Tuesday, May 3, 2016

Are Single-World Interpretations of Quantum Theory Inconsistent?

A recent eprint caught my atention: Single-world interpretations of quantum theory cannot be self-consistent by Daniela Frauchiger and Renato Renner. In the abstract we read
We find that, in such a scenario, no single-world interpretation can be logically consistent. This conclusion extends to deterministic hidden-variable theories, such as Bohmian mechanics, for they impose a single-world interpretation. 
The article contains an experiment based on Wigner's friend thought experiment, from which is deduced in a Theorem that there cannot exist a theory T that satisfies the following conditions:
(QT) Compliance with quantum theory: T forbids all measurement results that are forbidden by standard [non-relativistic] quantum theory (and this condition holds even if the measured system is large enough to contain itself an experimenter).
(SW) Single-world: T rules out the occurrence of more than one single outcome if an
experimenter measures a system once.
(SC) Self-consistency: T's statements about measurement outcomes are logically consistent (even if they are obtained by considering the perspectives of different experimenters).
A proof of the inconsistency of Bohmian mechanics (discovered by de Broglie and rediscovered and further developed by David Bohm) would already be a  big deal, because despite being rejected with enthusiasm by many quantum theorists, it was never actually refuted, neither by reasoning, nor by experiment. Bohmian mechanics is based on two objects: the pilot-wave, which is very similar to the standard wavefunction and evolves according to the Schrödinger equation, and the Bohmian trajectory, which is an integral curve of the current associated to the Schrödinger equation. While one would expect the Bohmian trajectory to be the trajectory of a physical particle, all observables and physical properties, including mass, charge, spin, properties like non-locality and contextuality, are attributes of the wave, and not of the Bohmian particle. This explains in part why BM is able to satisfy (QT). The pilot-wave itself evolves unitarily, not being subject to the collapse. Decoherence (first discovered by Bohm when developing this theory) plays a major role. The only role played by the Bohmian trajectory seems (to me at least) to be to point which outcome was obtained during an experiment. In other words, the pilot-wave behaves just like in the Many-Worlds Interpretation, and the Bohmian trajectory is used only to select a single-world. But the other single-worlds are equally justified, once we accepted all branches of the pilot-wave to be equally real, and the Bohmian trajectory really plays no role. I will come back later with a more detailed argumentation of what I said here about Bohmian mechanics, but I repeat, this is not a refutation of BM, rather some arguments coming from my personal taste and expectations of what a theory of QT should do. Anyway, if the result of the Frauchiger-Renner paper is correct, this will show not only that the Bohmian trajectory is not necessary, but also that it is impossible in the proposed experiment. This would be really strange, given that the Bohmian trajectory is just an integral curve of a vector field in the configuration space, and it is perfectly well defined for almost all initial configurations. This would be a counterexample given by Bohmian mechanics itself to the Frauchiger-Renner theorem. Or is the opposite true?

But when you read their paper you realize that any theory compatible with standard quantum theory (which satisfies QT and SW) has to be inconsistent, including therefore standard QT itself. Despite the fact that the paper analyzes all three options obtained by negating each of the three conditions, it is pretty transparent that the only alternative has to be Many-Worlds. In fact, even MW, where each world is interpreted as a single-world, seems to be ruled out. If correct, this may be the most important result in the foundations of QT in decades.

Recall that the Many-Worlds Interpretation is considered by most of its supporters as being the logical consequence of the Schrödinger equation, without needing to assume the wavefunction collapse. The reason is that the unitary evolution prescribed by the Schrödinger equation contains in it all possible results of the measurement of a quantum system, in superposition. And since each possible result lies in a branch of the wavefunction that can no longer interfere with the other branches, there will be independent branches behaving as separate worlds. Although there are some important open questions in the MWI, the official point of view is that the most important ones are already solved without assuming more than the Schrödinger equation. So perhaps for them this result would add nothing. But for the rest of us, it would really be important.

My first impulse was that there is a circularity in the proof of the Frauchiger-Renner theorem: they consider that it is possible to perform an experiment resulting in the superposition of two different classical states of a system. Here by "classical state" I understand of course still a quantum state, but one which effectively looks classical, as a measurement device is expected to be before and after the measurement. In other words, their experiment is designed so that an observer sees a superposition of a dead cat and an alive one. Their experiment is cleverly designed so that two such observations of "Schrödinger cats" lead to inconsistencies, if (SW) is assumed to be true. So my first thought was that this means they already assume MWI, by allowing an observer to observe a superposition between a classical state that "happened" and one that "didn't happen".

