# Lifting representation Heisenberg algebra

I (think) I've found the Heisenberg Lie algebra representation through quantization. Where we have $$q \mapsto q$$ and $$p \mapsto -i \hbar \frac{\partial}{\partial q}$$.

So this is only a Lie algebra representation. $$\rho_*: \mathfrak{h} \to \text{End}(V)$$.

Where $$V$$ is the hilbert space (representation space). And $$\mathfrak{h}$$ is the Heisenberg Lie algebra.

How do I lift this to a lie group representation $$\rho: H \to \text{Gl}(V)$$ . Where $$H$$ is the Heisenberg group? And how do I show it's unitary?

You are representing the three algebra elements $$q, p, 1\!\!1$$ by hermitian operators acting on functions of q. For simplicity, non-dimensionalize $$\hbar=1$$--only the clueless keep it unscaled.
Multiplying linear combinations of such operators by i and exponentiating yields the generic unitary group element representation for you , $$e^{ic + ibq + a\partial_q} ~ f(q) = e^{ic + iab/2} e^{ibq} e^{a\partial_q} ~f(q)= e^{ic + iab/2 +ibq}~ f(q+a),$$ for real coefficients a, b, c, the standard form used routinely in applications.