Friday, March 10, 2017

The Tablet of the Metalaw

This edition of the FQXi essay contest is called Wandering Towards a Goal. My entry is called The Tablet of the Metalaw. This is the abstract:

Reality presents to us in multiple forms, as a multiple level pyramid. Physics is the foundation, and should be made as solid and complete as possible. Suppose we will find the unified theory of the fundamental physical laws. Then what? Will we be able to deduce the higher levels, or they have their own life, not completely depending on the foundations? At the higher levels arise goals, life, and even consciousness, which seem to be more than mere constructs of the fundamental constituents. Are all these high level structures completely reducible to the basis, or by contrary, they also affect the lower levels? Are mathematics and logic enough to solve these puzzles? Are there questions objective science can't even define rigorously? Why is there something rather than nothing? What is the world made of?

At this time (2017-03-11 08.59 AM ET) my essay is in the top position, so I will immortalize this ephemeral moment in the picture below, since I expect the order will change dramatically, given that the votes will continue for nearly a month, and then the FQXi panel will add their choices:

Wednesday, February 15, 2017

The Standard Model Algebra

arXiv link:
A simple geometric algebra is shown to contain automatically the leptons and quarks of a generation of the Standard Model, and the electroweak and color gauge symmetries. The algebra is just the Clifford algebra of a complex six-dimensional vector space endowed with a preferred Witt decomposition, and it is already implicitly present in the mathematical structure of the Standard Model. The minimal left ideals determined by the Witt decomposition correspond naturally pairs of leptons or quarks whose left chiral components interact weakly. The Dirac algebra is a distinguished subalgebra acting on the ideals representing leptons and quarks. The resulting representations on the ideals are invariant to the electromagnetic and color symmetries, which are generated by the bivectors of the algebra. The electroweak symmetry is also present, and it is already broken by the geometry of the algebra. The model predicts a bare Weinberg angle θW given by sin2(θW)=0.25.

Wednesday, September 7, 2016

Quantum God (short story)

(link to pdf version
(link to Italian version, translation by Erica Mannoni)

2033 AD. The entire population of the planet was watching, most of them through the eyes of the media, waiting for Lord Q to perform a miracle and save the world. Thousands of people gathered around his tent, meditating, praying, praising him, and hoping for the miracle. The asteroid was heading toward the Earth. All previous attempts to destroy the asteroid failed, because it was a black hole. It was detected only by the way it bent the light and the trajectories of other asteroids in the Solar System. So the asteroid continued undisturbed to threaten the Earth, and Quentin, named by his followers Lord Q, was the only hope.

* * *

Roxanne joined Quentin’s group one year before, not as a believer, but as one of the last skeptics alive, by now a dying species. In the previous decades, scientific and technological progress continued to hunt God into the farthest and most obscure explanatory gaps, into oblivion. Until the emergence of Lord Q three years earlier, when everything was turned upside-down. Since then, he performed the most scientifically incredible miracles, normally attributed to a deity. Roxanne, a reputed physicist with a hobby of debunking pseudoscience, received a grant from a philanthropist who asked her to either find a scientific explanation for Quentin’s miracles, or prove that they are authentic.
When she joined the group, the believers disliked her for her skepticism, which remained unchanged even now, a year later. The only reason they tolerated her was because Quentin seemed to have a strange affection for her. She was allowed to be in his proximity all the time, and this made them dislike her even more, but they had to accept her. Quentin liked her, and was continuously amused by the suspicious look in her eyes, which was visible even when she was surprised by his miracles. For a year she followed him everywhere, witnessed him healing people, stopping natural catastrophes, wars, crime, and bringing back faith. She even saw him bringing back to life the president of the United States, killed by a rare form of cancer. But she continued to say that there must be a scientific explanation for everything.
Quentin never ceased to be intrigued by her disbelief and continued to watch her reaction as he performed his miracles. Once, he started to make flowers grow up out of nowhere and blossom in seconds, covering every piece of ground where Roxanne stepped. She was surprised, she blushed, and she told him that this is harassment. He didn’t know whether she was joking, so he stopped. It would have been the easiest thing for him to make her fall in love with him or even become a believer, but he didn’t want it to be like this. He loved her for being independent, and he would have never traded the chance to see her surprised and at the same time unimpressed by his powers for having an obedient Roxanne in place, a fervent adorer like the rest of his followers.

