Probably has got a fair way from a good book that explains it like i'm an idiot.

I've seen the argument that the easiest approach is the computational one. It helps to understand logic gates, and why Boolean logic, by itself, is irreversible. This is intuitive if the AND and OR gates have multiple inputs and one output. In quantum logic, the important feature is that n inputs have n outputs, information cannot be lost or erased, only converted (eventually to classical measurements). Unitarity is required, hence Hermitian operators (the Pauli operators), and hence matrix exponentiation. The global phase symmetry (the electric field can have any value in spacetime equivalent to the matter wave having any phase) means there is a factor $$ e^{i\theta} $$, for some phase angle $$ \theta $$.

If you have studied physics, you might have noticed that a lot of physical processes

*evolve* exponentially, which is to say, many physical systems can be expressed as a Taylor expansion. Say you have a tank full of water, and you open a valve at the bottom of the tank, the flow will begin strongly and taper off to a trickle--the force of the escaping water follows an exponential curve. So if you know all the physical parameters, you can write this flow as a Taylor expansion.

I've seen a claim that the universe is itself a Taylor expansion, and we don't know all the parameters in it just yet.

Anyway, the article by t'Hooft: this is in the Scientific American June 1980 ed. so it's quite old. But he discusses gauge theories for all the four forces, he goes into Yang-Mills theory, it's been one of those "I'll read that later" magazine articles, for me. So there's a public domain copy of the article

here