What is Size?
As far we can tell, the most fundamental particles may not have size (in the testable three spatial dimensions to which we are used to). But we have distance between particles, and that distance is what actually gives size to macro objects.
Now, there is some known distance between particles if we have some idea of where they are. And for we to know where they are we need to measure their interactions. For example: we can know where an electron is if we can make it collide with a photon, or we can know where proton is if we can make it collide with an electron, etc.
We have discovered that the position of particles is not deterministic. Yet, with enough measurements we have been able to create a model of their position probability distribution with virtually no error (as far as our current measurement instruments are concerned). Of course, reaching virtually no error is archived by adjusting the theory to the reality.
: By "we", I mean human kind.
: If particles are - or are made of - hypothetical strings that have size on other spatial dimension which we are currently unable to test is another thing.
But - you say - we know the diameter of electrons and protons...
No, we don't. The way we make these particles interact is by the electromagnetic force. That is, the particle don't really collide... they repel!
The way we have to estimate the diameter of particles is by the principles behind the Rutherfords Gold Foil experiment.
In detail, the experiment is as follows: shoot alpha particles to a thin gold foil, and detect where the alpha particles go after it reaches the gold foil.
From the experiment, it is observed that the alpha particles are scattered. The explanation is that the atom of gold has a nucleus that repels the alpha particle causing it change its direction. So, if we want to estimate the size of the nucleus we have to analyze how alpha particle behave near the nucleus.
Note: the link above explains in more detail and goes on how to calculate the diameter.
What is important to notice of this method of measurement is that it is based on the repulsive force caused by the charge of the particle. It is what we know as “charge radius”, and in practice is only serves as an upper bound to the size of the particle.
So, even if we know the charge radius of a particle, we only know that if the particle has any size it must be smaller than that. So, as said at the beginning: as far we can tell, the most fundamental particles may not have size.
But protons are made of quarks!
Yes... we call "quarks" the thing that protons are made of. But we cannot measure how far apart a quark from another quark is when they are bound. In fact, there are no free quarks. Even knowing the distance between quarks, we have the problem of the size of the Quarks themselves. Quarks just happen to be the perfect description of the interactions of the particles that we claim that are made of them.
Introduction to the photoelectric effect
When a photon interacts with an electron, it gives it energy that displaces the electron to another orbital/electron shell/bond.
If a photon doesn't have the energy required to displace the electron, then it simply doesn’t do it. That means that each material there is a given set of amounts of energies that a photon would have to have to interact with it.
If the electron that is displaced belongs to a bond, it breaks the bond (which is understandable if we consider a bond a shared orbital) in which case light is serving a catalyst of the chemical reaction that ensues.
Also, displacing an electron may move it to medium that allows it to move freely, guided by the local electric field. That is how light can create electric currents.
We will come back to the idea that photons interact with electrons.
But even if something interacts with light doesn't mean it is visible. It may be the case that you are unable to focus on it, or perhaps it is too dark (not enough light) for you to see. In fact, you want to use light of shorter wavelengths (higher frequency) to be able to focus on smaller object.
Here is where the shrinking part of the question becomes relevant: as you becomes smaller your eyes become smaller, and thus its curvature increases! And with it changes its focal length, also decreasing! A shorter focal length means that you can now focus on smaller things (in particular if we consider that now the eyes are closer together).
Yet, smaller eyes also mean less light entering the retina, so as you become smaller the world becomes darker.
What about the eyes? Let's see how shrinking works:
- As you decrease in size you lose mass. At some point you die, the reason being that your body can't no longer sustain itself. No.
- As you decrease in size you increase density, and at some point you will die because chemistry stop making sense (e.g.: hemoglobin in your blood being smaller than the oxygen you breathe).
- As you decrease in size you make atoms smaller. You push proton to proton and the nucleus become radioactive, the critical mass for nuclear chain reaction becomes smaller and smaller... until Boom. No.
- As you decrease in size the space from orbital to other orbital decreases (the nucleus remains intact). These happen :
- The energy required moving an electron from one orbital to another becomes smaller. If less energy is required, then you become sensible to lower light frequencies, which move the visible spectrum to shorter wavelengths .
