We propose a simple model for the origin and evolution of Ly alpha forest clouds based on cosmological conductive/cooling fronts. In this model the Ly alpha forest arises in the interfaces between the intergalactic medium and cold clouds that could be tentatively identified with protogalaxies. Most of the properties of the Ly alpha forest absorbers are reproduced with a very restricted number of assumptions. Among these are the correct range of HI column density, cloud sizes, and redshift, and HI column density distributions for the absorbers. Several predictions and implications of the model are briefly discussed.