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Chapter 5 Connections and Curvature

  • Mark J. D. Hamilton
Chapter
Part of the Universitext book series (UTX)

Abstract

From a mathematical and physical point of view it is very important that we can define on principal bundles certain fields, known as connection 1-forms. At least locally (after a choice of local gauge) we can interpret connection 1-forms as fields on spacetime (the base manifold) with values in the Lie algebra of the gauge group. These fields are often called gauge fields and correspond in the associated quantum field theory to gauge bosons.

In gauge theories, additional matter fields, like fermions or scalars, can be introduced using associated vector bundles. The crucial point is that connections (the gauge fields) define a covariant derivative on these associated vector bundles, leading to a coupling between gauge fields and matter fields (if the matter fields are charged, i.e. the vector bundles are associated to a non-trivial representation of the gauge group). In a gauge-invariant Lagrangian this results in terms of order higher than two in the matter and gauge fields, which are interpreted as interactions between the corresponding particles.

In non-abelian gauge theories, like quantum chromodynamics (QCD), there are also terms in the Lagrangian of order higher than two in the gauge fields themselves. This implies a direct interaction between gauge bosons (the gluons in QCD) that does not occur in abelian gauge theories like quantum electrodynamics (QED).

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Authors and Affiliations

  • Mark J. D. Hamilton
    • 1
  1. 1.Department of MathematicsLudwig-Maximilian University of MunichMunichGermany

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