Yang-Mills Field in the $\kappa$-space-time

In this paper, we construct $SU(N)$ Yang-Mills theory in the $\kappa$-space-time, valid up to first order in the deformation parameter $a$, using the generalisation of Feynman's approach. Using the $\kappa$-deformed Wong's equation derived, in the Jacobi identity involving velocities and coordinates of $\kappa$-deformed space-time, the $\kappa$-deformed homogeneous Yang-Mills equations are derived. We show the compatibility between the $\kappa$-deformed field strength derived using the Jacobi identity and the commutators of the gauge covariant derivative, up to first order in $a$. The $\kappa$-deformed field strength is covariant under $SU(N)$ gauge transformations. We then construct the Lagrangian for Yang-Mills theory in $\kappa$-deformed space-time and show that it is invariant under $SU(N)$ transformation and not under U(N) transformation. We also derive the expression for the force experienced by an isospin-carrying particle in the presence of Yang-Mills field in the $\kappa$-space-time.