But the things are not that simple, because even if a quantum state looks classical, it is still quantum. And there seem to be no absolute rule to forbid the superposition of two classical states. Einselection (environment-induced superselection) is a potential answer, but so far it is still an open problem, and at any rate, unlike the usual superselection rules, it is not an exact rule, but again an effective one (even if it would be proven to resolve the problem). So the standard formulation of QT doesn't actually forbid superpositions of classical states. Well, in Bohr's interpretation there are quantum and there are classical objects, and the distinction is unbreakable, so for him the extended Wigner's friend experiment proposed by Frauchiger and Renner would not make sense. But if we want to include the classical level in the quantum description, it seems that there is nothing to prevent the possibility, in principle, of this experiment.

Reading the Frauchiger-Renner paper made me think that there is an important open problem in QT, because it doesn't seem to prescribe how to deal with classical states:
Does QT allow quantum measurements of classical (macroscopic) systems, so that the resulting states are non-classical superpositions of their classical states?
I am not convinced that we are allowed to do this even in principle (in practice seems pretty clear it is impossible), but also I am not convinced why we are forbidden. To me, this is a big open problem. Can the answer to this question be derived logically from the principles of standard QT, or should it be added as an independent, new principle?

My guess is that we don't have a definitive solution yet. It is therefore a matter of choice: those accepting that we are allowed to perform any quantum measurements on classical states, perhaps already accept MWI, and consider that it is a logical consequence of the Schrödinger equation. Those who think that one can't perform on classical states quantum measurements that result in Schrödinger cats, will of course object to the result of the paper of Frauchiger and Renner and consider its proof circular.

I will not rush with the verdict about the Frauchiger-Renner paper. But I think at least the open problem I mentioned deserves more attention. Nevertheless, if their result is true, it will pose a big problem not only to Bohmian mechanics, but also to standard QT. And also to my own proposed interpretation, which is based on the possibility of a single-world unitary solution of the Schrödinger equation (see my recent paper On the Wavefunction Collapse and the references therein).

Monday, May 2, 2016

An attempt to refute my Big-Bang singularity solution

I learned recently about a paper which attempts to refute one of my papers. While being sure about my proofs, I confess that I was a bit worried, you never know when you made a mistake, a silly assumption that you overlooked. But as I was reading the refutation paper, my worries dissipated, and were replaced by amusement and I actually had a lot of fun. Because that so-called refutation was something like: "I will refute Pythagoras's Theorem by showing that it doesn't apply to triangles that are not right."

My paper in cause about Big-Bang singularities is arXiv:1112.4508 (The Friedmann-Lemaitre-Robertson-Walker Big Bang singularities are well behaved). As it is known, the main mathematical tool used in General Relativity is semi-Riemannian geometry, and this works only as long as the metric is regular. The metric ceases to be regular at singularities, but I developed the extension of semi-Riemannian geometry at some degenerate metrics, so it applies to a large class of singularities, in arxiv:1105.0201. And this allowed me to find descriptions of such singularities in terms of quantities that are still invariant, but as opposed to the usual ones, they remain finite at singularities. More about this can be found in my PhD thesis arxiv:1301.2231. In the paper arXiv:1112.4508, I give a theorem that shows that, if the scaling function of the FLRW universe is smooth at the Big-Bang singularity, then I can apply the tools I developed previously, and get a finite description of both the geometry, and the physical quantities involved.

The paper attempting to refute my result is arxiv:1603.02837 (Behavior of Friedmann-Lemaitre-Robertson-Walker Singularities, by L. Fernández-Jambrina). Both my paper and this one appeared this year in International Journal of Theoretical Physics. I think F-J is a good researcher and expert in singularities. But for some reason, he didn't like my paper, and he "refuted" it. The "refutation" simply takes the case that was explicitly not covered in my theorem, namely when the scaling function of the FLRW solution is not derivable at the singularity, and checks that indeed my tools don't work in this case. Now, while my result is much more humble than Pythagoras's Theorem, I will use it for comparison, since it is well-known by everybody. You can't refute Pythagoras's Theorem by taking triangles that are not right, and proving that the sum of squares of two sides is different than the square of the third. Simply because the Theorem makes clear in its hypothesis that it refers only to right triangles. My theorem also states clearly that the result doesn't refer to FLRW models whose scaling function is not derivable at the singularity. And F-J even copies the Theorem's enounce in his paper, so how could he miss this? So what F-J said is that my theorem can't be applied to some cases, which I made clear that I leave out (I don't claim my theorem solves everything, neither that it cures cancer). Now, is the case when the scaling function is not derivable important? Yes, at least historically, because some classical solutions fit here. But the cases covered by my theorem include what we know today about inflation. So I think that my result is not only correct, but also significant. In addition to this, F-J says that I actually don't remove the Big-Bang singularity. This is also true, and stated in my paper from the beginning. I don't remove the singularities, I just try to understand them to describe them in terms of finite quantities that make sense both geometrically and physically. But he wrote it as if I claim that I try to remove them and he proves that I don't, not that I accept them and provide a finite-quantities description of them.

