Reconstructing confined particles with complex singularities

TL;DR

This paper develops a rigorous method to reconstruct Minkowski space propagators from Euclidean propagators with complex singularities, preserving key physical symmetries but indicating confinement through zero-norm states.

Contribution

It introduces a novel reconstruction approach for Minkowski propagators with complex singularities, maintaining Lorentz symmetry and locality, and links complex singularities to confined states.

Findings

Reconstructed Wightman functions are holomorphic in the tube.

Lorentz symmetry and locality are preserved.

Reconstructed functions violate positivity and temperedness.

Abstract

Complex singularities have been suggested in propagators of confined particles, e.g., the Landau-gauge gluon propagator. We rigorously reconstruct Minkowski propagators from Euclidean propagators with complex singularities. As a result, the analytically continued Wightman function is holomorphic in the tube, and the Lorentz symmetry and locality are kept intact, whereas the reconstructed Wightman function violates the temperedness and the positivity condition. Moreover, we argue that complex singularities correspond to confined zero-norm states in an indefinite metric state space.