example we compute the polarization tensor of the colored plasma including a finite relaxation time for the fluctuations. We conclude in Section \[summ\] with some comments on the possible application of the developed formalism to the dynamics of color fields and instabilities in heavy ion collisions.

Theoretical setup {#setup}
=================

The system
----------

We are interested in obtaining an effective theory for the dynamics of a system of colored particles interacting with nonabelian classical gauge fields. In this work we will deal with scalar particles coming in three colors, which in our simple model would represent massless and spinless quarks. We shall therefore consider a classical Yang-Mills field coupled to conformal scalar matter in the fundamental representation of $SU(3)$.

In what follows, we will use $(\mu,\nu,\ldots)$ to denote world indices and $(a,b,\ldots)$ to denote internal (color) indices. We shall denote with $N=3$ the dimension of the fundamental representation, and use $n$ to indicate a generic dimension ($n=3$ or $n=8$ for the fundamental or adjoint representations, respectively). The generators $\mathbf{T}_a$ are traceless hermitian $n\times n$ matrices with commutation relations

=iC\^c\_[ab]{}\_c \[0\] and trace

\_a\_b=12\_[ab]{} \[01\] The Yang-Mills field is $\mathbf{A}_{\mu}=A^a_{\mu}\mathbf{T}_a$. The field tensor

\_=\_\_-\_\_-ig\[1\] belongs to the adjoint representation of the 