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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.
Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and L p {\displaystyle L ...
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.
In functional analysis and quantum measurement theory, a positive-operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVMs) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum ...
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.
Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2)), the special unitary group SU(3) and the Poincaré group. On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold , this action provides a measure of rigidity and ...
A maximal torus in the special unitary group SU(n) ⊂ U(n) is just the intersection of T and SU(n) which is a torus of dimension n − 1. A maximal torus in the special orthogonal group SO(2 n ) is given by the set of all simultaneous rotations in any fixed choice of n pairwise orthogonal planes (i.e., two dimensional vector spaces).