Consciousness and Physics from Scratch

Brain in a squeezed state [Source]

Max Tegmark’s claim that we are all mathematical structures taught me an important lesson: Do not take photos of yourself within 5 minutes of waking up. Since he has influenced my thinking so thoroughly, his newest paper on the physical basis of consciousness was mandatory reading.

Titled “Consciousness as a state of matter”, the paper scores at 30 pages in 10 pt font. The argument has some gaps that are filled with conjectures, but it is an interesting attempt to quantify and formalize the slippery notion of consciousness. I’ll not claim I understood it all, but my below summary should convey the general idea.

The title of Tegmark’s paper is somewhat misleading because except for the rather vague introduction, the idea that consciousness is a “state of matter” is not rigorously pursued. In fact the original title “Space, consciousness and the quantum factorization problem” would have been much more informative if less catchy. I recommend that before you upload your LaTeX file to the arXiv you remove all comments, including discarded title options.

Tegmark’s paper actually tackles two different problems. One is the question what properties a conscious system has and how to formalize them. The other is the question of how to identify macroscopic and mostly classical objects from a fundamental Hamiltonian and wavefunction that describes presumably everything. At least that is my reading of what Tegmark calls the “physics-from-scratch problem” though this left me to wonder where the rest of the mathematical universe has gone. Maybe I should have taken the blue pill.

So let us look at the question of consciousness first.

1. Consciousness

Tegmark builds on defining qualities for consciousness suggested by Giulio Tononi (never heard of him) according to which a conscious system needs to be able to a) store large amounts of information, and b) the information must be “integrated into a unified whole”. I’ll add my comments later, let me just say that though I don’t think these are very useful criteria, at least they are criteria. He later adds three more criteria c) dynamics (time-dependence), d) independence (dynamics is dominated by ‘forces from within’) and e) utility (records mainly information that is useful for it). The latter inches quite close to adaptive systems.

Tegmark then goes on to express the two criteria of information and integration in mathematical form and tries to derive conclusions about the conscious system from this. The approach that he uses is that he assumes the system is fundamentally a quantum system described by a Hamiltonian and a density matrix, and he performs various operations on the Hamiltonian that are supposed to bring it into a form where it is an ‘integrated whole’. For this, he essentially looks for a minimum of shared information between two subsystems under arbitrary unitary transformations. These subsystems are not local in any way, they are generic divisions of the Hilbert space.

He finds that arbitrary unitary transformations can dramatically lower the integrated information in a quantum system, basically by reducing entanglement between any two subsystems. Tegmarks uses a particular conjecture about the eigenvalues of the density matrix to make this point, and while the details may depend on this conjecture I don’t think this will be news for the folks in quantum information. It is basically the idea that Verlinde and Verlinde used in their solution to the firewall paradox, the same idea that I later used in my paper, that unitary operations can ‘disentangle’ subsystems. Tegmark concludes then that we have an “integration paradox […] No matter how large a quantum system we create, its state can never contain more than about a quarter of a bit of integrated information.”

A quarter of a bit is not much and if you can still follow my elaboration it’s probably not enough to explain your brain’s workings, so the criterion of integration does not seem particularly useful. Tegmark thus goes on to amend it by taking into account dynamics, ie the requirement to process information.

Comments: I don’t find it very plausible to require that the degree of integration a system possesses must be found by minimizing over all unitary transformations. Tegmark only acts with these transformation on the density matrix, so I am not sure whether the transformation is supposed to be an actual operation or whether it also should act on the Hamiltonian. In the latter case doing the transformation wouldn’t make a difference to observables, so why look for the minimum? Tegmark unfortunately doesn’t discuss observables at all. In the former case, if the unitary transformation is an actual change to the system then I think one should consider these different systems and again I don’t see why one should look for the minimum.

In any case, let us go on to the next point then, taking into account the dynamics. For this Tegmark now aims at finding a basis in the Hilbert space that minimizes the interaction terms in the Hamiltonian, thus maximizing what he calls separability. This leads to the second topic of the paper.

