$1+1D$ $U(1)$ gauge theory is a quantum mechanical system In article Exotic Symmetries, Duality, and Fractons in 2+1-Dimensional Quantum Field Theory there is statement (page 13):

Ordinary $1 + 1$-dimensional $U(1)$ gauge theory is effectively a quantum mechanical system of a single variable.

Such theory is defined as:
$$
L = -\frac{1}{4} F^{\mu\nu} F_{\mu\nu}
$$
In Lorentz gauge:
$$
\partial_\mu A^\mu = 0
$$
Residual freedom:
$$
A_\mu \to A_\mu +\partial_\mu \alpha 
$$
with $\Box \alpha = 0$.
Using this freedom on can remove all d.o.f. So 2d Maxwell theory haven't physical d.o.f. In colour of such consideration, this statement looks quite strange.
Could somebody explain how to prove/check this statement?
 A: *

*The E&M Hamiltonian Lagrangian density without matter in 1+1D is$^1$
$${\cal L}_H~=~-E_1 \dot{A}_1 ~-~{\cal H}_0 ~-~ A_0 \underbrace{\partial_1 E_1}_{\text{ Gauss law}}, \tag{i} $$
where the Hamiltonian density is
$${\cal H}_0~=~\frac{1}{2} E_1^2 -\theta E_1. \tag{ii}$$


*Next if we use the Coulomb gauge $\partial_1 A_1=0$ together with Gauss law $\partial_1 E_1=0$, we see that the canonical field pair $(A_1,E_1)$ does not depend on space $x^1$, only on time $t$.


*After eliminating the Lagrange multiplier field $A_0$, the field theory model (i) effectively becomes a point mechanical model with a single canonical pair $(A_1(t),E_1(t))$. This fact answers OP's title question.
--
$^1$ Eq. (i) is before gauge-fixing & after possibly ignoring spatial boundary terms. We use a Minkowski metric with positive spatial component and negative temporal component. The momentum variable to $A_1$ is minus $E_1$.
A: Maybe there is a more vivid and rigorous argument in the continuum theory, but I am not aware of them, so I 'll present an argument from lattice gauge theory.
The discrete version of gauge theory, which in continuum limit gives ordinary Maxwell (Yang-Mills action) is so called Wilson action:
$$
S_G = \sum_{\Box} \beta \ (1 - \frac{1}{N_c}\text{Tr } U_{\Box}) \qquad
U_{\Box} = U_\mu (x) U_\nu (x + \mu) U_\mu^{\dagger} (x + \nu) U_\nu^{\dagger} (x)
$$
Where $U_\mu (x)$ is the link poiniting in the $\mu$ direction from the site, located at $x$. The gauge transformations act on the links in following way:
$$
U_\mu^{'} (n) = \Omega (n) U_\mu (n) \Omega^{\dagger} (n + \mu)
$$
By an appropriate gauge transformation, any configuration can be put to so dubbed temporal gauge :
$$
U_0 (n) = \mathbb{1}
$$
Looking at the Wilson action, one sees, that the links on different timeslices decouple from each other, and we have independent 1-dimensional spin chains, which can be interpreted as (0 + 1) dimensional field theory in Euclidean space.
