Obtaining Euclidean action from Minkowski action

In summary, the Euclidean classical action for a scalar field is given by an integral over four-dimensional space of the field's derivative squared and its potential energy. This action can be obtained from the Minkowski space action for the same field by performing a Wick rotation and changing the sign of the action. The resulting Euclidean action is then equivalent to an integral over four-dimensional Euclidean space of the field's derivative squared and its potential energy.
  • #1
spaghetti3451
1,344
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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.
 
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  • #2
Continue from real [itex]x^{0}[/itex] to [itex]x^{0} = - i x_{4}[/itex], and from [itex]S_{Min}[/itex] so obtained get [itex]S_{Euc}[/itex] using [itex]S_{Euc} = - i S_{Min}[/itex]:
[tex]S_{M} = \int dx^0 \ d^3x \ \left( (1/2) (\frac{\partial \phi}{\partial x^{0}})^{2} - (1/2) (\nabla \phi)^{2} - U(\phi) \right) [/tex]
[tex]S_{M} = -i \int dx_{4} \ d^3x \ \left( - (1/2) (\frac{\partial \phi}{\partial x_{4}})^{2} - (1/2) (\nabla \phi)^{2} - U(\phi) \right) [/tex]
[tex]S_{M} = i \int dx_{4} \ d^3x \ \left( (1/2) (\frac{\partial \phi}{\partial x_{4}})^{2} + (1/2) (\nabla \phi)^{2} + U(\phi) \right) = i S_{E}[/tex]
[tex]S_{E} = \int d^{4}x_{E} \ \left( (1/2) (\frac{\partial \phi}{\partial x_{\mu}})^{2} + U(\phi) \right)[/tex]
 

1. How is Euclidean action related to Minkowski action?

Euclidean action and Minkowski action are two different mathematical formulations used in theoretical physics to describe the behavior of particles and fields. Minkowski action is based on the Minkowski spacetime, which has four dimensions (three for space and one for time). Euclidean action, on the other hand, is based on the Euclidean spacetime, which has four imaginary dimensions. Despite their differences, Euclidean action can be obtained from Minkowski action by performing a mathematical transformation known as Wick rotation.

2. Why is it useful to obtain Euclidean action from Minkowski action?

Obtaining Euclidean action from Minkowski action allows for easier mathematical calculations and analysis. In some cases, certain physical phenomena may be more easily understood in the Euclidean formulation. Additionally, Euclidean action can be used to calculate thermodynamic quantities such as partition functions, which are important in statistical mechanics and quantum field theory.

3. What is the Wick rotation and how is it performed?

The Wick rotation is a mathematical transformation that involves replacing the time coordinate in Minkowski spacetime with an imaginary time coordinate. This effectively rotates the Minkowski spacetime into a Euclidean spacetime. Mathematically, it is represented by t = -iτ, where τ is the imaginary time coordinate. This transformation is often used in theoretical physics to simplify calculations and relate Minkowski and Euclidean formulations.

4. Are there any limitations to obtaining Euclidean action from Minkowski action?

While obtaining Euclidean action from Minkowski action can be a useful tool, it is not always applicable in all scenarios. The Wick rotation only works for certain types of physical systems and may not be applicable in all cases. Additionally, the resulting Euclidean action may not always have a clear physical interpretation.

5. What are some applications of obtaining Euclidean action from Minkowski action?

One of the main applications of obtaining Euclidean action from Minkowski action is in quantum field theory. The Euclidean formulation allows for easier calculations of quantum field theory amplitudes, which are important in understanding particle interactions. Additionally, the Wick rotation can also be used in the study of statistical mechanics and condensed matter systems.

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