  The theory of modular flow has proved extremely useful for applying
thermodynamic reasoning to out-of-equilibrium states in quantum field theory.
However, the standard proofs of the fundamental theorems of modular flow use
machinery from Fourier analysis on Banach spaces, and as such are not
especially transparent to an audience of physicists. In this article, I present
a construction of modular flow that differs from existing treatments. The main
pedagogical contribution is that I start with thermal physics via the KMS
condition, and derive the modular operator as the only operator that could
generate a thermal time-evolution map, rather than starting with the modular
operator as the fundamental object of the theory. The main technical
contribution is a new proof of the fundamental theorem stating that modular
flow is a symmetry. The new proof circumvents the delicate issues of Fourier
analysis that appear in previous treatments, but is still mathematically
rigorous.
