Representation Coeffficients

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


Internal chargelike (“vertical”) operations come from compact groups, e.g., hypercharge U(1), isospin SU(2), and color SU(3). In the electroweak and strong standard model, they are implemented as gauge transformations that, via gauge fields in covariant derivatives, accompany the translations (see Chapter 6). External spacetimelike (“horizontal”) operations come from non-compact groups: For example, interaction-free vectors are acted on by Hilbert representations of flat space operation groups, e.g., of the Euclidean group \( {\bf SO}\left( 3\right) \vec \times \mathcal{R}^3 \) in nonrelativistic quantum mechanical scattering theory, or of the Poincaré (cover) group \( {\bf SL}\left( 3, \mathcal{C}\right) \vec \times \mathcal{R}^4 \) for elementary particles in relativistic quantum fields.

Each group determines its Hilbert spaces, finite- or infinite-dimensional, where its action can be represented by definite unitary automorphisms. A Hilbert space, e.g., the Fock space for translations and free particles, may not be appropriate for another group, e.g., for bound states or for the implementation of interactions.


Compact Group Parabolic Subgroup Spherical Function Cyclic Vector Plancherel Measure 
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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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