* * *

As Roxanne was waiting, worried to see what Quentin will do about the asteroid, Tom approached her.
– He will make it, don’t worry, he said.
– I know, she replied.
Tom was the first and only friend Roxanne made among Quentin’s followers. He used to be the leader of the third group of scientists sent by the James Randi Educational Foundation. The group tried to debunk Quentin’s miracles and find mistakes in the reports of the previous two groups, which declared that this was indeed the first authentic miracle recorded by the foundation. He saw no other choice but to accept Quentin’s miracles as true. The Foundation had to award the Psychic Prize for the first time in the history. They offered Quentin one billion dollars, which was the value of the prize at that time. Quentin donated the money to charity, of course. Tom became a believer, and in fact the most ardent one, given that he really looked for the smoke and mirrors and didn’t find any.
During one of their first conversations, Tom told Roxanne that he believed that all these miracles can be explained if Quentin had the ability to control quantum probabilities.
The behavior of particles is governed by quantum mechanics, which is very different from what we observe in our day-to-day life where things behave more like in classical mechanics. According to quantum mechanics, there is always an infinitesimally small probability for something apparently impossible for us to happen even in the real life. In the quantum world, if you observe that a particle – for example an atom – is in a small region of space, at a later time there is a small, but non-zero chance to find it in any other region of space. The reason is that quantum uncertainty makes the particle you know as being in a certain place to have an undetermined velocity, and therefore be able to move anywhere. Hence, at a later time, the particle will potentially be in all places simultaneously. When you observe it again, you will find it in one of these potential places. You can never know where it will be, only know the probability to find it in a given place. This probability is given by a formula called the Born rule. The probability to find the particle in a given place is small, nearly infinitesimally small, but not zero. The same works for more particles or atoms. It’s true that the probabilities become smaller and smaller as the number of involved atoms increases, but it never truly vanishes.
So Tom told Roxanne that he thinks Quentin performs his miracles by selecting which of the potential positions of a particle becomes true. If he can make a particle be where he wants, he can move objects. He can rearrange matter at will. Roxanne remembered that the neuroscientist Adam Hobson used to entertain similar ideas some years ago, but she always considered him a crackpot. She said:
– This happens only for quantum measurements, while Quentin, through his senses, makes classical observations, just like any of us. But, given that any observation we make is eventually a quantum one, maybe your hypothesis is true. However, the probability to control like this even a grain of sand is almost zero.
Tom said:
– Well, the chances are one in a billion of billions of billions… whatever, let’s just say a chance in a gazillion – but Quentin seems to bring those odds into existence.
– But even if he could do this, how can he influence larger objects, which behave classically due to decoherence?
Tom said:
– Decoherence suppresses the probability that the object behaves in a quantum way, but that probability never becomes truly zero. So there is always a very small chance that even larger objects behave in a quantum way, and apparently Quentin has a way to make this chance happen.
Roxanne replied that quantum probabilities are just those the Born rule says they are, and she doesn’t believe anyone can really break this law even a bit. So she can’t accept Tom’s suggestion, especially since it would mean Quentin breaking them. Tom said that for him the alternative is even worse because otherwise Quentin would break other laws, which are exact.
– Forced to choose between an exact physical law and a probabilistic law, Tom said, I would choose to sacrifice the probabilistic one.
– What about the many-worlds interpretation? she asked.
According to the many-worlds interpretation, every possible alternative result of a quantum observation is realized in an alternative world. This way, all possibilities already existing before the observation continue to exist in independent worlds, as if the world splits in many alternative histories.
Roxanne continued:
– Assuming the many-worlds interpretation is true, if Quentin’s miracles are explained because he controls the probabilities, this means that in the vast majority of the alternative worlds he doesn’t control them, just like any of us. And even for us, there is a very tiny chance that the possibility we wish becomes true, but that chance is so small, that it practically never happens. And even if miracles are just very improbable but still possible events, the Born rule has to remain valid in each of the alternative worlds. So the chances that Quentin remains in a world in which he always gets to make his miracles are the same – one in a gazillion.
Tom said he wants to think about this. Next day he said:
– What if he suppresses the possibility of the other worlds when he controls the probabilities?
Roxanne said she has to think about it.
Together, Roxanne and Tom analyzed every miracle made by Quentin, and indeed found that these could be achieved if Quentin had the ability to control quantum probabilities. Healing people, levitating, moving objects, all these seemed to them plausibly explainable if he would really control the quantum probabilities for the particles constituting the objects. But she was still not satisfied, she wanted to know how he does all these, and whether he really breaks the Born rule. She was sure that there must be a better explanation.

* * *

In the last minutes before the asteroid was about to hit Earth, Quentin got out of his tent holding a TV set. He put the TV on a table as the TV anchor was reporting the most recent news about the asteroid. Quentin sat on the grass and started to meditate. After several minutes, his peaceful face, his entire body started to glow. He began to levitate. He raised his eyes to the sky, then his hand, and smiled warmly. Shortly after, the anchor reported in an explosion of joy that the black hole changed its place, and it was no longer a threat to Earth. After a worldwide celebration, life on Earth continued to exist as before. 

* * *

A few weeks later, one night, Quentin was walking with Roxanne. After a full day of prodding him with all kinds of devices, she kept asking him all sorts of questions. He told her with a comforting voice to relax and just enjoy the night. Suddenly, she saw the stars moving in the sky, until they formed the image of her face. Really terrified, she yelled:
– Why did you do that?
Confused, he asked her what the problem was. She said:
– You just moved thousands of stars to impress me by drawing my portrait!
– So what? he said.
– You probably just killed dozens of civilizations!
She looked again in the sky, and the stars were back to their usual positions. He laughed:
– It was an illusion. I didn’t rearrange the stars in the sky, apparently I just bent the light rays coming from them. Or maybe I moved them back, I’m not sure…
They laughed, but she was still frightened.

