I'm still new to learning about these things, but is the viewpoint that a particle is a field excitation sort of the thing about starting with a lattice in the ground state with a field defined on the points of the lattice, then some excitations happen which cause the field to enter a particular "mode". This mode is the particle?
This captures the mathematical definition too which is just that an element x is idempotent if x applied to itself gives you back x. I.e, what you said that the function applied many times produces no change to the system.
