QFT take 3

I’m going to try to clean up some stuff from last time. I’ve realized that I was pretty sloppy. Given a classical field theory we can get to a QFT. To do this we just take the algebra $\mathcal{A}$ to be the universal *-algebra generated by the classical fields. Now to get the inner product to behave the way we want it, take any state $\omega: \mathcal{A}\to R$ such that $\omega^*=\omega$. Now we just define the inner product by $(a,b)=\omega(ab^*)$ and the module $D=\mathcal{A}\diagup\ker(\cdot, \cdot)$. If you want more details on why this works, it is known as the Gelfand-Naimark-Segal Construction.

Now we need to somehow get $\omega$ on the algebra $\mathcal{A}$. In physics people call this the Feynman integral. So we get something that is probably familiar to people that have taken a quantum mechanics class, $\displaystyle \int (\int\phi(x)^*f(x)dx) e^{i\int L(\phi)d^4x}d\phi$. Here we run into another of those math vs physics problems. Since our space could be infinite dimensional, we don’t have a well-defined notion of what a measure is there. We can’t even define a Radon measure and do our trick of using bump functions.

Let’s beat the problem this time by ignoring the how to define on things we don’t care about. Look at that Feynman integral. We need to define the integral of things that look like $\{Jet space forms\times e^{quadratics}\}$. We create what is called the Feynman measure, which is just a measure on this particular space. Now although this measure is not unique, it equates to picking a renormalization scheme, so we have a group acting on the space of F. measures and Lagrangians that preserve the QFT. This is essentially what gives rise to what are known as “anomalies.”

Well, I don’t think I want to go into the nitty-gritty of the specifics of everything. This is a pretty rough idea that I just threw out there in case people were wondering. If you’ve followed this and would like some more details, just comment. Otherwise, my guess is that for the most part people don’t really care and so I’ll move on to something else.

Edited: Well, darn. I’m sure I’m going to keep remembering things I left out since I took such a scatter brained approach to this. Well, at the absolute very least I should include the Wightman axioms. We want to extrapolate from what has been presented so far to get what the axioms of a QFT are.

1) We’ve seen this explicitly along with the rationale for it. We want $\mathcal{A}$ to be generated by $\phi(f)$ where f is a classical field with compact support.

2) The inner product is positive definite, for pretty intuitive reasons.

3) We have Lorentz and translation invariance. This is also intuitive since we want QFT to work with relativistically.

4)  An operator that pushes things forward in time is positive. This amounts to an operator that increases energy is positive ($(Ex, x)\geq 0$ for any $x\in H$).

5) We have a “locality” condition. This amounts to $\phi(f)\phi(g)-\phi(g)\phi(f)=0$ if the supports of f and g are spacelike separated (the commutator is 0).

6) There is a vacuum vector. This amounts to something being fixed by the Lorentz group.

There are some other minor axioms as well. Remember these things kind of vary depending on the situation. We are still not sure of the correct formalism. I should also emphasize that everything I’ve done so far is the free Hermition scalar QFT. OK. Hopefully that will cover me for now.