= Schrödinger representation of the Heisenberg group
{c}
= Schrödinger representation
{c}
{synonym}
Use the <Heisenberg group> law $(q,r,s)(q',r',s')=(q+q',r+r',s+s'+qr')$. For a fixed real $\lambda\ne0$, its Schrödinger representation on the <Hilbert space> $L^2(\mathbb R)$ is
$$
(\pi_\lambda(q,r,s)f)(t)=e^{i\lambda(s+rt)}f(t+q).
$$
Translations preserve the <Lebesgue measure> and the phase has modulus one, so these are <unitary operators>. Composing two such operators introduces the phase $\lambda qr'$, exactly the cross term in the <Heisenberg group> law. The <center of a group> therefore acts by $\pi_\lambda(0,0,s)=e^{i\lambda s}I$.
On smooth rapidly decreasing functions, the differentiated <Lie algebra representation> sends the coordinate generators to $d/dt$, multiplication by $i\lambda t$, and $i\lambda I$. Their <commutator> is $[d/dt,i\lambda t]=i\lambda I$. This is the representation behind the <canonical commutation relation> between <position operator> and <momentum operator>; their normalization fixes the central parameter.
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