Trinh @ Bath

MMath project 2020-21'

Divergent series and quantum physics

In the study of quantum theories (like quantum mechanics and quantum field theory), one of the key questions is how to grapple with the interpretation of divergent series. As a simple example, you have learned how to solve the regular Schrodinger harmonic oscillator, where the potential function looks like a quadratic, $$ V(x) = \frac{1}{2} x^2. $$ You learn that the permissible energies take discrete values like $E_n = (n + 1/2)$. In an effort to understand how this can be generalised, you might try to perturb the potential that it includes a small higher-order term, like this: $$ V(x) = \frac{1}{2} x^2 + \lambda x^4. $$

This is known as the potential for the quartic anharmonic oscillator. Then you try to write the energies as something that looks like this: $$ E_n = \left(n + \frac{1}{2}\right) + \lambda E_{n1} + \lambda^2 E_{n2} + \ldots = \sum_{j=0}^\infty \lambda^n E_{nj}. $$ But doing so, you quickly realise that the series is divergent and the terms in the series get very large very rapidly. How do you assign meaning to a divergent series? Perhaps put more sensationally: how could you sum the divergent series to obtain the actual answer of the energy? This problem lies at the heart of a lot of the puzzling issues about quantum theories—mainly that many of the quantities that you try to write down are divergent, and it is not clear how to extract meaning from them.

This project will be about learning about some of these techniques for studying divergent series.