### Life Cycle of a Theoretical Physicist

In 2001 Richard Borcherds gave a 15 week course on Quantum Field Theory, for which lecture notes [1] were taken by Alex Barnard — a student of Borcherds’. I’ve been reading them for a while now and I have to say, these notes are the best for those (mathematicians) who want to learn about QFT with little prior knowledge on the subject. So if you’re curious and you’d like to decipher all those mysterious expressions physicists use, here’s your chance. As a teaser let me just put a little excerpt here. “It’s funny ’cause it’s true…”

Life Cycle of a Theoretical Physicist

1. Write down a Lagrangian density $L$. This is a polynomial in fields $\psi$ and their derivatives. For example

$\displaystyle L[\psi]=\partial_\mu\psi\partial^\mu\psi-m^2\psi^2+\lambda\psi^4$

2. Write down the Feynman path integral. Roughly speaking this is

$\displaystyle \int e^{\mathrm{i}\int L[\psi]}\mathcal{D}\psi$

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” $\lambda$.

$\displaystyle a_0+a_1\lambda+a_2\lambda+\cdots$

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

$\displaystyle \int_\mathbb{R}\frac{1}{x^2}\ dx\longrightarrow \int_{|x|>\epsilon}\frac{1}{x^2}\ dx$

7. Now we have the series

$\displaystyle a_0(\epsilon)+a_1(\epsilon)\lambda+\cdots$

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…

[1] Borcherds, R.E., Barnard, A., Quantum Field Theory, Berkeley lecture notes, Fall 2001. arXiv:math-ph/0204014