Value of Mexican hat potential

[spaghetti3451]

Consider the Lagrangian

$$\mathcal{L} = \bar{\psi}(i\gamma^{\mu}{\partial_{\mu}}-y\phi)\psi +\frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi),$$

where the potential energy $V(\phi)$ is given by

$$V(\phi) = \frac{1}{2}\kappa^{2}\phi^{2} + \frac{\lambda}{24}\phi^{4}.$$

For $\kappa^{2}=-|\kappa^{2}|$, we obtain a mexican hat potential.

The stationary points of this potential give us the true vacua $\phi = \pm \sqrt{\frac{6|\kappa^{2}|}{\lambda}}$ of this potential as well as the unstable field configuration $\phi = 0$.

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Is there an alternative way, for example, by using the classical equations of motion of $\phi$ (setting $\psi$ to zero) to show that the
field configuration $\phi(x) = 0$ is unstable?

 

[radium]

You want to choose the solution that minimizes the free energy which you can just compute from the path integral. The classical limit means that you are evaluating it on the saddle point i.e., the equations of motion.

 

[spaghetti3451]

You want to choose the solution that minimizes the free energy which you can just compute from the path integral. The classical limit means that you are evaluating it on the saddle point i.e., the equations of motion.

Computing the free energy (from the path integral) and then minimising it and then evaluating it on the saddle point is overkill as I already have the classical equation of motion.

Do you have some way to use the classical equations of motion of $\phi$ (setting $\psi$ to zero) to show that the field configuration $\phi(x) = 0$ is unstable?

 

[radium]

Yes, what I wrote is how you would do it.

In general, you show that something is unstable by showing that it is actually a local maximum of the free energy. In this case case you evaluate the free energy from the path integral on the equations of motion in the classical limit, so this is how you would see that there is an instability using the equations of motion.

 

[spaghetti3451]

Can you provide a few steps to show how to do this?