Abstract
The functional formalism apparently fulfils the theorist’s dream: field theory reduced to quadratures! This and the following sections of the present chapter will be devoted to an introduction to field theory (especially, QCD) on a lattice, precisely the tool to implement such a programme.1
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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References
The formulation of field theory, and specifically QCD on a lattice, was given by Wilson (1975), who first proved confinement in the strong coupling limit. Application to actual calculations followed the pioneering work of Creutz. In our presentation we will follow mostly Wilson’s (1975) paper and Creutz’s (1983) text. Summaries of results of recent calculations may be found in the proceedings of specialized conferences; some are presented in Sect. 9.5.
We write the Lagrangian for a single quark flavour. For several flavours, replace m by m q , and sum over the flavours q.
Note that (9.1.4) uses a convention different from that of Sect. 2.1; now we set U(x) - exp(-+-i ∑ θa (x)ta). The notation (9.1.4) is forced by the fact that Lq corresponds to —L (see above).
That something like this had to happen is obvious if one realizes that the lattice regularization preserves dimension and gauge invariance (as will be seen) .
The trace is of course irrelevant for Abelian fields, but we write it to ease the transition to the non-Abelian case.
Other definitions of action, with the same continuum limit, are possible and have been used in the literature; see the treatise of Creutz (1983) and references therein.
Simplified and extended to actions other than Wilson’s by Dashen and Gross (1981) and González-Arroyo and Korthals Altes (1982), using the background field formalism. See also Kawai, Nakayama and Seo (1981) for the introduction of fermions. The values of the quark masses also differ between the lattice and the continuum; see González-Arroyo, Martinelli and Ynduráin (1982).
Of course, for QCD we have other reasons than the strong coupling lattice evaluation for believing in a linear potential and/or confinement.
All the same, it is a fact that lattice evaluations tend to give smallish values for ∧ — and for the light quark masses too; see Sects. 10.3, 4.
More accurately, two quarks and two antiquarks.
Anderson, Anderson, Gustafson and Peterson (1977, 1979) .
As opposed, for example, to the Higgs vacuum, that extends over all spacetime.
The results are due to J. P. Gilchrist, G. Schierholz and H. Schneider; see Schierholzs (1985) review, or, for more recent results, Aoki et al. (1998).
Bali et al. (1997); Luo et al. (1997) .
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© 1999 Springer-Verlag Berlin Heidelberg
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Ynduráin, F.J. (1999). Lattice QCD. In: The Theory of Quark and Gluon Interactions. Texts and Monographs in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03932-8_9
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DOI: https://doi.org/10.1007/978-3-662-03932-8_9
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