The Heisenberg-Weyl algebra or the Weyl algebra or the algebra of the Canonical Commutation Relations (CCR) is generated by the $p,q$ generators subject to the relation
$$
[q, p] = i \hbar I \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
$$
This is isomporphic to the algebra of the OP under the "change of base":
$$
A=\frac{1}{\sqrt{2}}(q-ip), \ \ \ \ B=\frac{1}{\sqrt{2}}(q+ip)
$$
A first simple and well known result, in the sense of a non-go theorem, can be derived by direct inspection of the commutation relation:
The CCR admit no finite dimensional representations.
This follows easily by a trace argument on both sides of (1) (or of $[A,B]=1$) as long as Planck's constant $\hbar$ is different than zero. So, from the beginning we have to focus in infinite dimensional representations of the Weyl algebra and since the problem is motivated by physical considerations it is natural to consider its generators acting on Hilbert spaces.
An old result (see 1, 2) tells us that when dealing with representations of (1) in Hilbert spaces, the generators $q$ and $p$ cannot both be bounded: at least one of them has to be unbounded.
Given that a self-adjoint (and thus symmetric), unbounded operator in a Hilbert space cannot be defined on the whole of the space but only on a dense subspace of it (this is a direct consequence of the Hellinger–Toeplitz theorem), one can see that the above remark poses delicate questions on the domains of the above operators (and thus the representations of the Weyl algebra) and at the same time underlines the importance of the use of unbounded operators in the formulation of QM.
Initiating from the above remark, Stone and von Neumann studied the integrated forms of $p,q$ i.e. the unitary, one-parameter groups
$$
\begin{array}{cccccc}
U(s) = exp(-i s p) & & & & & V(t) = exp(-i t q)
\end{array}
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)
$$
The above expressions are well defined because $p, \ q$
are self-adjoint (see 2,3).
$V(t)$ acts on $f(x)$ by multiplication with
$exp(-itx)$, while $U(s)$ acts by horizontal translations (shifts) of $f(x)$
i.e.
$$
\begin{array}{cccccc}
(U(s)f)(x) = f(x-s) & & & & & (V(t)f)(x) = e^{-itx}f(x)
\end{array} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)
$$
for any square integrable complex function of a real variable, i.e. for any
$f(x) \in L_{2}(- \infty, \infty)$. The operators $U(s)$,
$\ \forall s \in \mathbb{R}$ and $V(t)$, $\ \forall t \in \mathbb{R}$,
are unitary and thus bounded. Their domain is now the whole of the Hilbert space (and not only a dense subspace as was the case for the $p,q$ generators of the Weyl algebra).
Furthermore, $U(s)$, $V(t)$ satisfy
$$
U(s)V(t) = e^{ist} V(t)U(s) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)
$$
for all $s \in \mathbb{R}$ and for all $t \in \mathbb{R}$.
(4) is frequently called the integrated form of the Weyl relations or the integrated form of the CCR.
The Stone-von Neumann uniqueness theorem, determines conditions -the main one is (4)- which guarantee the uniqueness of the representation (3)
(and not directly the uniqueness of the usual Fock representation of $p,q$). In other words, it tells us that, given the operators
$U(s), \ V(t)$ on $L_{2}(- \infty, \infty)$, satisfying (4) (plus some technicalities on the domains) then their action is unitarily equivalent to the representation (2), (3).
There have been lots of studies (and references) on the Stone-von Neumann uniqueness theorem, its origins, its extensions, its descendants and on the exact relation between the Weyl generators and their integrated forms. If you are interested in such aspects, I could provide further references on such topics.
Remark 1: Formally, the transition between the CCR (1) and their integrated form (3), (4) can be understood as follows: $\bullet$ If we start from the CCR (1) together with (2) written in power series:
$$
\begin{array}{cccc}
U(s) = exp(-i s p) = \sum_{n=0}^{\infty}\frac{(-isp)^{n}}{n!} & & &
V(t) = exp(-i t q) = \sum_{n=0}^{\infty}\frac{(-i t q)^{n}}{n!} \\
\\
\end{array} \ \ \ \ \ \ \ (5)
$$
then one can deduce the integrated form of the CCR (3), (4) by applying the Baker–Campbell–Hausdorff formula. $\bullet$ For the converse, the CCR (1) follow formally on taking $\frac{\partial^2}{\partial t \partial s}$ of (4) at $s=t=0$.
However, this description is rather an indication than a strict proof of equivalence between the Weyl algebra and its integrated form: The reason is that $p, \ q$ are unbounded operators and hence the power series expressions cannot be valid on the whole of the space. Consequently, in general, the integrated form of the Weyl relations is not equivalent (at least not in some obvious way) with the Weyl relations themselves (i.e. the CCR).
Remark 2:
For the case of the infinite degrees of freedom Weyl algebra, there is no analogue -at least to my knowledge- of the Stone-von Neumann theorem (its applicability is limited to the finite generators case).
On the contrary, it has been shown (see 4) that, the CCR admit infinitely many, non-equivalent, irreducible representations on a Hilbert space, which are in bijection with the set of real numbers.
For interesting examples of irreducible CCR representations, which are inequivalent to the usual Fock space representation, one can look at books on the Foundations of Quantum Mechanics such as: 5, vol. II, p.251-252.
References:
A. Wintner, "The unboundedness of quantum mechanical
matrices", Phys. Rev., v.71, 1947, p. 738-739
C. R. Putnam, "Commutation Properties of Hilbert
Space Operators and Related Topics", Springer-Verlag, Berlin,
(1967)
M. Reed and B. Simon, "Methods of Modern
Mathematical Physics", vol.I, Academic Press (1975).
A. S. Wightman, S. S. Schweber, "Configuration
space methods in relativistic quantum field theory", Phys. Rev.,
v.98, 3, (1955), p.812-837
Galindo-Pascual, Quantum mechanics, v.I,
II, Springer, 1990, Translated from the spanish by: J.D.
Garcia, L. Alvarez-Gaume
