What is the impact of nonperturbative effects on time-dependent quantum theory?

[TrickyDicky]

If I haven't understood this tricky stuff very badly when the Hamiltonian is time independent, then Schrödinger’s equation implies that the time evolution of the quantum system is unitary, but for the time-dependent Hamiltonian one must add some mathematically "put by hand" assumptions (although they make physical common sense) like causality and independence of the time evolution operator on the state of the wavefunction to ensure the time evolution operator's unitarity demanded by QM's postulates and conserve the probability density.
Even so, we still often obtain divergent series like the Dyson series, that luckily for small coupling constants get close to experiment in the first terms.
Doesn't this perentorial need of perturbation theory suggest that it would be maybe more natural either a nonlinear or a non unitary approach to time-dependent quantum theory?

 

[atyy]

Divergent series are not necessarily bad. In some cases they are "asymptotic". An example of a useful asymptotic series is that used in some derivations of Stirling's approximation. However, I don't think it's been shown that the perturbation series in QED is asymptotic.

Divergent asymptotic series in Stirling's formula:
http://aofa.cs.princeton.edu/40asymptotic/
https://people.math.osu.edu/costin.9/pages/p117.pdf

Divergent asymptotic series in classical statistical mechanics:
http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/lec12.pdf

 

[The_Duck]

Even so, we still often obtain divergent series like the Dyson series, that luckily for small coupling constants get close to experiment in the first terms.
Doesn't this perentorial need of perturbation theory suggest that it would be maybe more natural either a nonlinear or a non unitary approach to time-dependent quantum theory?

The fact that the power series expansions of perturbation theory are often divergent doesn't mean anything bad about the underlying theory. It just means that perturbation theory isn't the whole story--there are "nonperturbative" effects that are invisible in pertubation theory. For example, quantum tunneling effects generally scale like $e^{-1/g^2}$ where $g$ is some coupling constant. You can never see such effects in a power series expansion in $g$ around $g = 0$, because $e^{-1/g^2}$ has no series expansion around $g=0$. Therefore the perturbation series expansion can't be convergent, because if it was it would converge to the wrong value.

But this is a failing of perturbation theory and not quantum mechanics. Nonperturbative treatments--for example, numerical solutions--show the nonpertubative effects like tunneling.

 

[TrickyDicky]

The fact that the power series expansions of perturbation theory are often divergent doesn't mean anything bad about the underlying theory. It just means that perturbation theory isn't the whole story--there are "nonperturbative" effects that are invisible in pertubation theory. For example, quantum tunneling effects generally scale like $e^{-1/g^2}$ where $g$ is some coupling constant. You can never see such effects in a power series expansion in $g$ around $g = 0$, because $e^{-1/g^2}$ has no series expansion around $g=0$. Therefore the perturbation series expansion can't be convergent, because if it was it would converge to the wrong value.

But this is a failing of perturbation theory and not quantum mechanics. Nonperturbative treatments--for example, numerical solutions--show the nonpertubative effects like tunneling.

I was not thinking in terms of what is "bad" or not for for a theory, but precisely pointing at the necessity of the nonperturbative effects and numerical methods you mention, and to me these effects and methods are quite often related to nonlinear effects and equations(say, in GR for instance where numerical methods are often the only possibility for BH hole problems, etc).

You mention quantum tunneling which seems a good example of an effect that resembles a nonlinear behaviour.