  It is well known that Yang-Mills theory in vacuum has a perturbative
instability to spontaneously form a large scale magnetic field (the Savvidy
mechanism) and that a constant field is unstable so that a possible ground
state has to be inhomogenous over the non-perturbative scale Lambda (the
Copenhagen vacuum). We argue that this spontaneous instability does not occur
at high temperature when the induced field strength gB ~ Lambda^2 is much
weaker than the magnetic mass squared (g^2T)^2. At high temperature
oscillations of gauge fields acquire a thermal mass M ~ gT and we show that
this mass stabilizes a magnetic field which is constant over length scales
shorter than the magnetic screening length (g^2T)^{-1}. We therefore conclude
that there is no indication for any spontaneous generation of weak non-abelian
magnetic fields in the early universe.
