Max Tegmark, a theoretical physicist at MIT, examined in a 2014 paper, the thinking of consciousness in mathematical terms, showing how they can be formulated in terms of quantum mechanics and information theory. The paper outlines how particular properties might arise from the physical laws that govern our universe.

Tegmark starts with assumption that some particle arrangements such as your brain feel conscious while other particle arrangements such as your pillow do not. The question he poses is, what properties of the particle arrangement make the difference between the two? The paper investigates the physical properties that a system must have to be conscious. Tegmark points out that we do not understand consciousness well enough to answer the question. The approach has applications for understanding how consciousness arises and also why the world appears the way it does to conscious observers.

To explore the hypothesis that consciousness can be understood as a state of matter, four basic principles that may distinguish conscious matter from other physical systems are examined. These are the information, integration, independence and dynamics principles. Tegmark suggests that consciousness and varying degrees of consciousness arise from a particular set of mathematical conditions, which create a separate state of matter. It is analogous to how certain conditions create different states of water, vapor, liquid and ice.

Tegmark’s analysis joins attributes of consciousness from neuroscience with physics, attempting to make these attributes calculatable. The neuroscience view of consciousness states that a conscious entity must be able to store information, efficiently retrieve and process it and exist as a unified whole. Tegmark uses the term perceptronium, defined as a substance that feels subjectively self-aware, for this special state of matter that enables consciousness.

The goal of Tegmark’s paper is to investigate the implications for physics of Guilio Tononi’s neuroscience work in “Consciousness as Integrated Information: a Provisional Manifesto” and further developed as Tonini’s Integrated Information Theory (ITT). In it Tononi argues that for a processing system to be conscious it needs to have two separate traits: (1) The ability to store a large amount of information and (2) the integration of that information as a unified whole that cannot be decomposed into nearly independent parts.

In order for ITT to extend our ability to detect consciousness to animals, computers and arbitrary physical systems, Tegmark argues its principles need to be grounded in fundamental physics. There are neural correlates of consciousness, mapped by brain activation patterns, that correspond to certain conscious experiences. Tegmark asks how we can look for physical correlates of consciousness defined as patterns of moving particles that are conscious. Tegmark points out that identification of physical parameters to identify conscious matter “from the outside” could be useful for controversial topics from artificial intelligence to determining when an animal, fetus or unresponsive patient can feel pain. In fundamental theoretical physics it would be useful to identify conscious observers in our universe through equations.

Tegmark concluded that “classical physics allows information to be essentially fully integrated using error-correcting codes, so that any subset containing up to about half the bits can be reconstructed from the remaining bits.” Information stored in Hopfield neural networks, a type of neural network model, is error-corrected naturally. By his calculations, if the human brain has approximately 10 11 neurons it would support only about 37 bits of integrated information. This brings forth the integration paradox that the information content of our conscious experience appears to be much larger than 37 bits. Tegmark interprets this to mean that at least one additional principle must supplement the integration principle.

Exploring the independence principle and the extent to which a Hilbert space factorization decomposes the Hamiltonian H into independent parts leads to the Quantum Zeno Paradox. That is if the universe is decomposed into maximally independent objects, then all change stops. But conscious observers do not perceive reality as static and unchanging, suggesting that integration and independence principles must be supplemented by at least one more principle.

The dynamics principle was explored according to the conscious system with a capacity to both store and process information. The energy coherence was used to measure dynamics. Maximizing dynamics gave periodic solutions that were unable to support complex information processing and reducing it enabled chaotic and complex dynamics, exploring the fully dimensionality of the Hilbert space. Tegmark found that high autonomy, which is a combination of dynamics and independence, can be attained even with a strong environment interaction.

Tegmark uses this way of thinking about consciousness as a way to study the quantum factorization problem, one of the fundamental problems of quantum mechanics. Tegmark argues that consciousness is relevant to the notions of observations and observers. The quantum factorization problem is that we as conscious observers perceive the world as a dynamic hierarchy of objects that are strongly integrated and relatively independent.

The quantum factorization problem arises because quantum mechanics describes the entire universe using three mathematical entities: a Hamiltonian describing the total energy of the system; a density matrix describing the relationship between all quantum states in the system; and Schrodinger’s equation describing how things change with time. This leads to an infinite number of mathematical solutions, with all possible quantum mechanical outcomes. It raises the question of why of the possible outcomes, we perceive the universe as a semi-classical, three dimensional world. For example, in a glass of iced water we perceive the liquid and solid ice cubes as independent even though they are part of the same system and intimately linked.