I'm new to Physics SE. I've seen a lot of interesting questions and answers, and thought it will be very useful to participate a little.

I'm currently stuck in a, probably, very simple matter, regarding the nature of the linear momentum $P_{\alpha}$ in field theory. I know it is commonly known as the "infinitesimal generator of translations", pointing an obvious relation with the Lie algebra generators. But which are the Lie group, and correspondingly Lie algebra, associated with $P_{\alpha}$?

I thought of something like the following: if $M$ is the Minkowski spacetime, let $G = (M,+)$ be the Lie group, therefore $G$ acts on $M$. So if $x^{\alpha}\in M$ and $g\in G$, then:

$g(x^{\alpha}) = x^{\alpha} + \delta^{\alpha}$,

but I've been struggling to mathematically write the relation between the algebra generators, the exponential map and the momentum operator $P^{\alpha}$.

Does anyone knows how to point me in the right direction? Thank you all!