  Propagators of confined particles, especially the Landau-gauge gluon
propagator, may have complex singularities as suggested by recent numerical
works as well as several theoretical models, e.g., motivated by the Gribov
problem. In this paper, we study formal aspects of propagators with complex
singularities in reconstructing Minkowski propagators starting from Euclidean
propagators by the analytic continuation. We derive the following properties
rigorously for propagators with arbitrary complex singularities satisfying some
boundedness condition. The two-point Schwinger function with complex
singularities violates the reflection positivity. In the presence of complex
singularities, while the holomorphy in the usual tube is maintained, the
reconstructed Wightman function on the Minkowski spacetime becomes a
non-tempered distribution and violates the positivity condition. On the other
hand, the Lorentz symmetry and locality are kept intact under this
reconstruction. Finally, we argue that complex singularities can be realized in
a state space with an indefinite metric and correspond to confined states. We
also discuss consequences of complex singularities in the BRST formalism. Our
results could open up a new way of understanding a confinement mechanism,
mainly in the Landau-gauge Yang-Mills theory.
