External and Internal Operations

Part of the Fundamental Theories of Physics book series (FTPH, volume 163)


Basic physical theories involve both external or spacetimelike and internal or chargelike degrees of freedom in complex representation vector spaces that are acted on, respectively, by operations from the Poincaré group, i.e., by Lorentz transformations and spacetime translations, and by electroweak and strong operations from the hypercharge, isospin, and color groups.

The external–internal dichotomy goes with a noncompact–compact distinction of the relevant groups. The properties of all basic interactions and particles are determined and characterized by invariants and eigenvalues for these operation groups. Although the product of external and internal operations in the acting group is direct, \( G_{{\rm ext}} \times G_{{\rm int}} \), the internal “chargelike” operations are coupled to the external “spacetimelike” ones: Any spacetime translation is accompanied by a chargelike operation. This is implemented by the gauge fields and the corresponding covariant derivatives for the particle fields in the standard model of electroweak and strong interactions. All interactions can be formulated, in a classical geometrical language, by connections of bundles, by a Riemannian connection for the tangent spaces of “horizontal” spacetime as the base, yielding the external interactions, i.e., gravity, and by connections of “vertical” complex vector spaces as fibers, yielding the internal interactions, e.g., the electroweak and strong ones.


Sterile Neutrino Lorentz Group Pure Gauge Bundle Versus Internal Operation 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.MPI für Physik Werner-Heisenberg-InstitutMünchenGermany

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