QFT Take 2

Let’s actually try to make some progress on QFT today. There are three parts to make a minimal definition. First, you need a module D over a *-commutative ring. So to get a few definitions on the table. A *-ring, R, is pretty easy. You just have a ring with an antiautomorphism and involutive mapping $*: R\to R$. This means that (i) $(a+b)^*=a^*+b^*$, (ii) $(ab)^*=b^*a^*$, (iii) $1^*=1$, and (iv) $(x^*)^*=x$. So if you’ve seen rings, this shouldn’t be out of grasp. An example would be complex numbers with complex conjugation. A <a href=”http://en.wikipedia.org/wiki/Module_(mathematics)”>module</a&gt; is basically a generalization of a vector space.

The second part is a Hermitian inner product $(\cdot, \cdot): D\times D\to \mathbb{R}$. So recall that Hermitian just means that it is self-adjoint. You could think of this as when you express the operator as a matrix the conjugate transpose is itself again. Lots of operators satisfy this, like the differential operator. Essentially the property Hermitian is in place, because if something is obsevable then it is Hermitian.

The last part is that we need a *-algebra, $\mathcal{A}$, of operators acting on D. Let’s jump out to a bigger picture for a second. The details here are sort of the details of getting around a problem. What we really want is basic. We want a Hilbert space H and an operator satisfying the axioms we want. So our field $\phi: \mathbb{R}\times M\to M$, and our operator defined at each $x\in M$ as $\phi(x)$ (an operator on H). The problem we are skirting is one of how to get around $\phi(x)\phi(y)$ when x and y get arbitrarily close (an uncertainty problem as you might guess).

So we do the standard trick of “smoothing out the singularities.” Instead of points we will use bump functions. A bump function on M is just a smooth function with compact support. We redefine the operator then to be $\phi(f)=\int \phi(x)f(x)d^nx$. Here is why I jumped out to the big picture we are skirting around. $\mathcal{A}$ is generated by $\phi(f)$.

Some examples will be instructive. Let G be a group and D an orthogonal representation. Then $\mathcal{A}$ is the group-ring of G, with “*” as $g^*=g^{-1}$. Or we could let L be a Lie algebra acting on a vector space D with an invariant symmetric inner product. The algebra can be $\mathcal{A}=U(L)$ with $a^*=a$. Or we could take $\mathcal{A}$ as any $C^*$-algebra or von Neumann algebra and D any Hilbert space that is a *-representation.

These three examples should make us notice something. These are not things physicists typically work with (unless they are doing mathematical foundations of QFT or something). So despite having a definition in place, we might need to make some restrictions or correlations to what computations are being made down the road. These three examples are QFT’s, but that is sort of weird, since we usually speak of “QFT” and not “a QFT” or “this QFT” as if there is only one.