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What are resurgent transseries?

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I have heard that so-called resurgent transseries can be used to describe certain non-perturbative aspects of quantum field theories.

What are resurgent transseries in the first place and what are some specific applications in quantum field theory or theoretical physics generally?

asked Aug 6, 2016 in Theoretical Physics by Dilaton (4,295 points) [ no revision ]

a direct answer here :  Introduction to Resurgence, Trans-Series and Alien Calculus -  arXiv:1411.3585 , and applications on arXiv:1511.05977
 

Here is a nice informal introduction to asymptotics beyond all orders:

http://iopscience.iop.org/article/10.1088/2058-7058/6/6/21/meta

1 Answer

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In quantum mechanics and quantum field theory, renormalized perturbation theory produces expansions in powers of $g$, where $g$ is a renormalized coupling constant. By appropriate resummation using the renormalization group one can also capture terms involving $\log g/g_0$. However, it is well-known that, e.g., instantons contribute terms of order $e^{-c/g}$ or $e^{-c/g^2}$, whose Taylor expansion is identically zero, so that they do not contribute at all to the power series expansion. These contributions are therefore intrinsically nonperturbative.

A transseries is an expansion of a function $f(x)$ as a power series in a vector $z$ whose components are fixed functions of $x$. In QM and QFT, the relevant case is $x=g$ or $x=g^2$ and $z_1=x$, $z_2=e^{-c/x}$, and in QFT usually also $z_3=\log x/x_0$. Clearly, transseries are more flexible than ordinary power series.

Resurgent functions are functions arising in the analysis of singular points of germs of analytic functions. (A germ is essentially a formal power series expansion.) Resurgent transseries are transseries that arise in methods for resummation that generalize Borel summation by looking at the obstructions for Borel summability such as renormalons.

It turns out that using resurgent transseries, one can obtain under certain conditions information from the nonperturbative sector, starting from the perturbative series alone. The basic idea is to construct from the perturbative series the terms appearing in the renormalization group equation (RGE) by Callan and Symanzik, to substitute into it the leading terms of an appropriate transseries, and to determine the coefficients so that they match the RGE to the order specified.

To check that the method really works one can apply it to simple examples form quantum mechanics where other methods can be used to compute nonperturbative information. For example, for the double well potential, this was done by Jentschura & Zinn-Justin (Phys. Lett. B 596 (2004), 138-144, and for some other cases by Dunne & Ünsal (arXiv:1401.5202). For applications to gauge theories and to strings, see arXiv:1206.6272 by Marino; of course, in the latter cases there is no known alternative way of checking the results.

A more mathematically oriented overview is presented in arXiv:1411.3585 by Dorigoni.

Thus it seems seems possible (and sometimes is claimed) that the perturbative series already contains all nonperturbative information. The future will tell.

answered Aug 6, 2016 by Arnold Neumaier (12,385 points) [ revision history ]

Thanks for these nice and very interesting explanations !

The PhD thesis by Lutz Klaczynski (2016) also contains material on resurgent transseries, in the context of (i) Haag's theorem and (ii) the question of Landau poles in QED.

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