I wanted to highlight a paper from 2004 by Jens Andersen that I came across that addressed an issue I have had ever since I began studying BECs. Before talking about thearticle’s contents, I feel the need to set the stage a bit before explaining what I like so much about it.

Having started my studies as a physicist with a stiff dose of Cliff Burgess’ perspective on EFT I have always found treatments of the interacting Bose gas to be a little opaque and arcane. Often results are derived using only two-body physics but then applied to the quantum many-body system. For instance the pseudo-potential is often derived in the context of scattering theory, and then immediately applied to a Bose gas in the $N\rightarrow\infty$ limit.

Books such as Abrikosov, Gorkov and Dzialoszynski’s text (as recommended by Gora Shlyapnikov at OIST’s summer school on coherent quantum dynamics) explicit and tackles the problem in the n-body problem showing, how the dilute-gas parameter $ n a_s^3$ emerges. The key details are a the separation of the fields into condensed and non-condensed components (this how one gets the n) and some ladder summations. This treatment, however is plagued with the technical difficulties associated with this latter point, and in my opinion this obscures the results.

Jens Andersen clears this up by taking advantage of the fact that he is writing in 2003 and not 1961, and so can take a more modern perspective. He emphasizes that we may construct a local effective field theory, and as is typical, the power counting scheme naturally identifies the leading order interaction term which is a contact potential. This renormalized coupling already has the infinite ladder sum accounted for. From here a Abrikosov and co. style separation of the condensed and fluctuating components could be performed.

This treatment also makes it clear why certain effects are universal such as the Lee-Huang-Yang energy, and why two-body physics can be used to accurately predict the behaviour of the many-body system. Once we have the effective Lagrangian it can reproduce observables in an Fock space, 2-body or N-body, and a matching calculation can be done in any of them.

All of these thoughts have been swirling about in my head for at least a year, but it was nice to see them laid out so clearly. Often when studying literature related to quantum gases one finds quotes such as

“As a consequence naive mean field approximations, which neglect the correlations between particles due to interactions, implicitly relying on the Born approximation…”

Yvan Castin – arXiv:cond-mat/0105058

which I am sure is technically correct, but is totally opaque to me. For instance I am unaware of any formal connection between mean-field theory and the Born-approximation. Furthermore, given the Born approximation is, in this context, essentially capturing two-body scattering behaviour, it is not obvious to me this is true. As a consequence, I have always struggled to understand statements like those quoted above.

I think Andersen’s treatment is really well laid out, and takes advantage of a more modern perspective on quantum field theory, which clears up certain conceptual issues and lets you see why these results do not explicitly rely on a ladder sum, nor the details of the original potential.