From Euclidean to Minkowski space with the Cauchy-Riemann equations

Abstract

We present an elementary method to obtain Green's functions in non-perturbative quantum field theory in Minkowski space from calculated Green's functions in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes is many times unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore we suggest to use the Cauchy-Riemann equations, that perform the analytical continuation without assuming global information on the function in the entire complex plane, only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge Quantum Chromodynamics, that is known from lattice and Dyson-Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy-Riemann equations to high-frequency noise, that makes difficult to achieve good accuracy. We also point out a few curiosities related to the Wick rotation.

12 pages in EPJ double-column format, 16 figures. This version: added paragraph, two references

References in corpus (4)

• Infrared Properties of QCD from Dyson-Schwinger equations
• The quark-gluon vertex in Landau gauge QCD: Its role in dynamical chiral symmetry breaking and quark confinement
• Scaling behavior and positivity violation of the gluon propagator in full QCD
• Dynamical chiral symmetry breaking with Minkowski space integral representations

Cited by in corpus (4)

• The quark-gluon vertex in Landau gauge bound-state studies
• Hadronic bound-states in SU(2) from Dyson--Schwinger Equations
• The analytic structure of three-point functions from contour deformations
• Spinning Pairs: Supporting ${}^3{P}_0$ Quark-Pair Creation from Landau Gauge Green's Functions