*Two of the key founders of quantum physics, Einstein and Schrödinger, were deeply sceptical of its implications about uncertainty and the nature of reality. Today, the orthodox reading is that uncertainty is indeed an inherent feature of quantum systems, not a reflection of our own lack of knowledge. But Oxford physicist Tim Palmer now argues that chaos theory shows that quantum uncertainty is in fact down to our own ignorance, not reality itself. This could have far-reaching consequences for our ability to marry quantum mechanics with general relativity. *

Everyone knows that long-range weather forecasts are uncertain. It’s because of those pesky butterflies. Unobserved, they flap their wings, causing unpredicted storms to appear weeks later. This is the metaphor used to describe the unpredictability of chaotic systems:tiny uncertainties in the initial conditions of a system grow and grow until they completely destroy the accuracy of any forecast. In this metaphor, the butterflies themselves aren’t uncertain about the state of their wings, it is us humans that are uncertain about them. Philosophers call this “epistemological” uncertainty – uncertainty to do with lack of knowledge.

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But according to the orthodox view about quantum mechanics, our most successfully tested theory of physics, uncertainty is not always of this epistemological type. Quantum mechanics is usually described as a theory of atoms and sub-atomic particles, but in truth it is believed to be a theory that underpins everything in the world, including the weather and the galaxies – all of reality. According to orthodox view, there is an *inherent* uncertainty about what happens to a quantum system when we attempt to observe it. At the moment of observation, the quantum state of a system collapses randomly from a superposition of possible states to some definite outcome. According to this account, randomness is incorporated into the basic equations of quantum mechanics. This in turn implies that quantum uncertainty is not purely epistemological, but is additionally “ontological”, meaning that reality is in itself inherently uncertain. But this orthodox view rests on a rarely questioned assumption. Chaos theory provides strong motivation for questioning the assumption. Rejecting it means that the uncertainty of quantum mechanics could, after all, be of the same epistemological type as that of chaos theory.

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Standard discussions about Bell’s inequality end by concluding that quantum uncertainty is not fundamentally due to our uncertainty about the quantum world, but due to the way quantum reality itself is.

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# Quantum Uncertainty and Bell’s Theorem

Two of the founding fathers of quantum mechanics thought the idea that reality was uncertain was ludicrous, and as a result refused to believe that quantum mechanics, in its current state, was the final word on the subject. Erwin Schrödinger devised his famous cat experiment to show that this interpretation of quantum mechanics leads to nonsensical cats which are half alive and half dead, and Albert Einstein famously remarked, in exasperation, that surely God does not play dice. And yet, despite this, most physicists today believe that quantum mechanics *is* the final word as far as quantum physics is concerned and that there *is* therefore an element of inherent ontological uncertainty about the world around us. Quantum uncertainty, these physicists would argue, has nothing to do with the butterfly effect: quantum uncertainty is much more radical.

So why does today’s consensus reject the concerns of Einstein and Schrödinger? The most important reason stems from a quantum phenomenon that Schrödinger himself named entanglement. Specifically, two particles can be emitted from a source, such that the properties of the two particles – e.g., their angular momenta (also known as spins) are correlated. This itself is not necessarily strange. However, the Northern Irish physicist John Bell showed that, under seemingly reasonable assumptions, these correlations, suitably combined, are limited in size. This is called Bell’s theorem. The 2022 Nobel Physics Prize was given to three physicists (Alain Aspect, John Clauser and Anton Zeilinger) who showed that in practice, the combined correlations can exceed this limit. Hence one or more of these seemingly reasonable assumptions must be wrong.

The standard interpretation of this experimental result is that it confirms that quantum uncertainty is ontological, not epistemological. That is, uncertainty is a feature of reality itself, not a reflection of the limits of our knowledge. Of course, this is such a startling conclusion that physicists have looked for other ways to explain Bell’s theorem. There is indeed an alternative interpretation, but it is too weird to be plausible. It assumes that the settings for the apparatus that measures the spin of one of the entangled particles somehow influence the measurement outcome for the other particle. It is a weird explanation because it implies what Einstein called “spooky action at a distance” – the idea that what happens to one particle can instantaneously influence another, distant particle. Einstein didn’t like spooky action at a distance, and neither do I, nor indeed most physicists I know. So standard discussions about Bell’s inequality end by concluding that quantum uncertainty is not fundamentally due to *our* uncertainty about the quantum world, but due to the way quantum reality itself is.

However, it turns out that there is another assumption in what is called Bell’s Theorem. It’s one that physicists intuitively think is true and therefore don’t tend to question. However, they should. The assumption concerns the validity of a way of thinking that is second nature to us: counterfactual reasoning. Below I will show that chaos theory casts doubt on an unquestioned belief in the validity of counterfactual reasoning.

