By L. C. Biedenharn
This 1985 textual content develops the idea of angular momentum from the point of view of a basic symmetry in nature and exhibits how this idea pertains to utilized components of study in smooth quantum physics
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Extra info for Angular momentum in quantum physics : theory and application
They will be described in the following chapters of the presented review. It is necessary to remark that many specific problems are not covered by the available methods. the classification of representations suggested by the theory often lead to difficult algebraic, combinatorial and analytic problems. Computer techniques have recently been successfully applied to the classification problems connected with finite groups. The classification of representations is usually reduced to the description of the "most simple" (indecomposable or irreducible) representations and to the study of their direct sums (integrals) and extensions.
This explains the role of the Fourier transform in the theory of differential equations: every differential equation with constant coefficients is reduced to an algebraic equation. I. Theory of Representations and Noncommutative Harmonic Analysis 27 The Fourier transform on a noncommutative group is designed to play a similar role. However, the translations on the group now generate a noncommutative algebra. Even so, the Fourier transform carries over this algebra into the algebra of multiplication by functions, this time, however, by operatorvalued functions instead of scalar valued functions.
5) f(g) = IGI ~ d>. (g)]. EG We have also (an analogue of the Plancherel formula): I. Theory of Representations and Noncommutative Harmonic Analysis (ft, /2)L2(G) = (ft,j2h2(G) := ~~~ ~ d>. )*]. 3 are reformulations of already proved facts. 3 with well-known facts of commutative harmonic analysis. Let G be a commutative group. Then all its irreducible representations are one-dimensional and coincide with their characters. The set G itself is a group with respect to the tensor product of representations, which coincides here with the usual multiplication of characters as functions on the group.
Angular momentum in quantum physics : theory and application by L. C. Biedenharn