Lagrangian of $\phi^4$ theory

Question

How do you derive the Lagrangian for a real scalar field ($\phi^4$ theory)?  I need to do so for spontaneous symmetry breaking. 

This question was asked by user34039 on Physics Stack Exchange but is now deleted there and has been restored from an archive.  

Comments

What do you mean by derive? Do you mean to ask what it's physical interpretation is, and/or why someone would want to study it? - joshphysics  

---

No, I mean to derive it from first principles. - user34039

---

Which first principles? The way this business works is that either you (1) find a system and then determine if it can be well-described by a field theory you care about or (2) you say "let's consider this field theory" and then see if there is a system whose dynamics are governed by it, or perhaps a hybrid of the two. Which of these are you looking for in this context? FYI I'm not the downvoter. - joshphysics    

---

hybrid of the two - user34039

---

I suspect that what you mean is that you have a $Z_2$ symmetric theory $\mathcal{L} = -\frac12 (\partial\phi)^2+\frac12\mu^2\phi^2-\lambda\phi^4 $ and you would like to compute the Lagrangian appropriate for fluctuations about the phase where the $Z_2$ is spontaneously broken. You would want to write $\phi(x,t)=\bar\Phi + \varphi(x,t) $ where $\bar\Phi$ is the minimum of the potential $V\left(\phi\right) = - \frac12\mu^2\phi^2 +\lambda\phi^4  $. You can then plug this decomposition into $\mathcal L$, and keep all the terms that depend on $\varphi$. It's just algebra. - Andrew 

Answer 1

I suspect that what you mean is that you have a $Z_2$ symmetric theory $\mathcal{L} = -\frac12 (\partial\phi)^2+\frac12\mu^2\phi^2-\lambda\phi^4 $ and you would like to compute the Lagrangian appropriate for fluctuations about the phase where the $Z_2$ is spontaneously broken. You would want to write $\phi(x,t)=\bar\Phi + \varphi(x,t) $ where $\bar\Phi$ is the minimum of the potential $V\left(\phi\right) = - \frac12\mu^2\phi^2 +\lambda\phi^4  $. You can then plug this decomposition into $\mathcal L$, and keep all the terms that depend on $\varphi$. It's just algebra.  

Comments

This answer actually did not exist in the original PSE question, however, Andrew's comment was basically an answer, so I converted it.