Editing 3D rotation group (section)

== Spherical harmonics ==
{{Main|Spherical harmonics}}
{{See also|Representation of a Lie group#An example: The rotation group SO.283.29{{!}}Representations of SO(3)}}

The group {{math|SO(3)}} of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space

:<math>L^2\left(\mathbf{S}^2\right) = \operatorname{span} \left\{ Y^\ell_m, \ell \in \N^+, -\ell \leq m \leq \ell \right\}, </math>

where <math>Y^\ell_m</math> are [[spherical harmonics]]. Its elements are square integrable complex-valued functions<ref group=nb>The elements of {{math|''L''<sup>2</sup>('''S'''<sup>2</sup>)}} are actually equivalence classes of functions. two functions are declared equivalent if they differ merely on a set of [[measure zero]]. The integral is the Lebesgue integral in order to obtain a ''complete'' inner product space.</ref> on the sphere. The inner product on this space is given by

{{NumBlk|:|<math>\langle f,g\rangle = \int_{\mathbf{S}^2}\overline{f}g\,d\Omega = \int_0^{2\pi} \int_0^\pi \overline{f}g \sin\theta \, d\theta \, d\phi.</math>|{{EquationRef|H1|H1}}}}

If {{mvar|f}} is an arbitrary square integrable function defined on the unit sphere {{math|'''S'''<sup>2</sup>}}, then it can be expressed as<ref name="Gelfand_M_S">{{harvnb|Gelfand|Minlos|Shapiro|1963}}</ref>

{{NumBlk|:|<math>|f\rangle = \sum_{\ell = 1}^\infty\sum_{m = -\ell}^{m = \ell} \left|Y_m^\ell\right\rangle\left\langle Y_m^\ell|f\right\rangle, \qquad f(\theta, \phi) = \sum_{\ell = 1}^\infty\sum_{m = -\ell}^{m = \ell} f_{\ell m} Y^\ell_m(\theta, \phi),</math>|{{EquationRef|H2|H2}}}}

where the expansion coefficients are given by

{{NumBlk|:|<math>f_{\ell m} = \left\langle Y_m^\ell, f \right\rangle = \int_{\mathbf{S}^2}\overline{{Y^\ell_m}}f \, d\Omega = \int_0^{2\pi} \int_0^\pi \overline{{Y_m^\ell}}(\theta, \phi)f(\theta, \phi)\sin \theta \, d\theta \, d\phi.</math>|{{EquationRef|H3|H3}}}}

The Lorentz group action restricts to that of {{math|SO(3)}} and is expressed as

{{NumBlk|:|<math>(\Pi(R)f)(\theta(x), \phi(x)) = \sum_{\ell = 1}^\infty\sum_{m = -\ell}^{m = \ell}\sum_{m' = -\ell}^{m' = \ell} D^{(\ell)}_{mm'} (R) f_{\ell m'}Y^\ell_m \left(\theta\left(R^{-1}x\right), \phi\left(R^{-1}x\right)\right), \qquad R \in \operatorname{SO}(3), \quad x \in \mathbf{S}^2.</math>|{{EquationRef|H4|H4}}}}

This action is unitary, meaning that

{{NumBlk|:|<math>\langle \Pi(R)f,\Pi(R)g\rangle = \langle f,g \rangle \qquad \forall f,g \in \mathbf{S}^2, \quad \forall R \in \operatorname{SO}(3).</math>|{{EquationRef|H5|H5}}}}

The {{math|''D''<sup>(''ℓ'')</sup>}} can be obtained from the {{math|''D''<sup>(''m'', ''n'')</sup>}} of above using [[Clebsch–Gordan coefficients|Clebsch–Gordan decomposition]], but they are more easily directly expressed as an exponential of an odd-dimensional {{math|'''su'''(2)}}-representation (the 3-dimensional one is exactly {{math|𝖘𝖔(3)}}).<ref>In ''Quantum Mechanics – non-relativistic theory'' by [[Course of Theoretical Physics|Landau and Lifshitz]] the lowest order {{math|''D''}} are calculated analytically.</ref><ref>{{harvnb|Curtright|Fairlie|Zachos|2014}} A formula for {{math|''D''<sup>(''ℓ'')</sup>}} valid for all ''ℓ'' is given.</ref> In this case the space {{math|''L''<sup>2</sup>('''S'''<sup>2</sup>)}} decomposes neatly into an infinite direct sum of irreducible odd finite-dimensional representations {{math|1=''V''<sub>2''i'' + 1</sub>, ''i'' = 0, 1, ...}} according to<ref>{{harvnb|Hall|2003}} Section 4.3.5.</ref>

{{NumBlk|:|<math>L^2\left(\mathbf{S}^2\right) = \sum_{i = 0}^\infty V_{2i + 1} \equiv \bigoplus_{i=0}^\infty \operatorname{span}\left\{Y_m^{2i+1}\right\}.</math>|{{EquationRef|H6|H6}}}}

This is characteristic of infinite-dimensional unitary representations of {{math|SO(3)}}. If {{mvar|Π}} is an infinite-dimensional unitary representation on a [[separable space|separable]]<ref group=nb>A Hilbert space is separable if and only if it has a countable basis. All separable Hilbert spaces are isomorphic.</ref> Hilbert space, then it decomposes as a direct sum of finite-dimensional unitary representations.<ref name=Gelfand_M_S/> Such a representation is thus never irreducible. All irreducible finite-dimensional representations {{math|(Π, ''V'')}} can be made unitary by an appropriate choice of inner product,<ref name=Gelfand_M_S/>

:<math>\langle f, g\rangle_U \equiv \int_{\operatorname{SO}(3)} \langle\Pi(R)f, \Pi(R)g\rangle \, dg = \frac{1}{8\pi^2} \int_0^{2\pi} \int_0^\pi \int_0^{2\pi} \langle \Pi(R)f, \Pi(R)g\rangle \sin \theta \, d\phi \, d\theta \, d\psi, \quad f,g \in V,</math>

where the integral is the unique invariant integral over {{math|SO(3)}} normalized to {{math|1}}, here expressed using the [[Euler angles]] parametrization. The inner product inside the integral is any inner product on {{math|''V''}}.