In his classic “Feynman lectures on physics”, Richard Feynman starts by saying:
If, in some cataclysm, all of scientific knowledge were to be destroyed, and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis that all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.
Of course atoms are not the basic unit, they are composed of nuclei surrounded by electrons. The nuclei are then composed of protons and neutrons (and short-lived virtual particles such as pions), and the protons and neutrons are themselves composed of quarks and gluons.
But what is the ultimate level? What, when one goes down to the most fundamental level, are things made of? While there are lots of opinions there is no accepted answer, and mulling it over for myself I realised that none of the options are attractive in the sense of aligning with intuition about what “physical stuff” would be made of. Here are some of the possibilities:
Particles: The concept of a particle is very useful (the above protons, neutrons, electrons are all examples of particles). The composite particles have a definite size, but the most fundamental particles, such as an electron, are conceptualised as being point-like (no spatial extent or internal structure) despite also having properties such as “spin”. That description of something as point-like seems pretty “mathematical” rather than being of a physical “object”.
Wave packets: Of course particles don’t behave as point-like entities, in that their influence has spatial extent. Electrons, for example, cannot get too close to each other, such that the effective “size” of such a particle can be summed up by the de Broglie wavelength. So perhaps we should think of the most basic “things” as being “wave packets”, spatially localised waves that behave, move around and evolve according to the equations of quantum mechanics. But surely the wave packet is a mathematical description of the physical thing, rather than being the physical thing itself?
Strings: One problem with point-like particles is that putting lengths of zero into the maths makes quantum mechanics stop working. For such reasons theorists have speculated that the basic “things” are not points but one-dimensional strings. This also has the advantage that strings naturally support waves,just as a guitar string does, in accordance with the wave-like maths of quantum mechanics. So far, though, string theory remains speculative, since empirical confirmation is beyond current technological capability.
Wave functions: Another option is to declare the quantum wave function itself to be what actually “exists”. But it is made up of complex numbers, not “real” ones, and can an “imaginary” number be physically extant? Also, formally the wave function extends to infinity, and there is only one wave function for the whole universe. So is there only one physical thing, one infinitely large object, in a complete reversal of Feynman’s account? Surely the wavefunction is a calculation tool, something that describes behaviour very well, but is not itself physically real? After all, isn’t the wave function “merely” maths?
Quantum fields: In quantum field theory particles are not seen as fundamental. Rather they are excitations of fields, being ripples and disturbances of a “field”, analogous to the waves caused by throwing a stone into a pond. But what is a “field”? It, again, is a mathematical construct. A classical field is a set of numbers, one number for each location in space. A quantum field is a mathematical operator at each location in space. That makes it a hugely abstract and mathematical concept, rather than something one would intuitively regard as physically real.
Space: So maybe we shouldn’t think of particles as being what ultimately exists, maybe we should think of space as the ultimate “thing”, with particles being ripples in space (the opposite of seeing space as an inert backdrop containing physically existing particles). But then there are proposals that “space” is itself a construct produced by the quantum entanglement of particles. So thinking of particles and space as distinct might be a mistake.
Which of the above should we go for? We already know, given quantum mechanics, that human intuition is a poor guide to reality at the micro level, so choosing based on accordance with our intuition would be unreliable.
The best option is perhaps to regard all of the above as instrumental, being models that work well and allow us to do calculations and make predictions, but not being how reality actually is. Perhaps such instrumental models are the best we can do?
One reason for doubting that any of the above is the final answer is that they are bound up with the nature of space itself, and we do not have a quantum theory of space (since we don’t yet have a theory merging quantum mechanics with gravitation). Thus we have a “known unknown” telling us that our current models are only instrumental approximations.
It does seem that the further we delve down into the most fundamental physics the more our descriptions seem mathematical rather than being about physical objects. Max Tegmark has taken this to the extreme, asserting that everything is ultimately made of mathematics. Our intuitive concept of what “physical stuff” is like may be appropriate only to the scale of our selves and our own senses, a vast number of orders of magnitude larger than the scale of the tiniest things (strings are hypothesized as being 1035 times smaller than a human). So we should surely expect any ultimate ontology to be pretty counter-intuitive. Indeed one could suggest that the fact that the character of the description changes from “physical” to “mathematical” is a sign that we’re approaching the underlying reality of what physical stuff is. Or we may just be approaching the limits of what humans can conceive.