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Symmetry Groups of Differential Equations

  • Petre P. Teodorescu
  • Nicolae-Alexandru P. Nicorovici
Part of the Fundamental Theories of Physics book series (FTPH, volume 140)

Abstract

Let us consider the set of functions f (θ, φ), defined on the surface of a sphere in ε3. The subset of square integrable uniform functions forms an infinite dimensional linear space, denoted by L2. This linear space becomes a unitary space if we introduce a scalar product defined by the relation
$$\left. {f\left\langle {,g} \right.} \right\rangle = \int {\overline {f\left( {\theta ,\varphi } \right)} g\left( {\theta ,\varphi } \right)d\Omega ,} $$
(3.1.1)
where dΩ=sinθ is the element of solid angle. The space L2 is complete, therefore it is also aHilbert space. To a rotation G of ε3 there corresponds an operator T(G) acting in L2, which is unitary, linear, and generates a representation of SO (3), defined by the formula \(T\left( G \right)f\left( {\theta ,\varphi } \right) = f\left( {{G^{ - 1}}\left( {\theta ,\varphi } \right)} \right) = f\left( {\theta ',\varphi '} \right)\)

Keywords

Symmetry Group Irreducible Representation Regular Representation Infinitesimal Generator Unitary Irreducible Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Petre P. Teodorescu
    • 1
  • Nicolae-Alexandru P. Nicorovici
    • 2
  1. 1.Faculty of MathematicsUniversity of BucharestBucharestRomania
  2. 2.School of PhysicsThe University of SydneyAustralia

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