According to my skript:
Quantum mechanic states $ψ ∈ \mathcal H$ changes under a rotation $R ∈ \text{SO(3)}, \vec{x} \rightarrow R\vec{x}$ according to $ψ \rightarrow U(R)ψ$, whereas $U(R)$ is a unitary representation of $\text{SO(3)}$ is, that means:
$$U: \text{SO(3)} \rightarrow \mathcal L(\mathcal H) = \{\text{linear transformation } \mathcal H \rightarrow \mathcal H\} = \text{GL}(\mathcal H,\mathcal H)$$
$$R \longmapsto U(R) $$
is a homomorphism, i.e. $U(R_1)U(R_2) = U(R_1R_2), U(1) = \mathbb I$ which is unitary $U(R)^{-1} = U(R)^*$.
Infinitesimal rotations are elements $Ω$ of the tangent space $T_{\mathbb I}SO(3) = \{\dot{γ}(0)|γ:[-ε,ε] \rightarrow \text{SO(3)}, γ(0) = \mathbb I\}$, where $γ(ε) = e^{εΩ} ∈ \text{SO(3)},γ(0) = e^{0Ω} = \mathbb I$, on $\text{SO(3)}$ at the point $\mathbb I$:
$$Ω = \frac{d}{dt}R(t)\bigg|_{t=0},$$
whereas $t \longmapsto R(t)$ is a differentiable curve in $\text{SO(3)}$ through $R(0) = \mathbb I$.
Every Lie group representation of $\text{SO(3)}$ on $\mathcal H$ corresponds to a Lie algebra representation of $\text{so(3)}$ (but not vice versa):
$$U(Ω):= \frac{d}{dt}U(R(t))\bigg|_{t=0}.$$
The transformation $Ω \longmapsto U(Ω)$ is a homomorphism of $\text{so(3)}$ $(\alpha_1,\alpha_2 ∈ ℝ)$: $$U(\alpha_1Ω_1 + \alpha_2Ω_2) = \alpha_1U(Ω_1) + \alpha_2U(Ω_2)$$ $$U([Ω_1,Ω_2]) = [U(Ω_1),U(Ω_2)],$$ whereas the last follows from $U(RΩR^{-1}) = U(R)U(Ω)U(R)^{-1} \quad (R ∈ \text{SO(3)}).$
I want to check the last statement, i.e. that $Ω \longmapsto U(Ω)$ is a homomorphism.
Calculation Questions:
1) If I just consider $\alpha_1U(Ω_1)$, is it then correct to state:
$$\alpha_1U(Ω_1) = \alpha_1 U(\frac{d}{dt}R_1(t)\bigg|_{t=0})$$ $$\alpha_1U(Ω_1) = \alpha_1\frac{d}{dt}U(R_1(t))\bigg|_{t=0},$$
by simply plugging in the definitions?
2) If this holds, can I then conclude: $$U(\alpha_1Ω_1 + \alpha_2Ω_2) = U(\frac{d}{dt}(R_1(\alpha_1t)R_2(\alpha_2t))\bigg|_{t=0})\\ \quad= \frac{d}{dt}U(R_1(\alpha_1t)R_2(\alpha_2t))\bigg|_{t=0} = \frac{d}{dt}(U(R_1(\alpha_1t))U(R_2(\alpha_2t)))\bigg|_{t=0} =\alpha_1Ω_1 + \alpha_2Ω_2.$$
3) I ask this, because someone else wrote down that:
$$U(Ω_1+\alphaΩ_2) := \frac{d}{dt}U(R_1(t)R_2(\alpha t)\bigg|_{t=0}.$$
4) So in general I am a bit confused about the notation, maybe one could clarify this a bit.
General Questions & Remarks:
I still don't get the main concept behind representations.
Let me place a few words into this room: Quantum Mechanics; Born's rule; Symmetries; Projective Representations; Wigner's Theorem; Irreducible Representations; Eigenstates; Composite Systems & Clebsch-Gordan-Coefficients; Wigner-Eckart-Theorem.
It would be great if one could show me his Idea in a few lines using the words given above. I myself will later try (so far I have only a long version in another language which I can poste later on).
Thank you in advance! :)
