Introduction to Instantons

Abstract

In the quasi-classical approximation, i.e. for small coupling constants, the complicated vacuum structure of QCD can be shown explicitly. Along with the trivial vacuum sector corresponding to the vanishing of vacuum fields A ^a _µ = 0¹ (small oscillations near A ^a _µ = 0 are accounted for by perturbation theory) there are infinitely many other sectors in which the vacuum field (A ^a _µ )_vac yields G ^a _µv = 0 and, still, cannot be reduced to A ^a _µ = 0 by any continuous gauge transformations. These additional sectors are labeled by integer numbers, the so-called winding numbers or topological charges. The corresponding classification of the non-equivalent vacuum sectors was first given by Belavin, Polyakov, Schwarz and Tyupkin, 1975, where the tunneling transition connecting the neighbouring sectors was also found. The field configurations interpolating between the classical vacua with different winding numbers is localized in space and in imaginary time.