  We analyze a recent proposal to map a massless scalar field theory onto a
Yang-Mills theory at classical level. It is seen that this mapping exists at a
perturbative level when the expansion is a gradient expansion. In this limit
the theories share the spectrum, at the leading order, that is the one of an
harmonic oscillator. Gradient expansion is exploited maintaining Lorentz
covariance by introducing a fifth coordinate and turning the theory to
Euclidean space. These expansions give common solutions to scalar and
Yang-Mills field equations that are so proved to exist by construction,
confirming that the selected components of the Yang-Mills field are indeed an
extremum of the corresponding action functional.
