Validity of perturbation theory

[spaghetti3451]

I was wondering why perturbation theory works in quantum mechanics. My lecturer said that no one really bothered why it worked anyway, until they found it gave problems in QFT and came back to non-relativistic quantum mechanics and found why it worked in this domain.

Can anybody explain?

 

[atyy]

He might be thinking of things like http://arxiv.org/abs/quant-ph/0503074.

He might also be thinking about divergent series which are asymptotic, and can give good approximations. One example of a meaningful divergent series is the asymptotic series that can be used to derive Stirling's approximation. http://en.wikipedia.org/wiki/Stirling's_approximation

However, the problem in QM and relativistic QFT in 3+1 dimensions is not identical. In QM there is a non-perturbative definition of the theory at all energies, so there is something to approximate. There is at present no non-perturbative definition of the theory at all energies in relativistic QFT in 3+1 dimensions.

 

[dextercioby]

Quantum Mechanics takes place in a single separable Hilbert space. Perturbation theory works because the Kato-Rellich theorem works. The mathematical foundations of (time-indep) perturbation theory were laid by Franz Rellich in the 1930s and reached maturity with T. Kato's work at the end of the 1940s and beginning of the 1950s.