Monday, January 8, 2024

Is your mind just a computation?

I made three videos, 46' together, about consciousness and computation.

In this series in three parts:
Can a computer have its own mind? Is your mind just a computation? We will see what Computer Science has to say. Don't worry, it's beginner level! DIY experiment so that you can verify what I say. The proof appeals to logic and experiment, not to phenomenal experience ("what is like", the "hard problem of consciousness", qualia, the experience of feelings, emotions, pain or pleasure etc) Based on my paper "Does a computer think if no one is around to see it?"  

In Episode 1, we will see that what we call computation is a convention, and it can be chosen in numerous ways.
 

 

In Episode 2, we will make an experiment to see that what we call computation is a convention, and it can be chosen in numerous ways. We will explore some implications.
 

 

In Episode 3, we will see that there is a way to know if your mind is just a computation.




Wednesday, November 29, 2023

Roy Kerr vs. the singularities

This preprint by Roy Kerr should be a hit (but I bet it will be ignored!) .

Kerr (yes, who found the well-known Kerr black hole solutions) disagrees with Penrose's singularity theorem and its variations. Namely these theorems prove the existence of geodesics that can't be extended beyond a finite affine length, but Kerr finds numerous examples of inextensible light rays that don't contain singularities. These geodesics go all the way to the null infinity, and yet the affine parameter remains finite. And there are such light rays through every point of the Kerr spacetime. Only some geodesics hit the ring singularity, but this region can be replaced by a nonsingular one, perhaps matter can do this. Kerr thinks that his perfectly symmetric vacuum solution doesn't happen in reality (despite the "no-hair theorem", which is in fact a conjecture improperly called "theorem", stating that all black holes evolve into a Kerr solution), even though he thinks that black holes exist.

Now, how is this possible? I mean the singularity theorems, now sealed forever by a Nobel prize, prove that there are conditions that necessarily lead to singularities. That if there's a black hole, there must be a singularity beyond its horizon. Or do they?

This is a bit of a word play. There are more meanings of the word "singularity". Normally singularity means a place where the metric blows up. Or its inverse. Or the curvature, or any field that we think it's physical. But then we can think of excluding these points from spacetime. If these points are "in the way" of the physical fields, if the evolution equations can't go beyond such a place but they should, this would be a problem even if we exclude them from spacetime. But if these singularities are somewhere at the "edge" of spacetime, and the spacetime admits a nice foliation so that the evolution equations work fine across the entire spacetime, why would this be a problem? And yet, the other definition of singularity, the one that is actually the object of the singularity theorems, includes such cases as well. That is, as a diagnostic method, it gives numerous false positives.

Here's what happened. And I don't say it's a plot against General Relativity, rather an accident, perhaps welcomed by many. If your spacetime contains singularities, we can think of excluding them from spacetime. But this, as I said, doesn't solve the problem. So maybe there is a way to detect this pathology even with the singularities removed, and talk about such a spacetime as being singular anyway. And here comes into play the redefinition of singular spacetime in terms of geodesic incompleteness. And it is said in the Hawking & Ellis bible, on page 258: 

I don't want to single out this great book, it explains well the adoption of this diagnosis, and others said similar things. But here I think lies the problem. Because this definition can be misunderstood (unintentionally I think) in a way that makes the singularity theorems seem about singularities even if there are no singularities in the interior of spacetime, even if the spacetime can be nicely foliated, offering a nice home to the evolution equations.

The singularity theorems prove (and they indeed prove this) that there are incomplete geodesics, where incomplete means they can't be extended beyond a finite affine length. Whether all of them deserve to be called "incomplete" is also questionable. If the affine length (which is not the same as geometric length anyway) of a timelike or null geodesic is finite, but it goes to the "real edge" of spacetime, as in Kerr's paper, why should it be called incomplete? This already seeds in our minds the idea that there's something wrong with them.

So, one on top of another, the meaning of words shifted so that now it's widely believed that General Relativity breaks down, due to the singularities. And Kerr gives nice rich counterexamples, all in the same spacetime of a Kerr black hole. I mean, his spacetime has a singularity, but the singularity theorem doesn't even predict that singularity. It predicts some singularities, but they are false positives, they don't occur on the geodesics up to the boundary of spacetime. It doesn't predict the ring singularity, because, as Kerr says, there is no trapped surface inside the inner horizon of the Kerr black hole. So, if we cut out the spacetime around that ring, and replace that region (and the "other universe" beyond the ring) with one without singularities, we get a spacetime without singularities (and from what we know matter may do this), and yet the singularity theorems as usually cited say it has singularities (outside that region)! 