- Since less energy is required to move your electrons, you will burn and die because all your body will be subject to uncontrolled chemical reactions.
- As you decrease in size you survive using unobtainium macguffins handwavium, or similar technology. You will be able to see thanks to the Sword of Omens or any other technology indistinguishable from magic that you may have at hand.
: I’m unsure of the order in which those happen – this is all speculative, since we can’t do this kind of shrinking.
: Larger wavelengths mean lower frequencies (to infra-red and radio waves). It should be noted that you need higher frequencies to be able to focus, and you will be seeing lower frequencies. So, it is all blurrier and blurrier.
Good! You got the most recent version of the unobtainium macguffins!
You first approach the size of an atom. Since you are a complex system and not a single particle (and you are using your unobtainium macguffin handwavium) you are not affected by decoherence. Also the atoms around you are pretty much stable, so no big deal. You continue to repel atoms. The only weird thing is that some particles would be able to tunnel through you.
What do you see? Well, there is light everywhere, it is making the electrons jump from an orbital to another, and then the electrons jump back and release the photon again. This is happening everywhere because things are agitated! I mean, temperature exists and it is not absolute zero!
So, everything would be bright, uniformly bright, no shape. Furthermore, repulsion happens by exchange of virtual photons, will you be able to see them? Who knows‽ Maybe you get into a bond and join a molecule.
Next you approach the charge radius of protons. I don’t believe you would see anything; the effect of light pushing you is greater than any illumination. At this scale the classical idea that light spreads uniformly from the source is no longer relevant. Instead a photon interacts with you, or not.
You next approach the size of your event horizon! In the past light had a chance to be reflected out of you, but now you are so small that light that would have miss you due to your size is trapped by your gravitational force. Furthermore light going out of you won’t go very far anyway (if something smaller than an atom can be considered far). You no longer see anyway.
I don’t know why we bother; you are a micro black hole anyway...
Finally you approach the Plank Length – The Plank Length is clever bit of calculation, but we don't really know what it means. Can something be smaller than the Plank Length? As far as we know there is no such thing. We don’t have means to measure it; in fact, we believe it can't be done. And for any practical purposes it doesn’t matter if something smaller is possible.
The Plank Length is a popular idea to cite, but saying “nothing can be smaller than the Plank Length” is just flat. What we know about the Plank Length is:
Writing this I came up with an interesting idea: any movement of less than a Plank Length could be considered to take no energy, and thus happens spontaneously, and instantaneously. Speed at Plank Length could be meaningless.
... You are a micro black hole anyway.
Another way to shrink
Let’s say that you can change the curvature of space-time in an unnatural way. That is, a way that doesn’t follow the curvatures that are described in Einstein’s equations, such that you can have a small area such that if you measure distance traversing it, it is less than you would calculate if you measure distance circumventing it. This is often referred to as “Tardis Space”.
Now, I do not pretend that there is a discontinuity in the curvature of space-time. Instead the curvature must be continuous. Looking from outside at a reasonable distance to the inside of the “Tardis” it would look as if everything inside is smaller, almost fitting correctly the apparent external size, you may think it is a miniature set. As you approach the inside start to appear larger, at the moment when you are at the door you can see the inside almost at full size. Now if you turn around and look to the outside, you see that it becomes giant as you walk in, but the inside is now apparently of normal size.
What I’m describing is an effect of gravitational lensing, exaggerated. Appropriate time dilation still applies. Times passes at a lower rate while you are inside.
Now I understand why in some RPG videogames when you walk on the over-world you are a giant, then you step on a tiny town and when you are inside it is of normal size… and the day cycle seems to stop.
It should be noted, that even with this weird curvature, what is being described is gravity. I find it hard to wrap my head around that idea that this thing pulls you in… but that is what General Relativity suggests. If that is true, then it is also true that any object entering experiences tidal forces, and thus you may not be able to survive the walk in.
And we want this effect at infinitum. Yeah, black hole, right there.
Note: I said the curvature for this does not follow Einstein's equations. You could get away with a curvature that doesn't extend outwards to infinity. That would be interesting.