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The properties of fractals can be exploited to try to explain why a whole class of counterfactual worlds might be inconsistent with the laws of physics.

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# Counterfactuals, Quantum Mechanics and Chaos Theory

Suppose you throw a rock, and it hits a window and smashes it. Did you cause the window to break? Perhaps, out of sight, someone threw a second rock moments earlier and it was that second rock that caused the window to break. You would be in no doubt that *you* caused the window to break if you could assert that in a counterfactual world where you didn’t throw the rock, the window didn’t break.

In this counterfactual world, the moons of Jupiter would orbit in the same way they do in the real world. All that’s different is that the rock isn’t thrown. Although it did not happen in reality, such a counterfactual world is nevertheless consistent with the laws of physics as represented by Newton’s laws of motion. This appeal to counterfactual possibilities is so deeply ingrained in our intuition that we rely on it all the time to infer causality in the real world.

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It turns out that Bell’s theorem depends on an assumption that certain hypothetical alternative spin measurements - ones that might conceivably have been performed on the entangled particles but weren’t - are allowed by the laws of quantum physics. That is to say, Bell’s Theorem assumes counterfactual quantum measurements are necessarily consistent with the laws of physics. This is the assumption that physicists don’t like to think might be wrong. If these counterfactual worlds turn out to be inconsistent with the laws of physics, then our intuitive ideas about causality will also turn out to be wrong.

Chaos theory provides a simple way to understand situations where counterfactual worlds are indeed *inconsistent* with the laws of physics. To make sense of this, I have to say a little more about the butterfly effect. The extraordinary thing that meteorologist Ed Lorenz discovered, back in the early 1960s, is that if you start his simple chaotic system from any initial state and let the state evolve, you will eventually see the state trace out a remarkable fractal geometry. A fractal geometry is one which has a structure which is never lost, no matter how far you zoom into the geometry. In particular, it has gaps which never disappear as you keep zooming. This is quite different to classical Euclidean geometry, like the surface of a sphere, which looks flat and boring if you zoom into it enough.

*A Lorenz Attractor*

These properties of fractals can be exploited to try to explain why a whole class of counterfactual worlds might be inconsistent with the laws of physics. But to understand this, we have to think big, very big indeed. We have to suppose that the whole universe, and literally everything there is in it, is collectively a chaotic system evolving precisely on some cosmic fractal geometry. In this picture, there is no guarantee that hypothetical counterfactual worlds that you simply cooked up in your head, will lie on this fractal geometry. If they don’t, then these counterfactual worlds will be inconsistent with the assumed geometric laws of physics.

In a series of technical papers I have developed a mathematical model where the counterfactual worlds which arise when you try to prove Bell’s theorem do *not* lie on the assumed fractal geometry of the universe. This means that quantum uncertainty *really could* be epistemological after all, and hence that reality is definite and concrete as we usually take it to be.

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A chaos-based model of quantum physics where uncertainty is epistemological may stand a much better chance of being married to general relativity than one where quantum uncertainty is ontological.

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# What does the non-existence of inherent uncertainty mean for physics and philosophy?

Whilst this is perhaps interesting philosophically, is it of any consequence for real physics? Maybe. The holy grail of theoretical physics is the unification of quantum and gravitational physics. Physicists have been trying to synthesise these two areas of physics for the last 70 years or more, so far without success. Some think that Einstein’s theory of general relativity needs a radical overhaul before such unification is possible. However, in my view, it’s the other way around. A chaos-based model of quantum physics where uncertainty is epistemological may stand a much better chance of being married to general relativity than one where quantum uncertainty is ontological.

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However, there is something else to draw from this discussion. If this fractal geometric model is right, it will signal the end of what’s called “methodological reductionism” in physics: the idea that to get a deeper understanding of the world around us, we need to probe smaller and smaller scales. This is the philosophy – successful to date - underlying the development of particle colliders. However, it may be that this philosophy has run its course and that to get a deeper understanding of the world, we instead need to probe not the smallest, but the very largest structures of the universe as a whole. The buzzword for the future may be one that in the past has been derided for its smell of new-age mumbo-jumbo mysticism: holism. And yet, if the laws of physics describe a fractal state space geometry on which states of the universe evolve, then indeed these laws will be profoundly holistic.

This, I believe, shows how important it is to understand the notion of uncertainty.

Tim Palmer's book *The Primacy of Doubt by Basic Books and Oxford University Press *is out on October 18^{th}, 2022.

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