I'd like to add that I was convinced as well, for a long time, that the singularity theorems imply the kind of metric singularities that are problematic. They were the reason why I worked to save General Relativity by reformulating it in a way that doesn't have infinities at the singularities. And I repeated numerous times the claim that the singularity theorems prove that the metric tensor has singularities, assuming that they are of this kind. And I might have regarded people who didn't believe in singularities as, let's say, not very serious. Despite being aware that there was a step in the proof of the singularity theorems that I never understood, namely exactly the step where from inextensibility we conclude the existence of such singularities. Despite never being able to find a place where this step is proved for a limited person like me. And that while knowing that I didn't understand that step, and being limited, I considered that I should trust the experts about it, or maybe just my limited understanding of what experts say. And now, after seeing Roy Kerr's counterexamples, I think I was wrong.

So yes, Kerr is right, to be able to say that General Relativity breaks down because of singularities we need a proof for exactly such singularities, and the singularity theorems alone don't do the job, and there are counterexamples showing this. But of course counterexamples are a no go mainly for the more mathematically inclined (and some of them noticed this at some times, but somehow the most spread interpretation of the singularity theorems remained unaffected). Many physicists may still use the confusion between the two notions of singular spacetimes (assuming they're aware of them) to reject classical General Relativity, and at the same time they would claim that quantum gravity doesn't have this problem, again without proof, without even a theory of quantum gravity! (The only argument is that quantum fields may violate a condition in the singularity theorems, but this doesn't prove that this avoids the alleged singularity)


But what if somebody takes notice now of Kerr's paper, and of the disambiguation of the term "singular spacetime", and finds a singularity theorem, with different conditions evidently, that is actually about such singularities? Even so, General Relativity can be formulated in terms of finite geometric objects, which can evolve beyond the singularities, as I showed some time ago https://arxiv.org/abs/1301.2231. This formulation is equivalent with the usual one outside the singularities, but it extends at the singularities too, at least in the usual cases.

So I see no reason why General Relativity is so often pronounced dead. I mean, sure, we need a quantum theory of gravity, but let's stop throwing the baby with the bathwater. There's no reason to treat like a stepchild one of the two babies, General Relativity and Quantum Theory, and favor the other one. The really naughty one ;)
 

Saturday, April 16, 2022

An underrated gem: WAY beyond conservation laws


I think the article Wigner-Araki-Yanase theorem beyond conservation laws by Mikko Tukiainen is an underrated gem (if we compare its content to the number of citations).


Here's why I think so.

First, I think the Wigner-Araki-Yanase theorem is underrated. It started with a paper by Wigner (here is an English translation.), who showed that you can't have an accurate ideal spin measurement which is also repeatable. By "repeatable" it's understood that, whatever result you get, by repeating the measurement you'll get the same result. In other words, accuracy requires that the measurement disturbs the system, so the spin is no longer what you measured it to be. You can avoid this by being satisfied with a less accurate result. Wigner also showed that repeatability can be obtained and the error can be made as small as wanted, if the measuring device is large enough so that the apparatus has large uncertainty for the conserved quantities.

Araki and Yanase generalized his result, and added some interesting observations, in particular that this limitation applies to the measurement device as well.

Wigner was brilliant enough to know how to give a more general proof, but he wanted the idea to be understood easily. He used the conservation of angular momentum along an axis to deduce the limitation of accuracy of spin measurement along an orthogonal axis. He only uses a conservation law, but all conservation laws contribute. He had to give a simple proof, without making too many assumptions about the evolution equation. So he probably thought, spin measurement is a simple example, and also entails the existence of other spin operators that are conserved by unitary evolution and don't commute with it.

While all conservation laws contribute limitations, on the one hand this is an expression of the symmetries, and on the other hand, in fact, they don't do anything. The limitation is in the transformation of the total state from the state before measurement into the state after the pre-measurement, that is, just before we invoke the collapse postulate (the collapse itself breaks the conservation laws). During pre-measurement the evolution is unitary, because collapse is invoked at the end. The evolution itself constraints the possible results of the measurement. Conservation laws were originally used as indications that we can use to find such limitations. A general proof in terms of general unitary transformations is very difficult, but you can look at a conserved quantity and deduce enough to know that the accuracy is limited if we want repeatability. So the conservation law was used to give a simple, although less general, proof. And to make it simpler, the conserved quantity had to be additive.
 
