In "The Quantum Theory of Fields" Weinberg says the following in the Appendix B to Chapter 2:

> To prove this theorem, let us recall the method by which we construct the operators corresponding to symmetry transformations. As described in Section 2.2, we introduce a set of real variables $\theta^a$ to parameterize these transformations, in such a way that the transformation satisfy the composition rule (2.2.15): $$T(\bar{\theta})T(\theta)=T(f(\bar{\theta},\theta)).$$
We want to construct operators $U(T(\theta))\equiv U[\theta]$ that satisfy the corresponding condition $$U[\bar{\theta}]U[\theta]=U\left[f(\bar{\theta},\theta)\right].\tag{2.B.1}$$
To do this, we lay down arbitrary 'standard' paths $\Theta_\theta^a(s)$ in group parameter space, running from the origin to each point $\theta$, with $\Theta^a_\theta(0)=0$ and $\Theta_\theta^a(1)=\theta^a$, and define $U_\theta(s)$ along each such path by the differential equation $$\dfrac{d}{ds}U_\theta(s)=it_aU_\theta(s) h^a_{\phantom{a}b}(\Theta_\theta(s))\dfrac{d\Theta^a_\theta(s)}{ds}\tag{2.B.2}$$ with the initial condition $$U_\theta(0)=1,$$ where $$[h^{-1}]^a_{\phantom{a}b}(\theta)=\left[\dfrac{\partial f^a(\bar{\theta},\theta)}{\partial \bar{\theta}^b}\right]_{\bar{\theta}=0}.$$

Just to summarize the notation, here one is dealing with a Lie group $G$, the function $T : \mathbb{R}^n\to G$ is a parameterization (i.e., the inverse of a coordinate chart) and the function $f$ is the coordinate-version of the multiplication operation. The matrix $h^a_{\phantom{a}b}$ seems to be the components of the inverse of the pushforward of the right translation map $h\mapsto hg$.

Now, if I understand, what Weinberg is doing is the following: we have some Hermitian operators $t_a$ (which I believe implicitly constitute a representation of the Lie algebra $\mathfrak{g}$) and want to build out of this a unitary representation of the group.

My issue though is where Eq. (2.B.2) comes from.

At first I thought Eq. (2.B.2) had something to do with the notion of [derived representation][1] defined through the relation on smooth vectors $U(\exp(sA)) = \exp(s dU(A))$ linking the group representation $U$ to the algebra representation $dU$. But this seems nothing like what Weinberg does (Weinberg is taking an arbitrary path on the group, not $\exp(sA)$).

Now where equation (2.B.2) comes from? In other words why the unitary operation $U[\Theta_\theta(s)]$ satisfies (2.B.2)?


  [1]: https://physics.stackexchange.com/q/324204/