2. Physics from Scratch

Tegmark interprets the “physics-from-scratch problem” as the question how to identify subsystems of the whole Hilbert space that can be separated as well as possible. These subsystems I believe are eventually supposed to give rise to the neatly separated (and almost classical) objects we experience, not to mention our own brains. He thus sets out to find a basis in which the interaction Hamiltonian between subspaces is minimized.

After another conjecture, this time about the energy eigenvalues of the Hamiltonian, he however finds that the minimal interaction Hamiltonian will always commute with the Hamiltonian of the subsystem, so there isn’t only little energy exchange, but actually none which then creates another paradox: “If we decompose our universe into maximally independent objects, then all change grinds to a halt.” This he finds does not describe reality and concludes “We have tried to understand the emergence of our observed semiclassical world, with its hierarchy of moving objects, by decomposing the world into maximally independent parts, but our attempts have failed dismally, producing merely a timeless world reminiscent of heat death.”

Then he goes on to weaken these requirements.

Comments: Recall that in Tegmark’s reading the physics-from-scratch problem includes the emergence of space and time. If that is so, I know neither what time nor what energy is supposed to mean and I have no clue how to interpret the equations. That there are unitary transformations which lead to a seemingly “timeless” picture is clear because one can shuffle the time-evolution from the wave-function into the operators. That of course does not affect observables, which brings me back to my earlier remark that it doesn’t seem very useful to try to quantify operators when no attention is paid to their expectation values.

Before reading Tegmark’s paper, I would have envisioned the physics-from-scratch procedure as follows. First you need to identify space and time from your Hamiltonian. Space and time are roughly the degrees of freedom that make the rest look as local as possible. Once you have that, you should be able to write down the Hamiltonian in a series of local, or almost local, operators of various dimensions. You need to define a vacuum state, then you can start building your Fock space. The rest is basically effective field theory. That, needless to say, is all “in principle”, not that anybody could do this in practice.

Just why the world we observe contains large things that are almost classical is probably not a question we can answer by looking at the properties of Hilbert-space decompositions in general, but it depends on the specific Hamiltonian. If we didn’t have confinement and if we didn’t have gravity our universe might just be a quantum soup.

After reading Tegmark’s paper, I am even more convinced that locality is a key requirement for the physics-from-scratch problem. Tegmark has some comments on this towards the end of the paper, but believes this requirement to be in conflict with the idea that space-time is emergent. I don’t think so, I think locality is what identifies space-time. Given that the objects that Tegmark wants to identify in the physics-from-scratch procedure are in practice very localized, I’d have expected this to be paid more attention to.

3. Summary

Having said that I don’t think Tegmark’s is a promising approach to the physics-from-scratch problem, let me come back to the topic of consciousness and the main premise that consciousness has to fulfill the above listed five criteria.

To begin with, these criteria I think are in the best case necessary but not sufficient criteria that you may want to look for in some system.

The problem is that “consciousness” is not in and by itself a thing, and it isn’t a state of something either. Consciousness is a noun that is shorthand for a verb much like, for example, the word “leadership”. Leadership isn’t a thing and it isn’t a property, it’s a relation. It’s somebody leading somebody. Consciousness too isn’t a thing, it’s a relation. It’s A being consciously aware of B. (Depending on whether you also want self-awareness B can be identical to A.) We call A conscious if we have evidence it is aware of many B’s. Just how many B’s you want is pretty arbitrary, I think it’s a sliding scale (just think about anesthesia or sleepwalking) and there is no sharp line where something becomes conscious.

Having said that, while I think Tegmark’s paper has some flaws, it is interesting and it provides a mathematical basis for further investigation. With some refinements of the criteria he has applied this can become a very fruitful approach to the physical basis of consciousness. All over the world neuroscientists are presently trying to build and program artificial brains. I am sure this mathematical approach with the possibility of quantification will one day become highly relevant to the study of artificial intelligence. It is a very courageous paper that pushes the boundaries of our knowledge and I hope that it will be influential. I really want to understand consciousness better, and for me the only proper way of understanding is by way of maths.

So what did I learn from this paper? I learned that you should not read papers about the physical basis of consciousness within five minutes of waking up. You might spend the rest of the day staring at your hand, in amazement of the fact that you have a hand, two of them even, and are able to stare, not to mention being able to think about staring. If you’ve stopped staring at your hand, let me know what you think about Tegmark’s idea.