Are projective representiations of a Lie group a representation of the semi-direct product of the group with $U(1)$ if the norm is preserved?

Question

Let's say we have a function $f(x_{\mu},t)$ that transforms under the action of an $N$-parameter group $G(a_{\nu})$. Then a projective representation of $G(a_\nu)$ in the $f(x_\mu,t)$ basis would follow

$$ T_G(a_\nu)f(x_\mu,t)=e^{i\theta(a_\nu)}f(x_\mu',t')$$

but $e^{i\theta}$ is also a representation of the $U(1)$ group. So let's say we have a group $G'(a_\nu)$ and $\theta(a_\nu)$ a function of $a_\nu$ that returns a a real number. We have a basis such that

$$ T_{G'}(a_\nu)f(x_\mu,t)=f(x_\mu',t') $$

and

$$ T_{U(1)}(\theta(a_\nu))f(x_\mu,t)=e^{i\theta(a_\nu)}f(x_\mu,t).$$

Then the action of both groups would be

$$ T_{U(1)}(\theta(a_\nu))T_{G'}(a_\nu)f(x_\mu,t)=e^{i\theta(a_\nu)}f(x_\mu',t')=T_G(a_\nu)f(x_\mu,t).$$

I'm thinking of this in the context of the wave-function where under the Galilean transformation

$$ T_{G}(\Delta\vec{x},\Delta t,R(\alpha,\beta, \gamma),v)|\psi(\vec{x},t)|^2=|\psi(\vec{x}',t')|^2$$

which means

$$ T_{G}(\Delta\vec{x},\Delta t,R,v)\psi(x,t)=e^{i\theta(\Delta\vec{x},\Delta t,R,v)}\psi(x',t').$$

Answer 1

If they were, then it would always be possible to obtain a true representation by restricting to $\mathbf 1\in \mathrm{U}(1)$, wouldn't it?

Let $G$ be a group and $V$ be a complex vector space. It is certainly true that if $\rho:G \rightarrow \mathrm{GL}(V)$ is a representation of $G$, then $$\varphi: g \mapsto e^{i\theta(g)} \rho(g)$$ constitutes a projective representation of $G$, because

$$\varphi(g_1)\varphi(g_2) = e^{i[\theta(g_1) +\theta(g_2)]}\rho(g_1)\rho(g_2) = e^{i[\theta(g_1) +\theta(g_2)]} \rho(g_1g_2)$$ $$= e^{i[\theta(g_1)+\theta(g_2)-\theta(g_1g_2)]} \varphi(g_1g_2)$$

However, any such $\varphi$ could be de-projectivized by simply multiplying it by the appropriate phase factor. Since not all projective representations can be de-projectivized into genuine representations, it follows that not all projective representations are of this form.

The classic example in physics is the irreducible projective representation of $\mathrm{SO}(3)$ on the vector space $\mathbb C^2$. Any $g\in \mathrm{SO}(3)$ can be written as

$$g = \mathrm{exp}\left[\sum_{n=1}^3\theta_n(g) L_n\right]$$ where $\theta_n(g)\in \mathbb R$ and $(L_1,L_2,L_3)$ are the standard three antisymmetric infinitesimal generators which constitute a basis of $\mathfrak{so}(3)$. The corresponding projective representation on $\mathbb C^2$ is given by

$$\varphi(g) = \exp\left[\frac{i}{2}\sum_{n=1}^3\ \theta_n(g) \sigma_n\right]$$

where the $\sigma_n$'s are the Pauli matrices. It is well-known that this representation cannot be de-projectivized (no faithful, irreducible representations of $\mathrm{SO}(3)$ exist on $\mathbb C^{2k}$).