Symmetry Groups of Differential Equations

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


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
$$\left. {f\left\langle {,g} \right.} \right\rangle = \int {\overline {f\left( {\theta ,\varphi } \right)} g\left( {\theta ,\varphi } \right)d\Omega ,} $$
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 \(T\left( G \right)f\left( {\theta ,\varphi } \right) = f\left( {{G^{ - 1}}\left( {\theta ,\varphi } \right)} \right) = f\left( {\theta ',\varphi '} \right)\)


Symmetry Group Irreducible Representation Regular Representation Infinitesimal Generator Unitary Irreducible Representation 
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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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