Euclideanization without Complexification of the Spacetime

Abstract

Minkowski spacetime can be mapped by a series of projections in a higher-dimensional spacetime to a Euclidean space, constituting a process of Euclideanization shown here in detail for two dimensions. The result allows regularizations and computations of integrals that appear in quantum field theory (QFT) without performing the standard Wick rotation of time to imaginary values. However, there is no physical spacetime transformation that produces a Wick rotation. In avoiding this complexification process, the new Euclidenization procedure has important advantages in the transformations of the action principles, including fermionic fields and theories at constant chemical potential. In all cases, complex-valued amplitudes of the form $\exp(iS/\hbar)$ are mapped to real statistical weights $\exp(-S_{\rm E}/\hbar)$ with a Euclidean action $S_{\rm E}$. The procedure is also amenable to fields on curved background spacetimes as well as gravitational interactions.