Abstract
In ordinary quantum mechanics, the WKB approximation consisits of expanding in powers of Planck’s constant, ħ.To zero order, we have the classical trajectory; higher orders yield the quantum corrections to this trajectory.The path integral formulation lends itself particularly well to the extension of the method to the fied-theoretic case.To accomplish this, we reintroduce ħ into the expression for the generating function of a field theory that, to simplify, we start by taking scalar.
Alice laughed. “There is no use trying”, she said, One cannot believe impossible things“.
”I daresay you haven’t had much practice“, said the Queen..., ”Why, sometimes I have believed as many as six impossible things before breakfast!“
Lewis Carroll, 1896
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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.
To check with (2.1.1), we have to identify U(x) = exp(+ i ex) i.e., the U of (8.2.9) is the inverse of that defined in Section 3.1.
More general ansatze have been described by Corrigan and Fairlie (1977) and Wilczek (1977).
With finite Minkowski action, but infinite Euclidean action.
“Semi-instantons” with finite Euclidean action and half-integer topological charge seem to have been found by Forgacs, Horvath, and Palla (1981). 6 This holds for any gauge group that is simple and contains an SU(2) subgroup.
The formulation of field theory and in particular QCD on a lattice was given by Wilson (1974), who first proved confinement in the strong coupling limit. Application to actual calculation of physical quantities followed the pioneering work of Creutz. In our presentations we will follow mostly Wilson’s (1975) paper and Creutz’s (1983) text. Summaries of recent results of calculations way be found in the proceedings of any high energy physics conference.
Note that (8.5.4) presents a U(x) with a definition opposite to that of Section (3.1); now we set U(x) _ exp(+ E 0„t“) The notation (8.5.4) is more convenient for our purposes here. 9 Other types of lattices, and boundary conditions have been considered in the literature but, in the author’s opinion, to little advantage. The interested reader may find references in Creutz (1983).
That something like that had to happen is obvious if one realizes that the lattice regularization preserves dimension and gauge invariance (as will be seen).
’1 This is irrelevant for abelian gauge theories, but basic for nonabelian ones since there the U will be noncommutative matrices.
The trace is of course irrelevant for the abelian case, but we write it for ease of transition to the nonabelian case. 13 Taking the real part is necessary because we only consider plaquettes with a given, counter clockwise orientation. Because the U are unitary, and a clockwise plaquette is the inverse of the counter clockwise one, Tr U (clock) = Tr U Li (c. clock) = Tr U *0(c. clock) = (Tr U o(c. clock))*: if we summed over both orientations independently we need not take the real part. The expression (8.6.12) presents the supplementary advantage over (8.6.11) that CP invariance is respected on the lattice, and not merely in the continuum limit.
Other definitions of action, with the same continuum limit, are possible. Cf. the treatise of Creutz (1983) and references therein. 15 For group integration, cf. Appendix J.
Simplifiied and extended to other actions than Wilson’s by Dashen and Gross (1981) and Gonzalez-Arroyo and Korthals Altes (1982), using the background field formalism. See also Kawai, Nakayama and Seo (1981) for the introduction of fermions.
Note that, for consistency with the sign choices in these sections about lattice QCD, we define the Euclidean Lagrangian with a sign opposite to the usual one.
Anderson, Gustafson and Peterson (1977, 1979).
The results are due to J. P. Gilchrist, G. Schierholtz and H. Schneider; see Schierholtz’s (1985) review.
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© 1993 Springer-Verlag Berlin Heidelberg
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Ynduráin, F.J. (1993). Nonperturbative Solutions. Lattice QCD. In: The Theory of Quark and Gluon Interactions. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02940-4_8
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DOI: https://doi.org/10.1007/978-3-662-02940-4_8
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