Sunday, March 27, 2016

Faster than light signaling leads to paradoxes

You may have encountered statements like this one made by Sabine:
Once you can send information faster than the speed of light, you can also send it back in time. If you can send information back in time, you can create inconsistent histories, that is, you can create various different pasts, a problem commonly known as “grandfather paradox:” What happens if you travel back in time and kill your grandpa? Will Marty McFly be born if he doesn’t get his mom to dance with his dad? Exactly this problem.
This is correct. Special relativity implies that, if faster than light signaling would be true, you would be able to signal to your own past, and this can lead to paradoxes. Here I will explain how exactly this can happen. This is rather elementary special relativity stuff, but I realized there is much confusion around it. First, I never saw a precise scenario in which faster than light (FTL) signaling can be used to signal back to your own past, so I will give one. Second, I have the feeling that when people make statements like this,
  • they either refer to the fact that, if an observer A sends FTL signals in her own future, for another observer B it may look like sending in back in time, in B's reference frame, as in this figure:
Orange lines represent light cones, blue represent timelike curves (observers), red represents the proper space of an observer, and green represents FTL signals. While the picture represents the proper space of A as a horizontal red line, the proper space of B is oblique, due to the Lorentz transformation (relativity of simultaneity).
The first scenario is not that paradoxical, because observer B can always reinterpret the signal from A to B as a signal going in his own future, from B to A. But even in this case, we will have the problem of who actually created the message in the first place.

  • or they refer to examples where the observer sends an FTL signal toward her own past, as in this figure:
The second scenario is the usual example of causality violation due to FTL you will find, but is refutable on the grounds that you are not allowed to send signals directly to your own past, or to receive signals directly coming from your own future.

Here is how FTL signaling would imply that one can signal back in time, using only signals sent in the future and received from the past, with respect to the proper reference frame:

The inertial observer A accelerates away from B, then sends an FTL signal at t₀. Observer B receives it at t'₀ in his proper time, then accelerates away from observer A, then sends it back, at t'₁. Observer A receives the signal at t₋₁, where t₋₁< t₀. 
So indeed FTL implies signaling back in your own past, even if FTL signals are sent only to the proper future and received only from the proper past.

Let us see how this allows paradoxes. Suppose that earlier A and B agreed on the following: if A receives the message "Yes", she sends the message "No", and if she receives "No", she sends "Yes". If B receives a signal, he just resends it without changing. Then, we have a paradox: does A send the message "Yes", or "No"? It is similar to the liar paradox, since if she sends "Yes", then she receives "Yes", so she sends "No", and so on. But it is also like grandfather's paradox, because B can send instead of a message, a killing FTL ray, to kill A or her grandfather before she was born.

So far there is no evidence of FTL signaling, except for some misunderstandings of the EPR "paradox". I don't know either of a fundamental physical law which prevents it, given that tachyonic solutions are mathematically consistent, both in special relativity, and in quantum field theory. But as we have seen, FTL would lead to time travel paradoxes.

Saturday, February 13, 2016

Gravitational waves, evidence of the fourth dimension of spacetime

Most of the headlines are right: gravitational waves are a long-known prediction of General Relativity, and their detection show that the theory is correct. I waited a bit to see if an important consequence of this fact will be uncovered, but it seems it did not, so let me tell you: This experiment refutes a great deal of alternatives to General Relativity proposed in the last decades. You perhaps already noticed that many physicists brag on social networks or even in online articles that the detection of gravitational waves confirmed not necessarily GR, but also the alternatives to GR they endorse. But in fact this experimental result refutes those alternative theories in which the background metric of spacetime is fixed, as well as those in which space is a three-dimensional thing that is not part of a four-dimensional spacetime, as in GR is. I will discuss first the latter. Many relativists would say that such theories were already refuted, but if you talk with a supporter of such a theory, you will hear that it is not necessarily so. The idea of a 3-dimensional space still could be defended, with the price of complicating the things. But in my opinion, LIGO just put the last nail in the coffin of such theories. Because gravitational waves are waves of spacetime, and not of space. They are waves of the Weyl curvature tensor, which simply vanishes in a space with less than four dimensions!

