Empty space as the soil of the universe: Part 2
In a provocative article Quantum mechanics and our part in creating reality, Blake Stacey writes:
[B]oth Alice and the atom participate in the measurement event; both partake in the creation of a new fact for the pair of them. And if the atom can participate in such ongoing acts of creation when the other player is an agent, surely it can do so when the other player is not. [...]
I love that line because it touches on something that sets QBism's idea of reality apart from all the rest: it is not just dead and inert matter, but matter with an extra dynamic: it must be able to (somehow) "participate" in its own creation. What does that mean? I don't know yet, but I think we need to start by changing our metaphors.
Traditionally, physics treats the building blocks of reality as things like Lego bricks: the bricks themselves are eternal and immutable, and it is only their reconfiguration into different arrangements that gives rise to dynamics and change. QBism says reality is more like soil: it allows things to grow out of it organically, which are essentially made out of the soil, but the soil itself is changed by the presence of what grows in it.
Quantum mechanics, it is often said, is not really a theory in itself but a set of abstract principles for constructing theories (indeed we could say the same thing about relativity). As of today, physicists have succeeded in constructing a quantum theory of every known type of matter: light, atoms, electricity, the individual parts atoms (protons and neutrons), and every fundamental particle known to exist. The quantum theory of all of these is known as the Standard Model of particle physics. There is only one thing left that we have not yet succeeded in applying quantum theory to, and that is space-time itself. For the moment, our best picture of space-time is still the one that Einstein gave us, which is kind of like a malleable, wobbly container in which matter exists (Einstein referred to it fondly as the "space-time mollusc").
But what happens when the matter inside the 'container' of space-time is quantum matter? We can think of space-time in general relativity as being like a still pond, and matter as things like rocks and fish that we can throw into the pond. Before quantum mechanics, having matter in space-time was just like throwing a rock into the pond: it would create a splash and some ripples, and the rock would sink in and remain there, perhaps being moved around by currents. Matter acts on space-time, and space-time acts on matter, but the two remain separate from each other, just like the rock remains solid and keeps its shape in the water. Adding quantum matter to space-time is like dropping one of those vitamin tablets into the pond: as soon as it enters the water is starts fizzing and bubbling, dissolving into the surrounding water, until you don't know where the tablet ends and the water begins.
Historically, the first sign of trouble with the vacuum came when Paul Dirac discovered his famous equation that tells us how to apply quantum mechanics to an electron in space-time. It was one of the first models of quantum matter in space-time that fully obeyed special relativity. However, the model seemed to have a critical flaw: according to it, the "vacuum state" was not the final resting place for an electron, after a long day of radiating away it's energy until it had nothing left. Rather, Dirac's vacuum was like a gaping hole in the universe: a gateway to an infinitely deep void of "negative energy states", into which the electron could keep falling, losing more and more energy, without end, like Alice falling down the rabbit hole.
Dirac was the first one to admit that there was something fishy about this, and he suggested "filling up" the vacuum with an infinite supply of negative-energy electrons, the infamous "Dirac sea". Since electrons are fermions, they would obey Pauli's famous "exclusion principle", and the electrons already present in the negative energy "sea" would prevent the positive energy electrons from falling to below zero energy.
Although the idea was regarded as suspect, Dirac's model couldn't simply be dismissed. For as strange as it sounded, the idea of the Dirac sea led immediately to a startling prediction: there could sometimes occur a 'hole' in the sea of negative energy electrons, which would then behave just like a particle of positive energy with the same mass and spin as an electron but positive instead of negative electric charge. At the time, no such particles were known to exist, but then in 1932 physicist Carl Anderson observed particle tracks in his cloud chamber that exactly fitted the description! Anderson discovered these tracks quite by accident, while he was looking for something else (cosmic rays), and he didn't even know about Dirac's "hole theory", so he had no idea what he had just seen. Many people at the time assumed that the particle Anderson had detected (named the 'positron') was in fact not a particle but a "hole" in Dirac's sea of negative electrons.
This view shifted with the seminal work of Pauli and Weisskopf in 1934. They were not happy with Dirac's model at all. One big problem was that Dirac's clever trick only worked for particles that obeyed Pauli's exclusion principle, such as electrons. But particles of light, called photons, did not obey the exclusion principle. To fix this, Pauli and Weisskopf set out to re-build the quantum theory of photons and electrons from scratch, and to do it in a way that would get rid of the negative energy particles for good. They finally succeeded in their task, and the result was the first example of what today we would call a 'quantum field theory'.
To pull it off, Pauli and Weisskopf made an ingenious innovation. At that time, quantum theories were constructed by first formulating a classical mechanical theory, and then performing a 'quantization procedure' on it, which would turn it into a quantum theory. Where Dirac had taken the classical theory of a single electron as the starting point for his quantum model, Pauli and Weisskopf decided to take the classical theory of a matter field as their starting point. After quantizing the field, they discovered that they no longer had a single particle with its wavefunction, but rather a quantum field which could have any positive number of particles associated to it, and the number of particles could fluctuate. Seemingly miraculously, this theory predicted that there was a minimum possible energy that the quantum field could have, if it happened to be in just the right state. And the state it had to be in was one in which there were zero particles; Pauli and Weisskopf called it the vacuum state. Since the vacuum state is the state of lowest possible energy, it just isn't possible for any field to exist in a state of lower energy: the gaping doorway to negative energy states that had haunted Dirac's model was now closed for good.
