Formulations

The most popular formulations of quantum field theory are

• Canonical Quantum Field Theory $\leftrightarrow$ Quantum Field Theory in Hilbert space,
• Path Integral Quantum Field Theory $\leftrightarrow$ Quantum Field Theory in Phase Space

Canonical quantum field theory can be summarized as follows:

The main components of quantum field theory are:

1. Theory of free fields. Here we neglect any interaction terms and only calculate what different fields do when they are on their own.
2. Theory of interacting fields. Here we derive the correct terms in our equations that describe interactions using gauge symmetry. Unfortunately, it turns out that we can't solve these equations in closed form but only approximately. There is a powerful mathematical machinery that allows us to calculate results approximately. At the heart of this machinery are the famous Feynman diagrams. In addition, an important trick (the interaction picture) allows us to reuse most of the things we derived for free fields.
3. Renormalization. The main problem in quantum field theory is that calulcations are technically extremely challenging. One reason for this is that when we calculate things naively we end always end up with the result "infinity". This is physically nonsensical and we need to take greater care. The mathematical machinery that allows us to tame these infinities is known as renormalization.

Lifecycle of a Quantum Field Theorist

1. Write down a Lagrangian density $L$. This is a polynomial in fields $\psi$ and their derivatives. For example $$L[\psi] = \partial_\mu\psi\partial^\mu\psi - m^2\psi^2 + \lambda\psi^4$$
2. Write down the Feynman path integral. Roughly speaking this is $$\int e^{i\int L[\psi]} D\psi$$ The value of this integral can be used to compute "cross sections" for various processes.
3. Calculate the Feynman path integral by expanding as a formal power series in the "coupling constant" $\lambda$. $$a_0 + a_1\lambda + a_2\lambda + \cdots$$ The $a_i$ are finite sums over Feynman diagrams. Feynman diagrams are a graphical shorthand for finite dimensional integrals.
4. Work out the integrals and add everything up.
5. Realise that the finite dimensional integrals do not converge.
6. Regularise the integrals by introducing a ``cutoff'' $\epsilon$ (there is usually an infinite dimensional space of possible regularisations). For example $$\int_\mathcal{R} {1\over x^2} dx \longrightarrow \int_{|x|>\epsilon} {1\over x^2} dx$$
7. Now we have the series $$a_0(\epsilon) + a_1(\epsilon)\lambda + \cdots$$ Amazing Idea: Make $\lambda$, $m$ and other parameters of the Lagrangian depend on $\epsilon$ in such a way that terms of the series are independent of $\epsilon$.
8. Realise that the new sum still diverges even though we have made all the individual $a_i$'s finite. No good way of fixing this is known. It appears that the resulting series is in some sense an asymptotic expansion.
9. Ignore step 8, take only the first few terms and compare with experiment.
10. Depending on the results to step 9: Collect a Nobel prize or return to step 1.

There are many problems that arise in the above steps

• [Problem 1] The Feynman integral is an integral over an infinite dimensional space and there is no analogue of Lebesgue measure.
• [Solution] Take what the physicists do to evaluate the integral as its definition.
• [Problem 2] There are many possible cutoffs. This means the value of the integral depends not only on the Lagrangian but also on the choice of cutoff.
• [Solution] There is a group $G$ called the group of finite renormalizations which acts on both Lagrangians and cutoffs.

QFT is unchanged by the action of $G$ and $G$ acts transitively on the space of cutoffs. So, we only have to worry about the space of Lagrangians.

• [Problem 3] The resulting formal power series (even after renormalization) does not converge.
• [Solution] Work in a formal power series ring.

Lectures on Quantum Field Theory by R. E. Borcherds, A. Barnard