- #1
spaghetti3451
- 1,344
- 33
The Euclidean classical action ##S_{\text{cl}}[\phi]## for a scalar field ##\phi## is given by
\begin{equation}
S_{\text{cl}}[\phi]=\int d^{4}x\ \bigg(\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi)\bigg).
\end{equation}
This can be obtained from the action ##S_{\text{Mn}}[\phi]## in Minkowski space for the same scalar field ##\phi## as given by
\begin{equation}
S_{\text{Mn}}[\phi]=\int d^{4}x\ \bigg(\frac{1}{2}(\partial_{\mu}\phi)^{2}-U(\phi)\bigg).
\end{equation}
To transform the action ##S_{\text{Mn}}[\phi]## in Minkowski space to the Euclidean classical action ##S_{\text{cl}}[\phi]##, one needs to perform a Wick rotation. I need someone to show the derivation.
\begin{equation}
S_{\text{cl}}[\phi]=\int d^{4}x\ \bigg(\frac{1}{2}(\partial_{\mu}\phi)^{2}+U(\phi)\bigg).
\end{equation}
This can be obtained from the action ##S_{\text{Mn}}[\phi]## in Minkowski space for the same scalar field ##\phi## as given by
\begin{equation}
S_{\text{Mn}}[\phi]=\int d^{4}x\ \bigg(\frac{1}{2}(\partial_{\mu}\phi)^{2}-U(\phi)\bigg).
\end{equation}
To transform the action ##S_{\text{Mn}}[\phi]## in Minkowski space to the Euclidean classical action ##S_{\text{cl}}[\phi]##, one needs to perform a Wick rotation. I need someone to show the derivation.