  It has been suggested that the Landau-gauge gluon propagator has complex
singularities, which invalidates the K\"all\'en-Lehmann spectral
representation. Since such singularities are beyond the standard formalism of
quantum field theory, the reconstruction of Minkowski propagators from
Euclidean propagators has to be carefully examined for their interpretation. In
this talk, we present rigorous results on this reconstruction in the presence
of complex singularities. As a result, the analytically continued Wightman
function is holomorphic in the usual tube, and the Lorentz symmetry and
locality are kept valid. On the other hand, the Wightman function on the
Minkowski spacetime is a non-tempered distribution and violates the positivity
condition. Finally, we discuss an interpretation and implications of complex
singularities in quantum theories, arguing that complex singularities
correspond to zero-norm confined states.
