Symmetry and Criticality

2,200 words

Sign up for the Blank Horizons mailing list here.

Header image taken from here.


Enlightenment redux

In a recent blogpost, I speculated that formless consciousness, the rapturous and sublime experience of the dissolution of the boundary between “self” and “other,” is the result of a phase transition in every neuronal network that encodes information about the content of conscious experience. In other words, we can imagine every such network as having an “on-off” switch; in the “on” state, it is generating some form of experiential content, and in the “off” state, it isn’t. Formless consciousness flips on every switch in the brain. Rather than being located at one point in “the state space of human consciousness,” i.e. the (extraordinarily vast) map of all conscious experiences, someone who experiences formless consciousness is located at multiple points at once, or perhaps even all points. (I stress that this discussion is all very speculative, since the science of formless consciousness has barely been developed.)

In this blogpost, I also wrote about the Zero Ontology, which claims that there is a seemingly paradoxical tradeoff between the information content of a system and its repertoire of possible states. [1] In particular, when this repertoire is maximized, the information content drops to zero. As I say in the blogpost,

“[Information is defined as] a reduction in uncertainty. Hence, if I were to specify the position of a rook on a chess board, for example, I would be conveying a nonzero amount (6 bits, in fact) of information. On the other hand, if I were to say that the rook could be anywhere on the chess board, I would be conveying zero information, since I am not reducing any of the uncertainty that someone would have about the rook.

By extension, as Pearce argues, a library that contains every permutation of letters and numbers, also known as the Library of Babel, would have zero information content. This example appears to create a paradox, as the Library of Babel would appear to have maximal information. But in fact, knowing that a book belongs in the Library of Babel doesn’t reduce any uncertainty about the letters and numbers that could appear in the book, since the Library of Babel implies that any configuration of characters is possible.”

Thus, if formless consciousness does turn on every information-encoding switch in the brain, then it is a state of zero information. Andrés Gomez Emilsson, director of the Qualia Research Institute (QRI), has noted that low-information states tend to be highly symmetrical. Symmetry implies that the state of an entire system can be inferred from knowledge of a single part. For instance, if a black-and-white image displays a symmetrical checkerboard pattern, then the color of a single pixel determines the color of every other pixel. But if a single component is sufficient to specify the the whole system, then the system also contains very little information.

The hypothesis that formless consciousness is a highly symmetric state (by virtue of containing zero information) dovetails neatly with QRI’s Symmetry Theory of Valence, which claims, roughly, that the happiest states of consciousness are the ones that display the greatest degree of symmetry. If it is true that formless consciousness is the most ecstatic experience known to humanity, then it follows directly from the theory that the substrate of formless consciousness – whether that’s brain activity or something else – is highly symmetrical.

All of this discussion, however, is abstract and philosophical. How can we rigorously verify the notion that formless consciousness is a maximally symmetric state? The answer hinges on the idea that formless consciousness is produced by a global “explosion” of phase transitions in the brain. In this essay, I will show that the dynamics of a phase transition can be derived mathematically from the symmetries of the corresponding system (in this case, the brain). Through this method, we can precisely measure the symmetries associated with formless consciousness and then compare them to those of ordinary experiences.

Symmetry and phase transitions

When a system undergoes a phase transition, the distinctions between the corresponding phases are sometimes characterized by a difference in symmetry. [2] For instance, consider the phase transition that occurs when liquid water turns into gas at the “critical” temperature of 100o C. As physicist Jeffrey Chang explains, the density of particles is homogeneous in the gaseous phase; “it doesn’t matter where in the box you look, there’s the same chance you’ll find a particle.” The density of the particles remains invariant when their spatial locations change. Hence, the particles exhibit translational invariance or symmetry. (Recall from earlier blogposts that symmetry refers to the general property of invariance under transformation, i.e. the properties of a system that stay the same when the system is changed.) On the other hand, the density of the particles becomes more inhomogeneous when water transitions into the liquid phase, so density varies depending on location. Relative to gas, liquid water displays broken translational symmetry. Thus, gas is the high-symmetry phase, whereas water is low-symmetry.

Mathematically, this relationship implies that the symmetries of the liquid phase form a subgroup of the symmetry group of the gaseous phase. [3] In the gaseous phase, as stated above, the density of water will remain the same no matter how much or where its particles are translated in space. The set of translations under which the density of the gas remains invariant, i.e. the symmetry group of the gas, is infinitely large; it consists of translations 1 unit to the right, 2 units up, 34 units down, and so on. The symmetry group of the liquid phase is smaller, though it is not zero. There are only certain translations that preserve the density of the liquid [4], which constitute a subgroup of the gaseous phase’s symmetry group.

How do we compute the transformations under which the high- and low-symmetry phases remain invariant for any physical system? Landau theory, developed by the Nobel Prize-winning physicist Lev Landau, yields a general method for answering this question. Landau theory seeks to describe changes in symmetry through the order parameter of the corresponding system. As I explained in a previous blogpost, the order parameter is a vector that characterizes the order of a physical system, and it changes, sometimes dramatically and spontaneously, when the corresponding system undergoes a phase transition. In the case of water, homogeneity could serve as an order parameter.

According to Landau theory, the order parameter is an irreducible representation of the high-symmetry phase of the system. Roughly, a representation describes how a group acts on a vector space; it maps the operations of a group onto a series of matrices. For instance, a representation of the group of symmetries of an equilateral triangle essentially amounts to a set of rotation and reflection matrices. (That is, when a vector is multiplied by one of these matrices, it will be rotated or reflected accordingly. I refer readers to one of last month’s blogposts for a very concrete and detailed explanation of the relevant math.) An “irreducible” representation is one that cannot be decomposed further into a direct sum of sub-representations.

