Non-perturbative effects: classical or quantum?

Question

Are non-perturbative effects (solitons) classical or quantum effects (corrections) ? (examples ?)

My confusion stems from the fact that, for instance, an instanton is a classical solution of the equations of motion. Why is it said to be a quantum correction then? At which point does $\hbar$ enter the game? (the problem, at least for me, stems from taking always $\hbar = 1$).

This post imported from StackExchange Physics at 2016-09-04 15:25 (UTC), posted by SE-user BLS

Answer 1

Instantons appear as classical solutions of Yang-Mills equation due to nontrivial topology of nonabelian gauge group. They may play some role in physics because of requirement of finiteness of vacuum energy. When we ask how explicitly instantons affect on physics, we must use quantum description: nontrivial homotopy group of nonabelian symmetry group and requirement of the finiteness of energy imply the statement that there are infinite number of different topological vacua which are labelled by discrete winding number $n$, and the true vacuum of theory is superposition of these vacua,

$$|\text{vac}\rangle \equiv \sum_{n}e^{in\theta}|n\rangle$$ This is exactly quantum approach. Instantons then acquire as quasiclassical amplitude of tunneling between vacua when we include extended field configurations in path integral (this is required by principle of cluster decomposition of S-matrix), $$\langle n - 1| \hat{S}|n\rangle \neq 0,$$ and the amplitude is exponent with degree $\sim \frac{1}{\hbar}$. That's where $\hbar$ arises.

This post imported from StackExchange Physics at 2016-09-04 15:25 (UTC), posted by SE-user Name YYY

Comments

Thank you, but what about generic solitons? Can you give me self-contained references where to find everything needed to understand solitons?

This post imported from StackExchange Physics at 2016-09-04 15:25 (UTC), posted by SE-user BLS

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@BLS: you may find the brief description of generic solitons in Weinberg's QFT Vol. 2 (the paragraph about extended field configurations).

This post imported from StackExchange Physics at 2016-09-04 15:25 (UTC), posted by SE-user Name YYY

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Ok, thank you. Do u know any other specific references on non-perturbative effects?

This post imported from StackExchange Physics at 2016-09-04 15:25 (UTC), posted by SE-user BLS

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@BLS : I also may recommend Rubakov book "Classical theory of gauge fields" (you may find there topological configurations theory as well as applications to the Standard model), Zahed and Brown article "The Skyrme model" (in which baryons are explained in terms of topological configurations called skyrmions).

This post imported from StackExchange Physics at 2016-09-04 15:25 (UTC), posted by SE-user Name YYY