This question is about the motivation for Weinberg's approach in "The Quantum Theory of Fields" to obtain unitary representations of Lie groups out of hermitian representations of its Lie algebra.

The notation is as follows: one is dealing with a Lie group $G$, $\{\theta^a\}$ are coordinates on a neighborhood of the identity, $T(\theta)$ is the group element with coordinates $\theta$. The function $f(\bar{\theta},\theta)$ is the coordinate-version of the multiplication operation defined by $$T(\bar{\theta})T(\theta)=T(f(\bar{\theta},\theta)).$$

The excerpt is from Appendix 2B:


> 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 $$\dfrac{d}{ds}U_\theta(s)=it_aU_\theta(s) h^a_{\phantom{a}b}(\Theta_\theta(s))\dfrac{d\Theta^b_\theta(s)}{ds}\tag{2.B.2}$$ with the initial condition $$U_\theta(0)=1,$$ where $$[h^{-1}]^a_{\phantom{a}b}(\theta)=\left[\dfrac{\partial f^a(\bar{\theta},\theta)}{\partial \bar{\theta}^b}\right]_{\bar{\theta}=0}.$$

The basic claim is that if we know the generators $t_a$ of the representation, i.e., if we know a Lie algebra representation, we can find the unitary representation $U[\theta]$ by defining $U_\theta(s)$ through Eq. (2.B.2) and identifying $U[\theta]=U_\theta(1)$.

What I'm used to is another approach: assume $G$ is connected, then any $g\in G$ is $$g = \exp(X_1)\cdots \exp (X_n),\quad X_i\in \mathfrak{g}$$

Now, if $D$ is the Lie algebra representation we define that $$U(\exp X)= \exp D(X)\,$$
then provided the so-defined $U$ is single-valued, it is just a matter of demanding $$U(\exp X \exp Y) = U(\exp X)U(\exp Y)$$ and we get $U$ on all of $G$ out of $D$.

This seems to have nothing to do with Weinberg's approach at first, in particular Weinberg studies $U$ on top of arbitrary paths connecting the identity to each group element, whereas in this approach one is considering the specific path $t\mapsto \exp(tX)$.

1. What is the motivation for Weinberg's approach? How one would even think about defining this $U_\theta(s)$ through (2.B.2) in order to obtain $U[\theta]$ out of the generators?

2. How Weinberg's approach relates to the approach through the exponential map I have outlined? Are they equivalent approaches?