From the order parameter, we can compute a set of polynomials that are invariant under the higher-symmetry group. Without getting too deep in the mathematical technicalities, these are polynomials that are generated by multiplying powers of the order parameter by arbitrary coefficients. One such polynomial could be:

p(X) = p0 + p1X + p2X2 + … + pmXm

Eq. 1

where p0pm are the arbitrary coefficients and X is the vector corresponding to the order parameter. The invariance of the polynomials implies that p(X) = p(gX), where g is a matrix that the representation maps onto; in other words, the polynomial stays the same even when the order parameter is multiplied by the representation corresponding to the group operation.

Thus, this method generates the transformations under which the (polynomials encoding the) order parameter remains invariant. How can we apply this method to the study of consciousness? First, we can use our intuition to narrow the range of symmetry-preserving transformations that would be relevant for consciousness. I do not think that the symmetries of consciousness are the same as the symmetries that govern the phenomena to which Landau theory has traditionally been applied, such as crystallization. When a system transitions into its crystalline phase (such as the solid phase of water), the corresponding high-symmetry group is the space group, which is the set of all 219 transformations in space (translations, reflections, rotations, and more) under which a three-dimensional configuration of objects remains invariant. [5] My intuition is that the spatial arrangement of neurons during phase transitions – or whatever units in the brain implement consciousness – has nothing to do with consciousness. In other words, a person’s state of consciousness is likely not affected by changes in the translational or rotational invariances of his neurons; the fact that the translational variance of neuron is constrained to, say, 1 mm in a particular direction after a phase transition should not correspond to a meaningful shift in a person’s conscious awareness. But who knows!

Free energy, psychedelics, and criticality

Robin Carhart-Harris’ Entropic Brain Theory (EBT) may serve as a fruitful launching pad for the discovery of symmetries that encode changes in consciousness. EBT claims that “the quality of any conscious state depends on the system’s entropy” or disorder. Physically, entropy in a network of neurons can be measured as the variance in the network’s intrinsic synchrony over time. Variance is low when the fluctuations in each neuron’s activity tends to synchronize with the mean signal of the network, and high when these oscillations are not coordinated with the overall network. Variance in synchrony therefore serves as an order parameter for a network of neurons. Additionally, EBT proposes that energy-inducing phenomena like psychedelics push the brain closer to the critical point between order and disorder. Indeed, researchers have found that high-level association networks in the brain, e.g. networks responsible for functions like cognition and attention rather than sensory processing, exhibit greater variance in synchrony under the influence of psilocybin, which is the psychoactive ingredient in magic mushrooms. 

Carhart-Harris’ notion of entropy is closely related to another neurobiological phenomenon, free energy. According to Karl Friston’s Free Energy Principle, free energy roughly corresponds to prediction error, i.e. the difference between the brain’s predictions about sensory inputs and the inputs themselves. Prediction error is a measure of uncertainty about the brain’s state, which is very similar to entropy. In fact, Carhart-Harris and Friston went on to collaborate with each other on a paper titled “REBUS and the Anarchic Brain: Toward a Unified Model of the Brain Action of Psychedelics”. [6] In the paper, they argue that psychedelics sensitize the neurons in the brain that encode expectations and beliefs to prediction errors, such that surprising events – events that are associated with a high degree of uncertainty – are more capable of altering deep-seated opinions, thought patterns, and so on.

Free energy is also very significant for the relationship between criticality and symmetry. According to physicist Harold Stokes, the polynomial described in Eq. 1 is the mathematical formulation of the system’s free energy. It is unclear to me how the free energy discussed by Stokes, also known as the “Landau potential,” is related to Friston’s free energy. But if we can show that the two are equivalent to each other, then we can directly calculate the transformations under which the order parameter of consciousness – namely, variance in the synchrony of neuronal networks – remains invariant.

Conclusion

Sublime states of consciousness, such as the one described in the beginning of this blogpost, correspond to high-symmetry phases of brain activity. Symmetry is defined as an aspect of the brain that remains the same when the brain is changed. In systems that exhibit criticality, like the brain, this invariant aspect is the order parameter. The relevant order parameter for consciousness, according to Carhart-Harris, is variance in the synchrony of neuronal networks. Hence, when the brain undergoes a phase transition into sublime states of consciousness, there is a larger range of transformations under which the variance in neuronal synchrony remains the same. We can determine these transformations by computing the free energy of the brain.


[1] Thank you to the Qualia Research Institute for informing me about the Zero Ontology.

[2] I credit the Qualia Research Institute for inspiring me to investigate the relationship between phase transitions/criticality and consciousness.

[3] Andrés notes (personal communication): “It is important to point out that this is a “statistical notion of symmetry” and that if you were to track every atom at a fine enough level of resolution, the gaseous state would in fact have no symmetries whereas the solid state would. It’s only at a statistical and high-level account of “density” that admits small perturbations where the gas state would be more symmetrical than the solid state.”

[4] In the case of the solid (crystal) phase of water, these translations would be “lattice vectors.” According to Chang, these are translation vectors of the form R = ha + kb + lc, where a, b, and c are the basis vectors of the crystal’s Bravais lattice. 

[5] According to Andrés, these space groups are a predominant aspect of DMT experiences, especially those that reach the “crystal worlds” phase transition. 

[6] I wrote about the REBUS model back in August 2020, and REBUS was central to the development of one of QRI’s core theories, Neural Annealing.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s