‘She’s a force of nature’ we might have said about Serena Williams at her unstoppable best. Physicists talk of four ‘fundamental forces’, though there is some quibbling about whether and how these can be ‘unified’. In school we learn of Newtons laws of motion, which are all about ‘forces’ pulling and pushing objects. But what ‘is’ force? What is happening when something ‘touches’ us?

Physicists say that ‘forces’ which are effect of the ‘exchange’ of some ‘force-mediating’ particles – and there are four types of such particles, sometimes technically called ‘gauge bosons’:

• Gravitons – which produce gravitational attraction between all things;
• Photons – which create electric (and magnetic) forces, and incidentally are also the substance of light – both visible and invisible (radio waves, infrared, microwaves, x-rays, … )
• W and Z particles – which are in some ways cousins of photons, but they have a solidity which makes them move more slowly than light;
• Gluons – which appear to act something like strings of glue or elastic, and hold the nuclei of atoms together – though they don’t feature much outside of an atomic nucleus and so have little effect on anything else.

It’s natural to wonder what this has to with the notion of force as conceived by Newton, the force of something pressing against our body, and forces marshalled by mechanical and civil engineers. Can we understand the forces between the components of an engine, or between the concrete and cables of a bridge, in terms of the effects of tiny particles tossing around other tiny little particles, like spacewalking astronauts pushing each other around by throwing tools to each other? Actually, we cannot. The most viscerally felt Newtonian forces, like the collision between your bed leg and your toe, can actually NOT be explained this way.

Now it’s not that physicists are totally confused – but scientists in many fields use language in very context specific ways, which can be confusing if we interpret them through our everyday lexicon rather than through the formal jargon. So this idea of ‘fundamental forces’ has to do with listing the logically independent ‘mechanisms’ of interaction that we can explicitly visualise as ‘a process’ – and it turns out that these mechanisms can be understood as various types of particles being tossed around between other particles. The ‘active’ particles recoil from the action of throwing or catching the others. It is fair to ask how it is that we can decide which particle is doing the tossing and which one is being tossed – and there are technical details which do make this distinction in a sensible way.

But this list of ‘processes’ does not, it turns out, explain everything, just by itself, in an otherwise dynamically empty universe. The particles which are linked to these fundamental forces, through their ability to emit and absorb photons and/or other ‘gauge bosons’, are, without exception, particles called ‘fermions’ – so named after the Nobel prize winning Enrico Fermi who was perhaps the last physicist to be a great pathfinder on both experimental and theoretical branches of physics. And all the fermions of a particular type (we are mainly concerned with electrons, which occupy most of the space that makes up objects we perceive as solid) form a great cohesive stew which senses itself in some way that produces innate forces for which we have no mechanistic explanations.

In schools it is taught that electrons are subject to a rule called the ‘Pauli exclusion principle’, which some of us are later told is a consequence of the fact that fermions obey ‘Fermi-Dirac’ statistics, but this is just talk and doesn’t get to the heart of the matter. The intriguing thing about fermions, physically, is that this ‘exclusion principle’ means you can’t penetrate them into each other, as it were. You have to place them side by side. Now fermions are not like tiny marbles which you can hold and stack up into little piles. They are, like all things, almost as slippery as photons, which are the fabric of electric and magnetic fields (including light) – fermions just have a teensy bit of mass, which means you can slow them down.

Photons, like all the other ‘gauge bosons’, can be piled up into and onto each other more or less indefinitely. In a model world where there are just photons, and none of the pesky distractions of the other fundamental forces, there is no limit to how intense you can make a laser beam, or how strong an electric field you can create. In the real world, all sorts of things happen when you put too much energy into a small space, but this is not the point.

Electrons, like other fermions, aren’t at all like little balls. They are quantum lumps of ‘electron fields’, the stuff of which is electrons – which we distinguish from ‘electric fields’, the stuff of which is photons. In a way, multiple electrons whose lumpiness is similar in size, like electrons described by ‘atomic orbitals’ around an atomic nucleus, can be stacked up into the same space – for example within the same atom. However, and this is the big point, they can’t be in the same ‘orbital’ – as per the ‘Pauli exclusion principle’. There is that little quibble about ‘they can be in the same orbital, if they have opposite spin’ – whatever we mean by spin –  but we can rework the wording so that whatever is meant by spin is absorbed into whatever is usually meant by orbital. The punchline is that you can’t just keep pumping copies of the same electron wave into a particular location – whereas you can do this with photons. Photons just ooze together with each other without hindering each other, and adding lots of them makes for strong electric or magnetic fields, or beams. When you stack many electrons into the same space, you have to keep adding extra bumps to the wave packets – you can’t just add copies of the same wave packet to build a stronger version of the original ‘electron field’.