Where did that leave the positron? If it is not a hole in the Dirac sea, then what could it be? According to quantum field theory, for every field of charged particles, such as the negatively charged electron field, there has to exist another kind of matter field whose properties are identical but whose charge is opposite. Thus every charged particle would imply the existence of an "anti-particle" of opposite charge. And if the two ever meet, they annihilate one another in a shower of pure energy. The positron is therefore not merely the absence of an electron: it is a real particle in its own right, with positive energy, which is the anti-particle of the electron.
Surprisingly, getting rid of the Dirac sea did not help to 'tame' the vacuum. Quite the opposite: as quantum field theory continued to be developed over the subsequent decades, and applied to more and more matter fields, the meaning of the vacuum only became more mysterious. To understand why, we need to delve a little deeper into the mechanics of quantum fields.
If you have a matter field in space-time, and you want to quantize it, the first thing you have to do is break it up into small mathematical pieces. In flat space-time (eg. little or no influence of gravity) it turns out that these pieces can be neatly separated into two groups, called the 'positive frequency' and the 'negative frequency' components of the field. When we quantize the field, the positive frequency bits become the electrons, and the negative frequency bits become positrons. Mathematically, every field has to have both types of components, so that is why every particle has to have an anti-particle.
This procedure is great when it works, but it has two basic limitations. The first is that, since we don't quantize space-time itself, the theory doesn't actually tell us what the energy of the vacuum state is. From astronomical observations of the rate of the universe's expansion, we have experimental evidence that the vacuum energy is very very small and positive. However, quantum field theory cannot even predict what its value should be. Basically, quantum field theory only allows us to predict energy differences relative to the vacuum, but the energy of the vacuum itself has to be "put in by hand", as physicists like to say.
The second limitation is that this procedure only works when the space-time is not too curved. In fact, it pretty much only works when the space-time is "globally hyperbolic" (remember the example of oil and water from the last post). In a general curved space-time background there is mathematically no way to cleanly separate the field's parts into positive and negative frequency components. This is related to the problem mentioned in the previous post, that in curved space-time we can't cleanly separate energy from geometry. So not only can we not quantize space-time itself, we also can't even quantize matter fields if they are in a space-time that is too curved!
This compounds the mystery of the vacuum, because it means that at our current level of understanding, we can't actually say very much about what the vacuum is. Quantum field theory does give us a few clues though.
The first clue is that we now have an actual mechanism by which particles with actual mass can be converted into (or created out of) pure energy. In general relativity it was ambigious whether vacuum energy could really become mass -- but in quantum field theory it is completely certain. When a particle and antiparticle annihilate, they turn into pure energy, and conversely, it is possible for pure energy to be converted into particle-antiparticle pairs. This is the whole principle of the Large Hadron Collider at CERN, in Geneva: by smashing together heavy particles we create so much energy concentrated in a small space that it spontaneously converts into matter. That allows us to see exactly what kinds of particles can possibly exist: we infuse the vacuum with energy and it produces particles like exotic fruits.
Here is another clue: the vacuum contains *possible* particles. In quantum field theory, in order to make calculations, we use a theoretical device called a 'perturbation'. The perturbation involves considering hypothetical processes that could take place in principle, in very short periods of time. For instance, an electron-positron pair might spontaneously be created out of the vacuum energy, and then a moment later annihilate back into pure energy. These are called 'virtual particle processes' and they are similar to the test-particle collisions of general relativity (remember the last post), because they do not represent processes that actually happen: all that matters is that they could happen. What is amazing is that in quantum mechanics, the mere fact that something could happen, can affect the things that do happen. When we calculate perturbations, in order to predict what will be observed, we need to take into account every possible "virtual" process that could happen, because just by being possible, they can affect what is actually observed.
The number of possible types of processes that could possibly happen in the vacuum is infinite. Perturbation theory is a theory of approximation: it classifies these types of processes into a hierarchy of "orders", such that the lowest orders have the biggest impact on our predictions, while higher orders have progressively less significance and also become progressively more complicated. It's a bit like having an infinite to-do list, where by good fortune the most important items are also the simplest ones to do! So far we haven't managed to go beyond order six or thereabouts, but it is enough to make extraodinarily accurate predictions about particle behaviour.
The vacuum of quantum theory has a mysterious energy, whose value we do not yet know how to predict, but whose effects we can see in the way it moves the stars. And the vacuum is teeming with possibilities: if we overload it with extra energy, it spontaneously converts them into real particles, popping out of the vacuum like bean sprouts!
That's the most incredible thing about how we discovered many of the particles we know today, including the Higgs' boson. We didn't just "discover" the Higgs particle by passively looking, we had to actively create the conditions that enabled the universe to produce it for us. It's like the difference between discovering different types of trees by going for a walk in the woods, versus discovering trees by fertilizing a patch of ground with compost and seeing what grows out of it. We don't merely observe nature anymore: now we cultivate it. The vacuum is not empty; it is the soil of the universe.