Abstract
In ordinary quantum mechanics, the WKB approximation is obtained by expanding in powers of Planck’s constant, ℏ. To zero order we have the classical trajectory; higher orders yield the quantum fluctuations around this trajectory. The path integral formulation lends itself particularly well to the extension of the method to the field-theoretic case. To accomplish this, we reintroduce ℏ into the expression for the generating functions of a field theory that, to simplify, we start by taking to be scalar. From (1.3.11) we then have,
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This is usually referred to as Euclidean QCD or, more generally, Euclidean field theory. We will distinguish Euclidean quantities from the corresponding Minkowskian ones by underlining the first. Also, sums over repeated space—time indices will be written explicitly, while sums over implicit colour indices will continue to be understood.
To check with (2.1.1) we have to identify U(x) = exp(-+i ∑ θata), i.e., the U of (8.2.9) is the inverse of the U defined in Sect. 2.1.
More general ansätze have been described by Corrigan and Fairlie (1977) and Wilczek (1977).
“Semi-instantons” with finite Euclidean action and half-integer topological charge seem to have been found by Forgács, Horváth and Palla (1981).
This holds for any gauge group that is simple and contains an SU(2) subgroup.
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© 1999 Springer-Verlag Berlin Heidelberg
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Ynduráin, F.J. (1999). Instantons. In: The Theory of Quark and Gluon Interactions. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03932-8_8
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DOI: https://doi.org/10.1007/978-3-662-03932-8_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-03934-2
Online ISBN: 978-3-662-03932-8
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