  We formulate the Migdal-Eliashberg theory of electron-phonon interactions in
terms of classical spins by mapping the free energy to a Heisenberg spin chain
in a Zeeman magnetic field. Spin components are energy-integrated normal and
anomalous Green's functions and sites of the chain are fermionic Matsubara
frequencies. The Zeeman field grows linearly with the spin coordinate and
competes with ferromagnetic spin-spin interaction that falls off as the square
of the inverse distance. The spin-chain representation makes a range of
previously unknown properties plain to see. In particular, infinitely many new
solutions of the Eliashberg equations both in the normal and superconducting
states emerge at strong coupling. These saddle points of the free-energy
functional correspond to spin flips. We argue that they are also fixed points
of kinetic equations and play an essential role in far from equilibrium
dynamics of strongly coupled superconductors. Up to an overall phase, the
frequency-dependent gap function that minimizes the free energy must be
non-negative. There are strong parallels between our Eliashberg spins and
Anderson pseudospins, though the two sets of spins never coincide.