The number of those trying to replace GR with other theories increased very much lately. The main reasons may be that they don't know how to handle singularities, or that they don't know how to enforce to gravity the few methods we know to quantize fields, so they come up with alternative theories. While I don't think it is easy to replace GR with something that explain as much starting from as little as GR does, I agree that these alternative should be explored (by others, of course). Related to whether there is a 3-dimensional space or a 4-dimensional spacetime, you can find reasons to doubt the fourth dimension too. First, even Galilean space and time can be joined in a four-dimensional spacetime, but not as tight as in Relativity. In Relativity, indeed, Lorentz transformations mix the time and space directions, leading to length contraction and time dilation, but some think that these are sort of due to the perspective of the observer, without needing a fourth dimension. In addition, many quantities become unified in the four-dimensional spacetime, such as energy and momentum, electric and magnetic fields etc. But maybe these are all just circumstantial evidence of the fourth dimension. You can take any theory and make it satisfy some four-dimensional transformations. Especially since the evolution equations are hyperbolic, you can do this. Also, you can express any equation in Physics in curvilinear coordinates, and this doesn't mean four dimensions, neither that the invariance to diffeomorphisms means something physical. So people cooked up or even revived various alternatives to GR, in which three-dimensional space is not part of a spacetime. If such a theory does not include curvature, it will not predict gravitational waves. Also, if it admits curvature, but only of the three-dimensional space, nothing in four dimensions, it still doesn't predict gravitational waves out of this curvature. So now the proponents of alternative to GR will have to adjust their theories. Maybe some predict naturally some sort of gravitational waves, but most don't, so they will put the waves by hand. The Cotton tensor, which is somewhat analogous to the Weyl tensor in three dimensions, because its vanishing means conformal flatness, is believed sometimes to give the gravitational waves. But the Cotton tensor vanishes in vacuum, where the Ricci tensor vanishes too. So this can't give gravitational waves in three-dimensional spacetime.

What about theories with more dimensions? For instance, Kaluza theory is an extension of GR to 5 dimensions, which is able to obtain the sourceless electromagnetic field from the extra dimension. You can also obtain other gauge theories as Kaluza-type theory. Such modification predict gravitational waves too.

What about String Theory? It is said that String Theory includes GR, so it must include gravitational waves too, isn't it? But the reason why is said to include GR is because it contains closed strings, which have spin 2, and they are identified with the still hypothetical gravitons (not even predicted by GR alone) just because they have spin 2. But if your theory has spin-2 particles, even if you call them gravitons, it doesn't mean you have included GR. String Theory usually works on fixed background, which usually is flat, or with constant curvature as in the anti-de Sitter spacetime. I am not aware of a successful way to include GR in String Theory such that gravity is an effect of spacetime curvature. If this can be done, can it predict gravitational waves in a natural way? Can it even include GR in a natural way?

To my surprise, the advocates of theories which don't have dynamical background, or are based on three-dimensional space, didn't take the chance to predict that there are no gravitational waves, as their theories imply. They should have done this, and they should have waited for the confirmation of their prediction by LIGO. My guess is that maybe they doubted that GR will be refuted - nobody wants to make predictions which contradict GR in regimes that can be experimentally verified. Whenever we could test the predictions of GR, they were always confirmed, so I think not even those supporting alternative theories actually believed that it will be refuted this time. So I guess that's why they didn't say that their theory predicts no gravitational waves, and that they really think that LIGO will show there are. Instead, now you can see that some claim that gravitational waves confirm their theories too. Like for examples they are waves of space alone, and not of spacetime, which is not true, unless you put them in your theory by hand (while in GR they are just there, not a mobile or replaceable part). So I expect to see a lot of papers in which it is explain that their theory was there too, along with GR, when gravitational waves were predicted.

Since the model was based on calculations made using GR applied to two colliding black holes, LIGO confirmed GR (again): it confirmed gravitational waves, and black holes (again). This does not exclude though the possibility that other modifications, alternatives or extensions of GR can work out similar predictions. So further experiments may be needed. But what I can say is that the theories that remained are modifications of GR that still explain gravity as spacetime curvature, and still make use of the four-dimensional spacetime. Theories that at purpose mimic most of GR.

