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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.
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 ...
The circle of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication.. In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group).
The Haar measure can be defined on any locally compact group and is a generalization of the Lebesgue measure (R n with addition is a locally compact group). The Hausdorff measure is a generalization of the Lebesgue measure that is useful for measuring the subsets of R n of lower dimensions than n , like submanifolds , for example, surfaces or ...
Haar measure can also be approached via Radon measures on locally compact spaces; these are non-negative functionals on the space of continuous functions of compact support. There is a unique (up to a scalar) non-zero left invariant Radon measure on a locally compact group.
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.