  The quasi-species equation describes the evolution of the probability that a
random individual in a population carries a given genome. Here we map the
quasi-species equation for individuals of a self-reproducing population to an
ensemble of scalar field elementary units undergoing a creation and
annihilation process. In this mapping, the individuals of the population are
mapped to field units and their genome to the field value. The selective
pressure is mapped to an inverse temperature \beta of the system regulating the
evolutionary dynamics of the fields. We show that the quasi-species equation if
applied to an ensemble of field units gives in the small \beta limit can be put
in relation with existing stochastic quantization approaches. The ensemble of
field units described by the quasi-species equation relaxes to the fundamental
state, describing an intrinsically dissipative dynamics. For a quadratic
dispersion relation the mean energy <U> of the system changes as a function of
the inverse temperature \beta. For small values of \beta the average energy <U>
takes a relativistic form, for large values of \beta, the average energy <U>
takes a classical form.
