Statement of the problem
I'm working with a function $\lambda : SU(n)\times SU(n)\times SU(n) \rightarrow \mathbb{C}$.
Given $U_1, U_2, U_3 \in SU(n)$, I'd like to know how to calculate $\lambda (\tilde{U}_1, \tilde{U}_2, \tilde{U}_3)$, where $\tilde{U}_i$ is some $SU(n)$ element in an arbitrarily small neighborhood of $U_i$.
The attempt at a solution
Two ideas:
- I know that any $U\in SU(n)$ that is in a small neighborhood of the identity can be written as
\begin{align} U = I + i \epsilon X + O(\epsilon^2) \end{align}
where $X \in \mathfrak{su}(n)$, the algebra of traceless hermitian matrices. So then is it correct to say that any element in a neighborhood of $U_1$ can be written as
\begin{align} U_1(I + i \epsilon X + O(\epsilon^2)) \end{align}
for some $X \in \mathfrak{su}(n)$?
- Define a 1-parameter family $U_1(t)= e^{i X_1 t}$ such that
\begin{align} U_1(0) = I, \hspace{5mm}U_1(1) = U_1 \end{align}
Claim: then any element in a neighborhood of $U_1$ is given by
\begin{align} \frac{d}{dt} U(t)\bigg|_{t=1} = e^{i X_1} ( i X_1) = i U_1 X_1 \end{align}
But $i U_1 X_1$ isn't unitary, so this must be wrong. Aside: why was this the wrong approach?