Appendix B: The Relation between Minkowski and Euclidean Actions

Abstract

The Minkowski action leads to canonical quantization and it is used to calculate matrix elements of the evolution operator e ^-iHt. The Euclidean action is used to calculate the partition function tre ^-βH. Lattice calculations are formulated in terms of the Euclidean action. In Minkowski space $g_{\mu \nu }=\left( 1,-1,-1,-1\right)$ and $\det g=-1$, whereas in Euclidean space $g_{\mu \nu }=\left( 1,1,1,1\right) =\delta_{\mu \nu }$ and $\det g=+1$. As a rule of the thumb, a Euclidean action can be transformed into a Minkowski action by the following substitutions.