= Solution
A unitary family with the required <Weyl relations> is
$$
\boxed{(W(x,y)f)(t)=e^{ixy+2iyt}f(t+x).}
$$
Translations preserve the $L^2$ <norm> and the multiplier has modulus one. Multiplication of two such operators leaves the phase $x_1y_2-y_1x_2$ after extracting $W(x_1+x_2,y_1+y_2)$, proving the relation. Translation and modulation are strongly continuous, first on compactly supported <smooth function>s by direct estimates, then on all of $L^2$ by density and unitarity.
If a <bounded operator> $A$ commutes with all $W$, it commutes with translations and every modulation $e^{2iyt}$. The <spectral theorem> for the multiplication coordinate therefore makes it commute with all bounded multiplication operators and interval projections. One can see its form directly: on a finite interval $I$, set $h_I=A1_I$. For bounded $\phi$ supported in $I$, commutation gives $A\phi=\phi h_I$. Testing characteristic functions shows $|h_I|\leq\|A\|$ almost everywhere. On increasing intervals these functions agree on overlaps and yield a single $h\in L^\infty$ with $A=M_h$.
Commutation with translations says $h(t+x)=h(t)$ almost everywhere for every $x$. Convolving with a smooth <approximate identity> produces <smooth function>s invariant under every translation, hence constants. Passing to the local $L^1$ limit makes $h$ constant almost everywhere. Therefore \b[$A$ is a scalar operator], and the unitary family acts irreducibly.
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