What's the connection between $\exp_\text{Lie}$ and $\exp_\text{Spectral}$? There are (at least) two places in quantum mechanics where we commonly exponentiate operators.

*

*Lie groups: We exponentiate elements of a Lie algebra, yielding elements of the associated Lie group. E.g., $e^{\sigma_i}$, where $\sigma_i$ is a Pauli matrix [a complex 2-dimensional representation of an element of $\mathfrak{so}(3)$] to yield a (unitary representation of) an element of $SO(3)$.


*Functional calculus/spectral theory: We exponentiate (possibly unbounded) "observable" operators on a (possibly infinite-dimensional) Hilbert space. E.g., $e^\hat H$, where $\hat H$ is the Hamiltonian.
(I'm leaving out the $i$'s that usually appear in the exponent in physics, but I don't think this changes anything about the question.)
These two usages of the exponential map seem to have completely different formal definitions: The Lie theory version can be stated as a map from an abstract Lie algebra to an abstract Lie group using the flow along a vector field. Let's call this $\exp_\text{Lie}$. The spectral theory version can be stated as an integral over a projection-valued measure associated with the operator. Let's call this $\exp_\text{Spectral}$.
Presumably in the finite-dimensional case, the two definitions agree in some sense, since $e^A$ can be computed using the series expansion of $e^x$ (for a particular finite-dimensional representation of the Lie algebra element). And in the infinite dimensional case, I suppose the Lie theory version doesn't apply (unless infinite-dimensional manifolds/Lie groups are sensible). But either way, what is the connection between these two definitions? Can one be viewed as a special case of the other? Is there a Lie group hiding in spectral theory, or a functional calculus hiding in Lie theory?
 A: I am not quite sure I follow everything in your question, but one way to see this is that when you have a Lie group $G$ you can consider a curve $x(t)\in G$ such that $x(0)=1$.
The Lie algebra is then the tangent space of $G$ at $1=x(0)$. So the Lie algebra consists of all the elements you can write as $x'(0)$.
In a handwavy way you can write, for small $\varepsilon$, $x(\varepsilon)\approx 1+\varepsilon x'(0)$, so close to the identity you can approximate an element of $G$ by $1$+ an element of your Lie algebra.
You see also that you can write $x(\varepsilon)\approx e^{\varepsilon x'(0)}$, so you can write an element of your $G$ as the exponential of something from the Lie algebra.
In fact if you write $t=\varepsilon n$ you can find that
$x(\varepsilon)^n \approx e^{t x'(0)} $, so you can write any element of your group (because $x(\varepsilon)^n$ is in $G$) as the exponential of something in the Lie algebra, and you can see this as successive iterations of elements of your Lie algebra.
This is linked e.g. in quantum mechanics where you have (ignoring imaginary stuff as you did)
$$
\frac{\mathrm{d}\psi}{\mathrm dt}=H\psi
$$
so that $\psi(t+dt) = (1+H \mathrm{d}t)\psi(t)$. Writing as before $dt = t/n$ you find $\psi(t) = (1+tH/n)^n\psi(0)\approx e^{tH}\psi(0)$.
A clearer picture comes if you use rotations, as you did. The Lie algebra of $SO(2)$, the 2-d rotations, is $so(2)$ and its elements read
$$
\left(\begin{matrix} 0 & -\lambda \\
 \lambda& 0 \end{matrix}\right)
$$,
and if you take a rotation matrix
$$R(\theta) = 
\left(\begin{matrix} \cos(\theta) & -\sin(\theta) \\
 \sin(\theta)& \cos(\theta) \end{matrix}\right)
$$,
you can easily see that
$$
R(\theta+\Delta \theta) - R(\theta) \approx \Delta \theta\left(\begin{matrix} 0 & -1 \\
 1& 0 \end{matrix}\right) \in so(2)
$$
for $\Delta \theta \ll 1$.
So if you have dynamics that look like
$$
\frac{\mathrm{d} \vec{x}}{\mathrm{d}t} = \omega \left(\begin{matrix} 0 & -1 \\
 1& 0 \end{matrix}\right) \vec{x}
$$
then the solution is given by an exponential of an $so(2)$ element, but this is only a rotation matrix.
If you think of your Lie group as something that transforms an object from its state at $t=0$ to its state at $t=T$ then the idea of the Lie algebra is to decompose that long time evolution into a multitude of successive transformations that are infinitesimally close to the identity, much like an ODE, and this results in an exponential.
