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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.
The representative functions A form a non-commutative algebra under convolution with respect to Haar measure μ. The analogue for a finite subgroup of Γ of SU(2) is the finite-dimensional group algebra C[Γ] From the Clebsch-Gordan rules, the convolution algebra A is isomorphic to a direct sum of n × n matrices, with n = 2j + 1 and j ≥ 0.
Compact groups all carry a Haar measure, [6] which will be invariant by both left and right translation (the modulus function must be a continuous homomorphism to positive reals (R +, ×), and so 1). In other words, these groups are unimodular. Haar measure is easily normalized to be a probability measure, analogous to dθ/2π on the circle.
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 ...
If G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure.Using the Haar measure, one can define a convolution operation on the space C c (G) of complex-valued continuous functions on G with compact support; C c (G) can then be given any of various norms and the completion will be a group algebra.
Here is the left invariant Haar measure on . A vector | ψ {\displaystyle |\psi \rangle } for which c ( ψ ) < ∞ {\displaystyle c(\psi )<\infty } is said to be admissible , and it can be shown that the existence of one such vector guarantees the existence of an entire dense set of such vectors in H {\displaystyle {\mathfrak {H}}} .
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 ...