  Infinitely many distinct trait values may arise in populations bearing
quantitative traits, and modelling their population dynamics is thus a
formidable task. While classical models assume fixed or infinite population
size, models in which the total population size fluctuates due to demographic
noise in births and deaths can behave qualitatively differently from constant
or infinite population models due to density-dependent dynamics. In this paper,
I present a stochastic field theory for the eco-evolutionary dynamics of finite
populations bearing one-dimensional quantitative traits. I derive stochastic
field equations that describe the evolution of population densities, trait
frequencies, and the mean value of any trait in the population. These equations
recover well-known results such as the replicator-mutator equation, Price
equation, and gradient dynamics in the infinite population limit. For finite
populations, the equations describe the intricate interplay between natural
selection, noise-induced selection, eco-evolutionary feedback, and neutral
genetic drift in determining evolutionary trajectories. My methods use ideas
from statistical physics and present an alternative to some recently proposed
measure-theoretic frameworks.
