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Consider a complex scalar field theory described by the Lagrangian

$$\mathcal{L}=\frac{1}{2}\partial_\mu\phi^*\partial^\mu\phi-U(\phi).$$

In the literature, I often see the authors apply a change of variables in order to "rescale" the potential $U(\phi)$. For example, if the potential is of the form

$$U(\phi)=\frac{1}{2}m^2|\phi|^2-\frac{1}{3}a|\phi|^3+\frac{1}{4}b|\phi|^4.$$

one could apply the "rescaling" \begin{align} \phi &\rightarrow \frac{m^2}{a}\phi\\ x &\rightarrow \frac{x}{m} \end{align} which leads to the corresponding "rescaled" Lagrangian $$ \mathcal{L}=\frac{1}{2}\partial_\mu\phi^*\partial^\mu\phi-\frac{1}{2}|\phi|^2+\frac{1}{3}|\phi|^3-\frac{1}{4}B|\phi|^4 $$ and equation of motion $\partial_t^2\phi-\partial_x^2\phi+\phi-|\phi|\phi+B|\phi|^2\phi$, where $B\equiv bm^2/\alpha^2$.

The above example was taken from equations (1)-(6) of this paper, for example. I know that rescaling of an ODE is a useful trick in order to reduce the parameter degrees of freedom from the equations (as explained in this answer). However, there are a few things that are causing me confusion:

How does one decide how to choose the appropriate rescaled variables? Is this done by dimensional analysis? If so, what are the units of the scalar field $\phi$? And how does one deal with the fact that the time derivative term and space derivative term have (ostensibly) different units?

I am particularly confused by how the $\frac{1}{2}\partial_\mu\phi^*\partial^\mu\phi$ term transforms under the rescaling (I tried to apply the rescaling myself by hand, and I can almost arrive at the correct equation of motion, but I end up with an additional $1/m^2$ factor in front of the $\partial_t^2$ term). Is there some trick to this?

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How does one decide how to choose the appropriate rescaled variables? Is this done by dimensional analysis? If so, what are the units of the scalar field $\phi$? And how does one deal with the fact that the time derivative term and space derivative term have (ostensibly) different units?

The time and space derivative are the same because we're working in natural units, with $c = 1$.

There is no single rule one uses to choose rescaled variables, you can perform whatever rescaling you want in order to simplify the equation. In this case, the authors just wanted to end up with simple coefficients in front of the quadratic and cubic terms. They can modify those coefficients by scaling $\phi$ and scaling $x$. Then by considering a general scaling and demanding it do that, they figured out what specific scale factors they need to achieve it.

I am particularly confused by how the $\frac{1}{2}\partial_\mu\phi^*\partial^\mu\phi$ term transforms under the rescaling (I tried to apply the rescaling myself by hand, and I can almost arrive at the correct equation of motion, but I end up with an additional $1/m^2$ factor in front of the $\partial_t^2$ term). Is there some trick to this?

Again, we're working in natural units and using Lorentz invariance. When they talk about scaling $x$, they really mean scaling both time and space.

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  • $\begingroup$ Ah, I see. So essentially there is also the unstated scaling $t\rightarrow t/m$? This makes sense to me: if I explicitly wrote out the factor of $1/c^2$ in front of the $\partial^2/\partial t^2$ term, in which case we have $\partial^2/ \partial (ct)^2$ which would scale just like the spatial derivative $\endgroup$
    – Superbee
    Mar 29 '20 at 21:56
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    $\begingroup$ @Superbee Sure, but it's not "unstated". When people write $x$ in relativistic QFT papers, they always mean spacetime. If they wanted to include just space, they'd have written $\mathbf{x}$ instead. $\endgroup$
    – knzhou
    Mar 29 '20 at 21:57

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