Desbois.Les.Houches. Impurities and quantum Hall effect
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Extra resources for Desbois.Les.Houches. Impurities and quantum Hall effect
6. But note that this transformation is not a bundle automorphism from Trautman’s group G, nor is it generated by an element of the structure group G. Indeed, it leaves the intrinsic geometric structure of the bundle itself unchanged, affecting only the way its connection is taken to be represented on the base space M. In the ﬁber bundle formulation, the (electro)magnetic ﬁeld is represented by the curvature of the principal ﬁber bundle P(M, U(1)). A precise deﬁnition of this notion is given in appendix B, but the basic idea of curvature may be simply explained geometrically.
It is tempting to call such a transformation an active gauge transformation, in contrast to a passive gauge transformation, consisting of a mere change of bundle section that preserves the bundle’s intrinsic structure and leaves the connection unchanged. But this terminology should be resisted at this point, since to employ it is tantamount to endorsing a particular interpretation of the ﬁber bundle formalism. 18 1 what is a gauge theory? 2 Electromagnetic interactions of matter ﬁelds The previous section showed how to represent the action of a classical electromagnetic ﬁeld on quantum particles using ﬁber bundles.
2 a ﬁber bundle formulation 17 by Aμ = (ϕ, −A) or Aμ = (ϕ , −A ) respectively. In this way, a choice of potential for electromagnetism goes along with a choice of phase at each point for the particle’s wave-function, and this simultaneous choice corresponds to a choice of section for the principal bundle P. 6, which also illustrates how both principal ﬁber bundle and associated vector bundle share a common base space representing space-time. A change from one section of the principal bundle P to another may be taken to represent a gauge transformation.