# II. Dualities

Dualities are common in science. In the field of linear programming (LP), for example, a mathematical optimization problem involving 20 variables and 10 constraints in its primal form can be transformed into its dual form consisting of 10 variables and 20 constraints. Sometimes LP problems are easier to solve in their dual rather than primal form. Since they are dual to each other, if you solve one you solve the other. This technique is the secret behind Support Vector Machines in machine learning, and how they overcome the so called “curse of dimensionality“. Certain machine learning problems involving, say, hundreds of thousands of input variables (e.g. pixels), but far fewer examples, can be solved in their dual form much more easily.

Figure 3: Black hole diagram showing event horizon and singularity at the center – the subject of the AdS/CFT duality from UCSD Center for Astrophysics and Space Sciences

Einstein‘s theory of relativity is essentially a theory about dual descriptions of the world depending on one’s reference frame. A clock moving at near the speed of light will appear to slow to a crawl to a stationary observer, but an observer moving along in the same reference frame as the clock will see time pass normally. Reconciling these two reference frames is the work of the Lorentz transformations and they always do reconcile. Today, one of the most exciting dualities in science is the research in theoretical physics concerning the AdS/CFT duality. In this duality, one can use quantum field theory (Conformal Field Theory or CFT) to describe the quantum entanglement of particles on the boundary surface of a spacetime – like at the event horizon of a black hole, or, one can use gravity to describe spacetime curvature (Anti-de Sitter space or AdS) in the interior (the Bulk) and get the exact same results (see this video by Leonard Susskind for a simple explanation). Coincidence? The two theories are independent from each other, and have stumped physicists for nearly one hundred years as to how to unify them. But, now we are getting our first hints of how they come together – it now seems they offer descriptions of the Universe that are dual to each other. Two ways of describing the same thing. Two perspectives on the same phenomena. Certain problems in black hole research are easier to solve from the gravity perspective, and others are easier from the quantum field theory perspective. Since they are dual to each other, if you solve one you solve them both.

Descartes’ was the first to talk about a duality as it relates to the subjective 1st person experience (see Mind-Body Dualism). In his description, there was mind “stuff” (res cogitans) and matter “stuff” (res extensa). He believed that humans possessed this mind stuff but that animals were mindless machines saying: “without minds to direct their bodily movements…animals must be regarded as unthinking, unfeeling, machines that move like clockwork.”. In the argument we will follow in this essay, we are not looking at the 1st person/3rd person duality in the Cartesian sense, i.e. we are not talking about a dualism between a non-material spirit and matter as two separate things, but, rather, are looking at the problem as reconciling two views of the same underlying physical matter and processes, governed always by the laws of physics. One of these views is internal and subjective, the other external and objective, and represent, like the dualities in linear programming, Relativity Theory, and AdS/CFT, two ways of describing the same thing.

When we speak to others we have a subjective, 1st person experience of what it is like to be us speaking to them. We are perhaps happy, nervous, relieved, angry, or some complex mixture of these emotions and more. The conversation may be an easy one to have or a difficult one. Those listening to us have a 3rd person description of us speaking too. They may notice what we said, how we appeared while speaking. They may describe our speech as sad, excited, or some complex mixture of adjectives. If they attach electrodes to the neurons of our brain, they can provide an even more detailed objective description of us, but it still will be only that – an objective description. And, all this works vice-versa as well: our objective description of them, and, for each, their subjective experience of being themselves. These two descriptions of the same phenomenon, are dual to each other – they are two alternative ways of describing the same interaction. A subjective/objective, internal/external, 1st person/3rd person duality. Furthermore, we readily trust that others have this 1st person perspective even though we will never be able to experience it for ourselves. We will never be able to be them. Nor will we be able to prove that they are indeed having a subjective experience, yet it does not seem like a reach to believe it. In the same sense, we hope to make a plausible, circumstantial, and compelling case for free will even if we will never be able to prove it.

The problem of bridging the gap between the 1st and 3rd person perspectives is known as the “hard problem of consciousness” and was coined by the philosopher David Chalmers in his book “The Conscious Mind: In Search of a Fundamental Theory” (1996). However, here we are specifically not talking about the deep and complex subject of human consciousness. Human consciousness is characterized, I think we can all agree, by several things including: a 1st person perspective, choice and a sense of free will, a sense of self/self-awareness, a memory of the past, expectations of the future, qualia of the senses, emotions, a feeling of life, an awareness of death, and so on. Here, for starters, we want to approach a watered-down version of the hard problem of consciousness and only talk about the first two items in the list: a 1st person perspective and choice.

But, before we can dive in, we will need to talk about quantum mechanics and indeterminism. The scientific world outside of quantum mechanics is entirely deterministic. Take the case of the planets in our solar system: once we specify the initial coordinates, momenta, and a gravitational constant, the future orbits of all those planets are precisely determined. That is because the General Theory of Relativity, which describes gravity, is entirely deterministic. At the macroscopic level, the level of electrical circuits and bar magnets, determinism applies to electromagnetism as well. The equations of chemistry are mostly deterministic too. But, when we look really closely at the natural world, down to the level of quantum mechanics, we see a world that is indeterminant, governed by probabilities. On macroscopic scales these quantum probabilities average out and so the world appears deterministic – for example the trajectory of a baseball, or a rocket, appears precisely predictable. But, look up close at the atom and we can only calculate probabilities of finding, say, an electron, in a certain location, and nothing more than probabilities. The picture of the atom is not one of electrons orbiting a nucleus analogous to our solar system. Rather, fluctuating quantum waves of the spherical harmonics (along with a radial component) describe the location of the electrons and do so only probabilistically. It doesn’t appear there are any hidden variables either (see Bell’s theorem for more). That is, it is not that our description of the atom is incomplete. It is simply that, at its most fundamental level, the Universe is indeterminant. Still, these probabilities are governed precisely by quantum mechanical wave functions and it is not clear, even with indeterminism, how free will might enter into the laws of physics. In fact, physicists have argued specifically that the indeterminism of quantum mechanics does not imply free will. Their argument is based on the belief that (a.) quantum fluctuations are too small to affect decision making in the brain, and (b.) that there is no freedom in quantum mechanical laws – the outcomes of experiments are generated randomly as specified by the wave function in a probabilistic sense. However, recent theoretical and experimental developments in quantum biology, and a new perspective on randomness, which we will describe here, will open the door and allow free will to exist and to be understood in compliance with physical law. To proceed further, however, we first must introduce the reader to some of the concepts of quantum mechanics and how they may impact biological systems.