Momentum space from principal bundle

Question

I wish to consider a principal bundle which has a fibre $F$ of spatial $d$-dimensional momenta over a base manifold of $\mathbb{R}$. So, naturally the momenta $k^i$ is the generator of the structure group which is the group of translations with the Lie algebra $[k^i, k^j] = 0$. Also, the generators of this group are also its elements. So, a connection on this bundle should be$$Dk^i = A^i_j k^j dt$$ where $dt$ is the 1-form on $\mathbb{R}$. Since, $k^i$ is a constant at every point so the connection should be a constant as well so I chose it to be $A^i_j = |k^i|\delta^i_j$. Over this bundle I define the sections $\phi(k), \phi(-k)$ so that $$D\phi(\pm k) = d\phi(k)\pm|k^i|\phi(\pm k)dt$$. Using this I can define the momentum space action for a $d+1$-dimensional scalar as $$S= \int \text{Tr}(D\phi \wedge *D\phi) = \int d^dk~ D\phi(k)\wedge *D\phi(-k)$$ Since, I am relatively new to fibre bundles can someone explain to me if I have applied the definitions of the principal bundle correctly to this case? And whether my definitions are sound? Also, please suggest how may I improve upon my application if necessary.