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Quantum mechanical formalism for biological evolution

We study the evolution of sexual and asexual populations in general fitness landscapes. We find deep relations between the mathematics of biological evolution and the formalism of quantum mechanics. We give the general structure of the evolution of populations which is in general an off-equilibrium process that can be expressed by path integrals over phylogenies. These phylogenies are sums of linear lineages for asexual populations. For sexual populations instead, each lineage is a tree of branching ratio two and the path integral describing the evolving population is given by a sum over these trees. Finally, we show that the Bose-Einstein and the Fermi-Dirac distributions describe the stationary state of biological populations in simple cases.