This is a deep mystery. There is a formal theorem about this, which in some sense ‘explains’ something about it. The great Richard Feynman said that the complexity of this theoretical argument suggests we are missing something conceptual. Seen another way, it could be argued that it’s the photons that are weird. When we stack up everyday objects, we can’t have them all be in the same place. The first book on the shelf is not in the same place as the next, and eventually the shelf is full. A marble stack has a size and shape that that grows as we add marbles. If photons and other ‘gauge bosons’ have some sort of physical substance, how is it that we can just keep adding more and more of them into the same little space, without theoretical limit, until the sparks fly? That is the other half of the deep mystery. To say that ‘quantum field theory’ ‘explains’ all this feels a bit like saying that newtons laws explain the atomic structure of steel balls just because they describe their motion. Quantum field theory is a stunning description of the world at a certain range of size scales – but it is not really a fundamental explanation in any usual sense of the word explanation.

We may seem to be straying from the point – which is to explain how common everyday forces arise out of the microscopic structure of matter and the fundamental processes which science has enumerated – and how this requires us to understand something more than the four conventional ‘fundamental forces’ – but we are making progress.

The punchline of the microscopic view is that fermions, which we think of as the main building blocks of ‘solid matter’ that experiences forces, need these variations in lumps and bumps if they are to coexist in the same space. These bumps are in fact what determines the shapes of the atomic orbitals we learn about in school – spherical S orbitals, dumbbell shaped D orbitals, and clover leaf D orbitals (and the list goes on). The more ‘lobes’ we add, the more energy the orbital implies. A typical analogy is the guitar string. Striking an open string makes one sound – it sounds the note after which the string is named. Striking the string while gently touching the middle produces a sharper pattern of vibration: the string then vibrates with two lobes of maximum movement (at a quarter string length from either end, with minimal motion in the middle) rather than just the one lobe of maximum movement that at the centre when the string is struck while fully free. The shorter wavelength makes a sound with twice the frequency – an octave higher. This higher frequency is a faster buzzing around in a confined space. This helps us visualise more energetically excited electrons, and that’s what happens when we add electrons to atoms – the more we add, the more buzzing-around they need to exhibit, in order to live in the same little space.

So, if you squish a bunch of electrons together, this problem of coexisting in a smaller and smaller space creates the need for more smaller and smaller vibrational lobes, like sliding your finger down the fretboard of a guitar. These smaller orbital lobes mean more energy. This is quite different from the experience we have of compressing the air in a pump. What we feel when we compress air is ‘just’ the effort required to push back against gas molecules rattling around and bumping against the piston of the pump – it has nothing to do with gas molecules fundamentally struggling to occupy the same space – at least not at anything resembling the temperatures and pressures which we experience in everyday life. In order to compress the clouds of electrons in atoms, we fundamentally have no choice but to give them extra energy, in order for them to coexist within the rules that make fermions what they are – an ooze in which one quantum lump of electron is all you get, for each kind of ‘orbital’ that your space can accommodate.

It’s as if a guitar string can only be struck with a single inherent power for its base note, and some other specific inherent power for each harmonic – so that in order to make it louder, you have to stimulate the higher harmonics. You simply cannot make the base note itself any louder or softer, and each harmonic only hums at one natural loudness. Or you can imagine a flute which can only be blown at one volume, unless you add overtones. This is certainly not the behaviour of any ordinary sound wave, or electric field, for which we have some intuition from everyday experience – but it really is how electrons manifest, for reasons which we frankly don’t understand.

So here’s what happens then the atoms of your toe get smooshed together with the atoms of the edge of your bed. All these atoms are driven by the motion of your foot, relative to the bed, into each other’s space, and this requires lots of energy to be pumped into the electrons of the atoms at the surfaces of your toe and the bed, so that the electrons there can occupy the same space. But this energy is not really available, so the compression can barely happen, and your foot grinds to a halt.

Now you might try to argue that this collision between your toe and the bed is explained by the fact that the electrons on both sides of the collision are all negatively charged, and like charges repel each other. But the electrons inside all atoms already live with each other quite happily, in really close quarters. The negatively charged electrons bring along with them the positively charged protons in the atomic nucleus, and this makes them net neutral, so that the net electromagnetic forces between atoms are really very tiny. The crux of the matter is the intrinsic requirement that in order to make an electron cloud more dense you have to provide a extra energy to the electrons – and so they effectively push back – not by tossing around photons, but by simply having nowhere to go unless the extra energy is provided. It’s been called ‘Pauli blocking’.

Sometimes there are heated little arguments about whether this is a ‘Pauli force’, or maybe actually ‘just’ a ‘pseudo’ or ‘superficially apparent’ force. But really, this behaviour of electrons, through their fermionic nature, does in fact explain the origin of normal everyday forces that objects exert on each other when, as we see it from our perspective, they touch. What we experience as contact is electrons running out of space – somehow sensing and avoiding each other without any interactions which we can describe as a ‘process’. In our theories, this is built into abstract mathematics that hold the space of our fundamental ignorance of what electrons, or electron fields, really ‘are’.

Think of how little we can learn from Newton’s laws about the microscopic fabric of steel, even while these laws do a good job of describing the rolling and collision of steel balls. We can learn nothing at all – after all, Newton’s laws say precisely the same thing when the balls are made of brass, or plastic. As impressive as our insights are, into how electrons behave, and how everyday materials, built from atoms, which are largely electron clouds, interact – these insights tell us exactly nothing about what electrons really are, or how the world’s electron ocean senses itself, demands energy in exchange for compression, and generates what we all know as the force of touch.