this is my mental picture for how they travel without a medium, how (like water waves) some can't stay still, why they have wave and particle properties, energy/mass equivalence, conservation, etc. it might capture uncertainty too -- i've heard that all waves have an uncertainty relation (say in their power spectrum), but i don't get why -- it seems like we can discuss waves with absolute precision.
Answer
Actually that's not too far off the mark, although I'm not sure "knot" or "kink" is the best word. Quantum field theory, the best theory we currently have to describe particles, says that particles correspond to excitations of a field, which are kind of like waves in water; you could consider the surface of a pond "excited" whenever it's not flat. Just as with water waves, there's an infinite variety of "shapes" you can have for these excitations. For example, you could have a repeating wave, in which the surface of the water cycles up and down over a large area, or you could have just one wave front that just propagates across the water without spreading out very much. The former case is pretty typical for things like light waves, and the latter case is pretty typical for particles of matter, although technically any kind of field (whether it's the electromagnetic field for light, or a quark field in matter, or whatever) can have any of the different types of excitations.
By the way, according to special (and general) relativity, even an object that is standing still is moving through time. So all of these excitations move through spacetime in one way or another. But only certain ones (the excitations in fields corresponding to massless particles) can move through space in such a way that they appear to us to be traveling at the speed of light.
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