2.3 Quantum fields
In classical (i.e., not quantum) physics, a field is a quantity, such as the force on an electron, that has a value at each point in space and time. An example is the electromagnetic field, and its value at any point determines the motion of an electric charge at that point. If the value of the field is constantly changing with time, which will generally be the case, then that value can be expressed as the sum of a large number of separate, smaller, changing or oscillating values, in the same way that the human voice can be broken down into many separate, pure vibrations, or oscillations, of different frequencies. Now, if you apply quantum mechanics to each pure oscillation in the quantum field, then you find that the energy of each oscillation can only increase or decrease in discrete jumps. Unlike the classical field, the quantum field doesn’t have an actual value anywhere or at any time: it is Schrödinger’s equation that determines the probability that the outcome of a measurement or interaction of the field will have a given value at any particular point in space and time. If any of the oscillations is given enough extra energy to result in a jump, then, in quantum field theory, that jump represents a particle. In the quantum electromagnetic field, for example, such a jump would be a photon. This is how the creation and annihilation of photons is manifest in the quantum electromagnetic field.
For each quantum field, there is a corresponding particle: the particle of the quantum electromagnetic field is the photon, the particle of the quantum electron field is the electron, the particle of the quark field is the quark (there are actually six types of quarks and quark fields), and so on, and it is the mutual interaction of these fields that determines particle interaction, creation and annihilation. Since particles are excitations of quantum fields, quantum fields are generally regarded as more fundamental than the particles themselves.
Quantum fields are completely mathematical structures, and so they are completely specified using mathematics: nothing is “left out”. Quantum fields account for all of the observable particles and three of the four known forces. (It’s true that gravity, and the gravitational puzzles of dark energy and dark matter – which we shall discuss later – don’t yet fit tidily into the quantum field scheme, but, as far as we know, the behaviour of light and matter in gravitational fields and, indeed, the very “shape” of our universe – see later – are completely specified by the mathematical structures of general relativity, and there is no reason to think that dark energy and dark matter will turn out to be any different.)
Now here is where it gets tricky. Since our universe appears to be completely specified by quantum fields and general relativity, which are ultimately purely mathematical structures, this implies that the universe itself is purely a mathematical structure. On the other hand, in seeming contradiction to the mathematical structure picture, it appears that there are several ways to demonstrate the “physical reality” of quantum fields. I have put this in inverted commas because, of course, that begs the question of how you define “physical reality”, and reality itself. (As a spoiler, I hope you will eventually be convinced that reality within a mathematical structure is actually the structure itself, but just keep that in the back of your mind for the moment, while I tell you about the case for the “physical reality” of quantum fields.)
Out of several contenders, possibly the most vivid piece of evidence for the “physical reality” of quantum fields is the Casimir effect. This phenomenon was named after the Dutch physicist, Hendrik Casimir, who proposed its existence in 1948. He said that, if you take two flat parallel metal plates and place them very close together in a vacuum, they will be “sucked” towards each other, even when the plates have no electric charges on them to account for the “suction”. The source of this force between the two plates is the vacuum itself! Recall how the quantum electromagnetic field can be broken down into a sum of oscillations: well, these waves extend through space as well as in time, and quantum theory predicts that, even in a vacuum, the waves don’t disappear entirely. Because the metal plates are electrically conducting, the electric field at the surfaces of the two plates must be zero, meaning that only “standing waves” of the quantum electromagnetic field are allowed to resonate between the plates, like standing waves in a guitar string where both ends of the string are fixed – these fixed points are called nodes.
The distance between nodes in a standing wave is half a wavelength, so that only whole numbers of half-wavelengths can oscillate between the plates. Among others, this excludes all waves whose half-wavelength is greater than the distance between the plates – they won’t fit between them. So, at smaller plate separations, more waves are excluded. However, each wave has a fixed quantum energy, and so the energy between the plates is smaller when the plates are closer together.
This results in a force pushing the plates together. To see this, imagine slowly moving the plates apart. The plates start with a small separation, where the energy between the plates is low, and they finish up at a slightly larger separation, where the energy between the plates is greater. So, the energy of the system was increased by moving the plates apart, and that energy must have been supplied by work required to move the plates apart. Therefore, there must be a force trying to keep the plates together, and that force is the Casimir force.
Casimir effect – half-wavelengths greater than the plate separation are excluded between the plates. Illustration from Wikipedia
Equally, you can view the compressing force as resulting from the momentum imparted by waves bouncing off the two plates. Each standing wave between the plates can be broken into two waves travelling in opposite directions, one towards a plate and one away from it, so that the waves are in effect bouncing off the plates. The oncoming wave pushes the plate in its direction of travel, transferring its momentum to the plate. However, there are more waves bouncing off the plates coming from outside pushing the plates inwards than there are waves between the plates pushing them outwards, so that the overall effect is again to push the plates together.
The force between the plates is very small at distances of millimetres and above, but it becomes significant at plate separations below a micrometre, and, despite the practical difficulties of measuring such a force, the results of such experiments agree well with theory. Since the origin of this tangible force is the quantum electromagnetic field, we say that quantum fields have a “physical reality”. Nevertheless, when you drill down into the theory of quantum fields, you are still left only with the mathematics, like the smile on the Cheshire cat. How do we reconcile the “reality” of the quantum field with it being apparently a purely mathematical structure?
The answer is in the same way that we regard all interactions between quantum fields – for instance, two electrons approach each other and then move apart because of their mutual repulsion, which is completely described by a four-dimensional mathematical structure that we would interpret as quantum fields interacting over time. Ultimately, you can analyse the attraction between Casimir’s plates in the same way.
In the final analysis, since particles and their interactions are ultimately configurations of quantum fields and their interactions, and since quantum fields are purely mathematical structures, then the space and time within which these quantum fields operate must also be purely mathematical structures! The space and time parameters of the quantum fields are intrinsic to their structure. When you think about it, it couldn’t be any other way: there can’t be a “physical” space within, say, a “physical” bowl that contains a mathematical structure that we call a quantum field. Instead, we have to regard the four-dimensional block universe purely as a mathematical structure comprising space and time parameters in which quantum fields can operate. Notice that, if we regard a purely mathematical structure as its own reality, then every substructure within it is also real (where a substructure could in this case be the two electrons bouncing off each other).
The mathematical structure of our block universe is not trivial. As we observed earlier, the time parameter is treated differently from the three spatial parameters in that its square has the opposite sign from those of the squares of the spatial parameters. (Strictly, although this was first raised in the context of Minkowski geometry, it is incorporated in Riemann geometry, which includes gravity, as we shall discuss later.) One of the important results of this is that the “laws” of physics are the same for all observers, regardless of whether they are moving or stationary with respect to another observer. An example of this is the set of Maxwell’s equations, dealing with electric and magnetic fields: the form of Maxwell’s equations is the same for all observers, and, indeed, shows that the speed of light (which may be regarded as a combination of fluctuating electric and magnetic fields) is independent of the motion of the observer, which Einstein made a principle of relativity. If the mathematical structure of the block universe didn’t have a limit on relative speeds, then there would be the kind of absolute frame of reference that Isaac Newton envisaged. Within that frame, whenever your speed would approach that of light, electromagnetic interaction between atoms and electrons would change, and chemistry (and you, and all matter) would crumble.