* * *

For days, Roxanne kept studying Quentin with various high-tech devices. The money was not a problem for her sponsor. She scanned him, monitored his brain activity, recorded everything, and sent the data to specialized laboratories, for more thorough analysis. She found that he had a device implanted in his brain, following a car crash that happened three years earlier. Quentin refused to talk about the implant. But the implant seemed to do nothing relevant that would explain his abilities. It was simply a device that monitored his brain activity, collecting data from a number of places on his cortex. The implant was also stimulating some regions of his brain from time to time, but nothing relevant.
She also found something that surprised her even more: everywhere in Quentin’s body, there were billions of tiny spheres. She didn’t see them initially, they were too small, but she eventually found them after a more detailed analysis of Quentin’s tissues. She collected several of them and sent them to a laboratory. Just like the implant in his brain, the spheres seemed to be useless, or at least they didn’t exchange energy or information with the body at all, so she was very curious about their role.
When the result came back, she was perplexed. The tiny spheres were nanobombs, this was their only functionality. Remote controlled nanobombs, programmed to blow if they receive a certain signal, but obviously never detonated because that signal was never sent. Why on Earth would he have an implant that collects brain activity and never does anything with it, and why have billions of nanobombs also doing nothing at all? Anyway, the presence of nanobombs prevented her from trying to disable Quentin’s brain implant to see if it was the source of his powers.
* * *
Roxanne told Quentin what she found, and insisted that he should give her more explanations. He said that he will tell her more, if she promised to keep the secret.
– I have something to confess, he said. I am not the first one with these powers. Professor Adam Hobson, my uncle, who saved me after the car crash, he used to have them as well.
– You mean Adam Hobson, the guy with that crazy theory about the quantum brain, who disappeared a few years ago and was never found? Roxanne said.
Quentin told her how he had the car accident, and Adam saved his life with his highly advanced surgical robots. He said he had to implant a device in Quentin’s brain. Then Adam personally conducted the recovery therapy, teaching Quentin how to gain control of his body again and how to control his thoughts. After the recovery, Adam revealed more to Quentin. He said that both of them had implants, and that the implants allowed their minds to control matter.
Quentin continued:
– Uncle Adam taught me how to make miracles, by wishing them and by thinking at the changes I should be observing in the world after every miracle. He told me that he can already do this, and I will soon be able to do it too, and that both of us are godlike beings. He said there is no room for two gods, and that he will soon leave this world for a better one. He also said that I will leave this world soon too, and to tell everyone who cares about me that I will go to a better world. Then Uncle Adam activated my implant, and then he vanished in a bright explosion. It was the last time I saw him. I don’t know how this device functions. I just wish for things to happen, visualize what to expect once they happen until I have that feeling that they will do, and then they happen exactly as I visualized them. I learned that this way I can do pretty much anything.
Roxanne said:
– But this doesn’t make sense. The implant doesn’t do anything, it just collects your brain activity and stimulates it to make you feel happy. I just don’t understand…

* * *

Quentin was doing his morning meditation, when Roxanne came with a desperate look on her face, yelling from afar:
– You have to stop doing any miracle right now! This is gonna kill you!
– What?! said Quentin. What do you mean? What happened?
– You know Tom’s hypothesis that the way you do your miracles is by controlling quantum probabilities? Well, you can’t control them, nobody can!
– OK… so what? Quentin said.
– I know what’s in your head, what that implant does to you, she said.
– Well, good to know you finally got it. I’m all ears. Sit down here with me…
Roxanne caught her breath, but didn’t sit on the grass near Quentin. Instead she kept circling him, explaining:
– Whenever there’s a choice between more quantum alternatives, new worlds are created, in which each of these possibilities become reality. You can’t control which world is ours, you exist in all of them. Including in those in which your miracle doesn’t work. But when you make a miracle, you visualize the desired result, and your brain implant collects this information from your brain. Then, it compares it with what you observe afterwards. If your wish becomes real, then nothing happens.
– I don’t get it, Quentin said. Nothing happens, so the device does nothing. Then how can this explain anything?
Roxanne grabbed his shoulders and looked at him frantically:
– But if your wish doesn’t come true, which is almost always the case, then the implant detonates the billions of bombs in your body. You explode into a bright light, and you die.
– I never died…
– You will always find yourself, of course, in a world in which you are not killed, hence where your wish came true. Your implant is a quantum suicide device, inspired by the quantum suicide thought experiment proposed in the eighties to test the many-worlds interpretation. Gazillions of worlds are created whenever you make a miracle, and gazillions of copies of you are killed in all of these worlds! Gazillions of copies of us are left in tears… In all worlds, except in those very, very, very rare in which your wish comes true!
Seeing her crying, Quentin dispersed the clouds in the sky and made out of thin air a rain of flower petals.
– See, nothing happened, dummy…

Cristi Stoica, May 17, 2016

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.