Representation Theory of the Rotation Group
Before looking for all (finite-dimensional) irreducible representations of the Lorentz group we treat the same problem for the rotation group SO(3,R). There are four reasons for this.
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Representation Theory of the Rotation Group
Before looking for all (finite-dimensional) irreducible representations of the Lorentz group we treat the same problem for the rotation group SO(3,R). There are four reasons for this. • The general methods are easy to demonstrate here.
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• The isomorphism between and the complex rotation group SO(3,C) mentioned in sect. 6.5 leaves us with the expectation that some analytic continuation of the representations of SO(3,R) will lead to representations of the Lorentz group. (It will turn out that we do not get all representations this way, but the remaining ones are then easily found.) • The unitary representations of the rotation group play an important role in the quantum mechanics of angular momentum, so that connections between the abstractly treated problems and physical applications are easily established. • The unitary irreducible representations of SO(3,R) will be directly required in chap. 9 for the representation theory of the Poincare group. The finite-dimensional irreducible representations of the rotation group SO(3,R) may be classified and constructed by elementary means; one can also prove full reducibility for reducible representations and carry out the reduction; finally one can extend the results from SO(3,R) to SO(3,C) and thus to the restricted Lorentz group This route to the finite-dimensional representations of the rotation and Lorentz group is described, e.g., in Cartan (1966). However, the way how the principle of relativity is realized in quantum mechanics requires the construction of representations of the Poincare group in a space of quantum states, i.e., in a Hilbert space (see sect. 9.2), which in general will be of infinite dimension. For a mathematically rigorous treatment of this, deeper considerations from functional analysis together with the theory of integration on groups would be necessary. It would be impossible within the bounds of this book even to define all concepts precisely, let alone to prove the fundamental theorems. We shall therefore simply quote some of these theorems and work, as far as infinite-dimensional representations are concerned, with formal analogies to the finite-dimensional case, whose precise meaning can be given only by constructions from functional analysis. For the rotation group-just as for any other compact topological group-the general theory tells us that all continuous representations in a Hilbert space are equivalent to unitary ones and thus are completely reducible, and that the irreducible representations are all finite-dimensional. We shall therefore introduce the concept of unitary representation and construct the irreducible unitary representations. Methodologically, we will go beyond previous chapters by making systematic use of group
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R. U. Sexl et al., Relativity, Groups, Particles © Springer-Verlag Wien 2001
7 Representation Theory of SO (3)
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elements 'infinitely' close to the unit element, assigned to which are 'infinitesimal' transformations. This way of proceeding is not necess
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