Space is dead, long live spacetime!

Tuesday, January 12, 2016

Wavefunction collapse vs. unitary evolution, superdeterminism vs. free-will

Today appeared Quanta's special issue dedicated to Feynman. It is a very cool new open access journal on Quantum Mechanics.


I am happy because it contains my article, On the Wavefunction Collapse, edited by two excellent quantum theorists, Eliahu Cohen and Matt Leifer. In the paper, I discuss the possibility that the unitary evolution, governed by Schrödinger's equation, allows for the apparent wavefunction collapse. I first wrote about this idea some years ago, and its implications on free-will triggered some interesting developments. There are several great difficulties with this, mostly due to the fact that quantum measurement introduce strong constraints on the solutions of Schrödinger's equation. But I hope my arguments that these constraints are not incompatible with unitary evolution are more convincing now. The article had three completely different versions. The first one was based on integral curves in the configuration space, those called by some Bohmian trajectories. I consider the idea of interpreting these integral curves as point-particles interesting, but in order to survive, Bohmians had to transfer more and more of the physical properties initially attributed to point-particles moving along these trajectories to the pilot wave, and I think that eventually only the pilot wave matters. So in fact the wavefunction does all the job. The second version of my paper was based on Feynman's path integrals, but I realized that my original approach, to use Schrödinger's equation, is better suited.

Note that unitary evolution is deterministic. Moreover, trying to assign reality to the wavefunction leads to non-locality, as Bell's Theorem shows, and to contextualism, as the Bell-Kochen-Specker Theorem shows. And last year I published a simple proof that maintaining unitary evolution implies very fine-tuned initial conditions of the observed system and the measurement apparatus. This amounts to what is called superdeterminism. But since nobody can see the complete initial data of the wavefunction, it is also possible to consider that the initial conditions are initially not fixed, and they are more and more constrained with each measurement. While superdeterminism forces us to admit that the property we will choose to measure one day was determined from the Big-Bang, leaving the initial conditions free, and fixing them with each measurement, allows us to choose freely what to measure. And this doesn't break causality, because you can't change the observed past, only the "yet undecided" past. The required consistency between the initial conditions can also be seen, when thinking in terms of the four-dimensional block world picture from Relativity, as a global consistency principle, where "global" refers to the entire spacetime. So we have a timeless picture, based on the block world, but which does not contradict free-will, and a temporal picture, based on the delayed choice of initial conditions. These two pictures provide alternative interpretations of superdeterminism which are compatible with free-will (whatever "free-will" means).

But if there is such a thing as free-will, the free agent should at least partially be somehow above the world and outside of time, to be able to choose among the possible deterministic solutions describing the world itself. Because if it would be completely part of the solution, it could not have free-will. It is easy from here to speculate about an immortal soul and even the possibility that it is part of a supreme being, and I don't want to do this, especially since I consider myself free-will-agnostics. However, this implicit connection may be the reason why so many people are firmly either for, or against free-will.

Completely independent on this, yesterday, Sabine Hossenfelder wrote on her blog a post called Free will is dead, let’s bury it, in which she made some strong affirmations against free-will and people who believe in it. That free-will is bad science. I think that we know too little about this to call it science, but this can be said also about many things which we know exist and we would want to understand better. Then she said that people who believe in free-will have existential worries and hidden agendas. I agree that when we speak about believing in something, even in a physical law, we arrive at that belief in part because of our past experiences. Otherwise, how can we explain that people can change their opinion even about physical laws? So indeed, subjectivity is involved, but this happens all the time, not only with respect to believing in free-will. Then she said "I am afraid the politically correct believe in free will hinders progress on the foundations of physics". I think that if physicists reject their peers' papers or throw away their own results for not being consistent with free will, this is rather the exception, and they do this for many other reasons, including sex, race, or simply because they have different views. At the end, she wrote "buying into the collapse of the wave-function seems a small price to pay compared to the collapse of civilization". This is a nice pun, but quantum theorists who believe in collapse do so because they can't make sense of the outcomes of measurements without collapse, not because they want to support free-will. Many of them don't even believe in free-will, while others don't believe in collapse, but still don't reject free-will. But the reason they don't accept easily alternatives to QM (in particular hidden-variable superdeterministic theories and my unitary collapse approach) is simply that standard QM works much better, and not because they want to save their illusion of free will.