I think you raise a very important question, but I think you make it sound more trivial than it is. The point is: a lot of physicists would *like* to have alternative expansions, but it is very difficult to come up with one. If you've got some suggestions, don't hesitate to put it forward.

The standard expansion starts from the time evolution operator $\mathcal{U}(t,t_0)$ and a Hamiltonian $\hat{H}$, which together form the Schroedinger equation:

$$ -i\partial_t \mathcal{U} = \hat{H}\mathcal{U}$$

Integrating this gives,
$$ \mathcal{U}(t,t_0) = 1 - i\int_{t_0}^t dt_1 \hat{H}(t_1)\mathcal{U}(t_1,t_0)$$
and by iterating, i.e. substituing this expression for $\mathcal{U}$ on the right hand side, you can come up with a formal power series for $\mathcal{U}$ called Dyson's series. You can modify it in some ways, like splitting the Hamiltonian into a solvable and perturbative part, and correspondingly for the time evolution operators. In the end you'll end up with expressing correlation functions that you *want* in terms of a series of correlation functions of some model that you *know*. And it's natural for this series to be an expansion in terms of the coupling constant of the perturbative part.

So can you get around this expansion? Well, sometimes there are some non-perturbative approaches available. You have, for instance, the realm of exactly solvable models. These rely on the presence of severe symmetry constraints. Examples are certain 2D conformal field theories, in which correlation functions satisfy differential equations. These equations arise due to restriction on the operator algebra (the presence of so-called null-states) and Ward identities associated with the symmetry algebra, which includes the conformal structure. Powerful stuff.

Other examples are the Bethe ansatz and the algebraic Bethe ansatz. As far as I understand these models are based on constructing a full set of eigenstates in some Hilbert space + extension, without an explicit reference to the Hamiltonian (meaning the Hamiltonian is subject to some restrictions, but need not be explicitly known). This is a very powerful technique and valid for the entire range of the coupling constant. But requiring integrability can be quite a constraint.

AdS/CFT was also mentioned, which is a marvellous weak/strong coupling duality. This makes use of the idea that correlation functions are the same for two seemingly different theories, which differ in dimensionality and the presence of gravity. Lattice regularization works also quite well, as far as I know.

An alternative expansion to Dyson's series which comes to mind is the Magnus expansion (see also here). The biggest advantage to this expansion is that it stays unitary once you cut the series off somewhere. But is it a strong alternative..?

My view on the matter is that a new expansion or approach could very well be the next best thing since sliced bread.

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