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Since right Haar measure is well-defined up to a positive scaling factor, this equation shows the modular function is independent of the choice of right Haar measure in the above equation. The modular function is a continuous group homomorphism from G to the multiplicative group of positive real numbers .
Since every operator in SU(2) is a rotation of the Bloch sphere, the Haar measure for spin-1/2 particles is invariant under all rotations of the Bloch sphere. This implies that the Haar measure is the rotationally invariant measure on the Bloch sphere, which can be thought of as a constant density distribution over the surface of the sphere.
Any compact group is locally compact.. In particular the circle group T of complex numbers of unit modulus under multiplication is compact, and therefore locally compact. The circle group historically served as the first topologically nontrivial group to also have the property of local compactness, and as such motivated the search for the more general theory, presented here.
The representations of the group are found by considering representations of (), the Lie algebra of SU(2).Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation; [1] we will give an explicit construction of the representations at the group level below.
which would say the left translate of a right Haar measure is a multiple of a left Haar measure. However, the left translate of a right Haar measure is also a right Haar measure so what you're looking for is an assertion that a right Haar measure is a multiple of a left Haar measure, e.g. is itself left Haar. That's only true for unimodular groups.
Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to SO(3); consequently SO(3) is isomorphic to the quotient group SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere S 3, and SU(2) is the universal cover of ...
[2] The Hilbert cube carries the product Lebesgue measure [3] and the compact topological group given by the Tychonoff product of an infinite number of copies of the circle group is infinite-dimensional and carries a Haar measure that is translation-invariant. These two spaces can be mapped onto each other in a measure-preserving way by ...
The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate measurable functions defined on G. The Lebesgue measure on the real line is a special case of this.