Instantons and Borel Resummation

+ 5 like - 0 dislike
270 views

As explained in Weinberg's The Quantum Theory of Fields, Volume 2, Chapter 20.7 Renormalons, instantons are a known source of poles in the Borel transform of the perturbative series. These poles are on the negative real axis, and the series remains Borel-summable as long as the coupling constant is not too large.

However, instantons are objects in the Euclidean version of QFT. What's the significance of the above Borel resummation in the Minkowski theory?

This post has been migrated from (A51.SE)
retagged Jan 1, 2016
You seem to be thinking that the perturbation theory of the Euclidean and Minkowskian theories are unrelated, but in fact they are related. I suppose a good analogy here is complex analysis, where you can compute an integral by contour deformation and the position of the poles in complex plane is important.

This post has been migrated from (A51.SE)
@SidiousLord: good point. But I'm more interested in seeing an application in a specific problem, which Weinberg's book hasn't done.

This post has been migrated from (A51.SE)

Borel summation provides analytic resummed functions. Analytic continuation of the $n$-point functions of a Euclidean quantum field theory with reflection positivity to real time gives the $n$-point functions of a Minkowski quantum field theory. The instantons of a Euclidean quantum field theory lead to a nonuniqueness of the Minkowski vacuum and corresponding $\theta$ angles (parameterizing this) in the Minkowski quantum field theory.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysics$\varnothing$verflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.