Demystifier does have a point though, in that anti-matter is to an extent an interpretation grafted onto the mathematics of the problem. What does rear its ugly head is the existence of negative energy solutions. As the hamiltonian is the generator of time translations, these particles would go backwards in time. The fix of reinterpreting this as a different particle traveling forwards in time is known as the Feynman-Stuckelberg prescription. (For completeness, you also have to reverse the direction of the 3-momentum.)(Coming clean: similar solutions also exist at a classical level! There's a good reason that escapes me now as to why we can ignore them classically but not in QFT... I'll try and get back to you when I remember it, or where I read what it was. I seem to recall it having to do with the necessity of summing over states, or because it ends up being unavoidable that transitions to these states would occur, or somesuch quantum quirk, but can't remember properly or work it out right now.)
Spin is something I'm about do some work on understanding myself, so there's less guarantee of correctness in what follows than is usual even in my posts
The important point seems to be that it's a measure of how states transform under representations of the Lorentz group. Obviously, if you're messing around with spacetime you have to make allowances for the possibility that this will have some, albeit an indirect, effect on the state vector of some system that's supposed to live in that space. Furthermore, it makes sense that if you perform two successive transformations, equivalent to some third, single lorentz transformation, then the transformation you impose on your hilbert space shouldn't distinguish between physically equivalent states. My group theory isn't up to letting me progress much beyond that elementary level (which I'll hopefully be doing something about later today, or at least this week) but it at least makes sense to me that we need a bookkeeping device that tells us what representation our state transforms under. Spin-0 just means we have the trivial representation of the Lorentz group, i.e. the case where it actually makes no difference. That works for the scalar field, so if there was nothing in the universe apart from the Higgs, there'd be no need for the construction. Luckily for us, the universe turns out to be a more interesting place than that.
