Rigorous reconstruction of gluon propagator in the presence of complex singularities

TL;DR

This paper rigorously examines the reconstruction of gluon propagators with complex singularities, showing that while Lorentz symmetry and locality are preserved, positivity is violated, implying complex singularities relate to confined states.

Contribution

It provides a rigorous analysis of Minkowski propagator reconstruction in the presence of complex singularities, highlighting their implications for quantum field theory.

Findings

Wightman function remains holomorphic in the usual tube.

Lorentz symmetry and locality are preserved.

Positivity condition is violated, indicating confined states.

Abstract

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.