ClebschGordanegyütthatókkal

ClebschGordane is a hypothetical mathematical construct used in theoretical discussions to generalize the notion of angular-momentum coupling beyond the standard Clebsch–Gordan coefficients. Named here to illustrate how tensor products of irreducible representations might be decomposed in a broader symmetry setting, ClebschGordane coefficients are defined for two initial representations of a Lie group G and express the expansion of their tensor product into a direct sum of irreducible components.

Formally, if |J1 m1> and |J2 m2> denote basis states of two irreducible representations with labels J1, J2 and magnetic quantum numbers m1, m2, the ClebschGordane coefficients C^{J1 J2 J}_{m1 m2 m} are defined by the relation |J1 m1> ⊗ |J2 m2> = sum_{J,m} C^{J1 J2 J}_{m1 m2 m} |J m>, where the sum runs over the irreducible components J that appear in the decomposition and m labels their internal states. These coefficients satisfy orthogonality and completeness relations analogous to standard Clebsch–Gordan coefficients: sum_{m1 m2} C^{J1 J2 J}_{m1 m2 m} C^{J1 J2 J'}_{m1 m2 m'} = δ_{J J'} δ_{m m'}, and sum_{J m} C^{J1 J2 J}_{m1 m2 m} C^{J1 J2 J}_{m1' m2' m} = δ_{m1 m1'} δ_{m2 m2'}.

In practice, ClebschGordane is a theoretical tool used to discuss how coupling rules might extend to more complex groups, such as SU(3) or higher-rank algebras, and to illustrate the role of selection rules and symmetry in composite systems. While not part of standard reference literature, the concept helps students compare the algebraic structure of CGCs with more general coefficient systems used in representation theory and particle-physics model building.