormhole) is the same Hilbert space as the Hilbert space of two faraway black holes (that are allowed to be entangled). Clearly, these two pictures of the same Hilbert space envision completely different background spacetimes – the spacetimes have different topologies, in fact. So the definitions of field operators in the black hole interior(s) are clearly different in these two pictures. In other words, the definition of local field operators depends on whether you describe the same Hilbert space as two black holes that can get entangled later (but you're "expanding" around the microstates for which the entanglement is low and the black holes are assumed to be independent to start with) or the Hilbert space of a single Einstein-Rosen bridge with just "one interior" (you're expanding around a particular microstate for which the entanglement entropy is maximized; note that there can be many such maximally entangled microstates for which the bridge is correspondingly "twisted"). In other words, the definition of the local field operators is background-dependent, i.e. dependent on the choice of the spacetime background you have to make manually and subjectively before you start your calculations. It's clear because the local operators depend even the topology which is totally different in the two choices. The two black holes have two interiors while the Einstein-Rosen bridge only has one component of the interior. For various situations or classes of microstates, one of the two descriptions is more convenient or practical than the other description, but there can't be a universal law that would make one description more correct than the other one a priori. You must predecide how many components the interior(s) has (have) before you start to talk about the field operators in the interior(s).

Finally, I must say that I believe that most of what I wrote above aren't my exclusive original insights but just a reinterpretation of some insights made by Papadodimas and Raju which uses different words. If this is not a legit way to describe what they concluded, they will tell me and I will inform you, too.

I like to think about the ER-EPR correspondence but again, I believe it is just a more specific, visualizable "Ansatz" how to write the field operators at different places and the general, non-visualized principles for the operators were already found by Raju and Papadodimas (and perhaps others whom I may have slightly overlooked). The Raju-Papadodimas conditions for the mutual relations between the field operators start to break down once you arrive to short enough distances where the Einstein-Rosen bridges with the Hawking radiation become visible.

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Two-sigmaish CMS multilepton excesses with a \(\tau\)

A possible hint of third-generation superpartners

Matt Strassler mentioned an interesting anomaly reported by CMS at a SUSY conference this week:

A Discrepancy to Keep an Eye On (Prof Dr RNDr Matt Strassler PhD CSc DrSc Dot COM)

It's small enough so that you may assume that it's just another example of a fluctuation that will go away with more data. But it's large enough for many of us to gain the right to be intrigued. ;-)



The excesses have something to do with multileptons. If you search this blog for multileptons, you find many articles, mostly from late year 2011 and early year 2012. The words "year" were inserted for you to notice that there were many hyperlinks in the previous sentence. It's plausible that those flukes have gone way during the 1.5-2.0 years.

What are the overrepresented events this time?




The CMS folks have performed a search for some signatures – at this point, it's not quite clear what comprehensive theory beyond the Standard Model they are testing but multileptons of course do naturally appear in long enough decay chains of supersymmetric particles (they are a strong hint that at least a pair of new particles was produced because they're almost absent in the Standard Model and in the decay of one new particle) – for events with four leptons and some of the subsets of these events exhibit modest but tantalizing excesses.




We're talking about events with four leptons including

• at least one \(\tau^\pm\) that decays to hadrons; I start with that to point out that the signal could have something to do with the third generation of fermions and their partners
• one light (\(e^\pm\) or \(\mu^\pm\)) lepton-antilepton pair (this is what OSSF1, opposite-sign-same-flavor-one, stands for) that seems not to originate from a Z-boson decay (because the Z-boson is assumed to be understood and boring; the condition is called "off-Z")
• the total energy in the lepton pair plus a few other jets quantified by a variabled called \(H_T\) should be surprisingly small relatively to expected LHC energies; this condition indicates a threshold above which a new massive particle was barely created
• another charged lepton, \(e^\pm\), \(\mu^\pm\), or \(\tau^\pm\)
• some missing momentum – which surely gets a contribution from the neutrino(s) that is (are) produced along with the \(\tau\) lepton but may also include the LSP etc.

A table that Matt reproduces for us has about 48 entries and 3 of them contain noticeable excesses. All of them are in the OSSF1, \(H_T\lt 200\GeV\), off-Z, \(N(\tau_h)=1\), \(N_{b\rm -jets}=0\) – bins with other options for these variables are really consistent with the Standard Model predictions.

The three bins with an excess only differ in the missing transverse energy:\[

\begin{array}{|c|c|c|}
\hline
E_T^{\rm miss} & {\rm predicted} & \text{observed}\\
& \text{events}& \text{events}\\
\hline
(0,50) & 7.5\pm 2 & 15\\
\hline
(50,100) & 2.1\pm 0.5& 4 \\
\hline
(100,\infty)& 0.6\pm 0.24 & 3\\
\hline
{\rm total} & 10.2\pm 2.1 & 22\\
\hline
\end{array}

\] Sorry if the error margins shouldn't have been added in quadrature; even if it is roughly correct, it's not the totally accurate way to account for the error margins. I guess that the right error should be between 3 and 4 events. I suspect that the errors in the table are just the systematic ones and the Poisson-type statistical ones \(\sqrt{N}\) must be added in quadrature manually. If you can clarify these issues, it would be appreciated.

At any rate, when you combine these three similar channels – naturally ignoring the magnitude of the transverse energy – you get something that may look like a greater than (or at least approximately equal to) \(3\sigma\) excess (because \(10.2\) and \(22\) are really far from each other). I don't want to say \(5\sigma\) which the sloppy calculation above could hint at because I don't believe this can be the right result.

I would say that the missing energy may be arbitrarily low in the events above so if an LSP is created, it should be a light one, safely below \(50\GeV\) – possibly the \(8.6\GeV\) dark matter particle suggested by CDMS II-silicon and others. And the superpartners created at the beginning of the reaction are likely to have something to do with the third generation – like staus. Or sbottoms...

This is not an excess you should think about every night at 3 a.m. so far. But it's an excess that you may return to at some point in the future and describe by the words that you already knew about it on August 29th, 2013, before the revolution got started.

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