But these are just assumptions Wigner made to prove the result, and this made me initially think that there is nothing special or metaphysical about conservation laws in this context, despite Wigner's other very important realizations about the role of symmetry. But there is a very important lesson about symmetries (which, as we know from Emmy Noether, are the reason behind the conservation laws), as elucidated by the works of Ozawa, Loveridge, Busch, Miyadera and others.

Conservation laws are often used to deduce things without solving equations. But they don't constrain, they express the constraints of the system, since these constraints restrict the symmetries, and therefore the conservation laws. On the other hand, the symmetries of the system really capture an important aspect of the constraints, as explained in this wonderful article by Loveridge, Busch, and Miyadera.

The reason why I consider Mikko Tukiainen's paper important is that it seems to indicate another deeper aspect, that seems to go beyond that. He not only it gave a more general proof, but in that proof, conservation laws play no role (you can read it for free here). He used instead the idea of quantum incompatibility, which is a way to understand the major features of quantum mechanics that distinguish it from classical mechanics (although the most useful examples are still given by conservation laws). This is neat, complements the idea based on symmetry, and it's in some sense more general.

Both the symmetries and quantum incompatibility go deep, but maybe there is a deeper reason than both of these - the full range of such limitations of measurements is still unknown. And maybe there is no general characterization of this. But anyway, I think there's more to be learned about this.

Since both the WAY papers together have together a relatively small number of citations (hundreds), I consider them underrated too. This is another mystery to me.

Saturday, March 28, 2020

The negative way to sentience (comments welcome!)

I wrote an essay about sentience and its relations to physics. For the moment, I  keep it on ResearchGate, and I am welcoming comments.

The negative way to sentience (comments welcome!)


Abstract. While the materialist paradigm is credited for the incredible success of science in describing the world, to some scientists and philosophers there seems to be something about subjective experience that is left out, in an apparently irreconcilable way. I show that indeed a scientific description of reality faces a serious limitation, which explains this position. On the other hand, to remain in the realm of science, I explore the problem of sentient experience in an indirect way, through its possible physical correlates. This can only be done in a negative way, which consists in the falsification of various hypotheses and the derivation of no-go results. The general approach I use here is based on simple mathematical proofs about dynamical systems, which I then particularize to several types of physical theories and interpretations of quantum mechanics. Despite choosing this scientifically-prudent approach, it turns out that various possibilities to consider sentience as fundamental make empirical predictions, ranging from some that can only be verified on a subjective basis to some about the physical correlates of sentience, which are independently falsifiable by objective means.


Sunday, October 27, 2019

Representation of the wave function on the three-dimensional space

My last paper

Representation of the wave function on the three-dimensional space

One of the major concerns of Schrödinger, Lorentz, Einstein, and many others about the wave function is that it is defined on the 3N-dimensional configuration space, rather than on the three-dimensional (3D) physical space. This gives the impression that quantum mechanics cannot have a 3D space or space-time ontology, even in the absence of quantum measurements. In particular, this seems to affect interpretations which take the wave function as a physical entity, in particular, the many-worlds and the spontaneous collapse interpretations, and some versions of the pilot wave theory. Here, a representation of the many-particle states is given, as multilayered fields defined on the three-dimensional physical space. This representation is equivalent to the usual representation on the configuration space, but it makes it explicit that it is possible to interpret the wave functions as defined on the physical space. As long as only unitary evolution is involved, the interactions are local. I intended this representation to capture and formalize the nonexplicit and informal intuition of many working quantum physicists, who, by considering the wave function sometimes to be defined on the configuration space and sometimes on the physical space, may seem to researchers in the foundations of quantum theory as adopting an inconsistent view about its ontology. This representation does not aim to solve the measurement problem, and it allows for Schrödinger cats just like the usual one. But, it may help various interpretations to solve these problems, through inclusion of the wave function as (part of) their primitive ontology. In appendices, it is shown how the multilayered field representation can be extended to quantum field theory.

https://journals.aps.org/pra/abstract/10.1103/PhysRevA.100.042115