- #1

spaghetti3451

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- 34

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?

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

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?

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