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On wikipedia (see this link), the Lagrangians of the $\phi^4$ equation for real AND complex scalar fields are given. One may derive the Klein-Gordon equation by inserting into the Euler-Lagrange-equations. This yields for the real scalar field

$$\partial_{\mu} \partial^{\mu} \varphi + m^2 \varphi + \frac{\lambda}{6} \varphi^3 = 0 \tag{1}$$

and for the complex scalar field

$$\partial_{\mu} \partial^{\mu} \phi + m^2 \phi + 2 \lambda |\phi|^2 \, \phi = 0 \tag{2}$$

(as well as the same equation with the replacement $\phi \longrightarrow \phi^{\ast}$). Interestingly, equation (2) does not become equation (1) when assuming that $\mathrm{Im} \phi = 0$.

1.) Is there any particular reason for taking these differing coefficients?

2.) The real part of $\phi$ satisfies the same equation (2) as the imaginary part of $\phi$. Assuming some uniqueness condition on the equation holds, one has $\mathrm{Re}\phi = \mathrm{Im} \phi$ for coinciding initial conditions. Am I missing something here or is solving the equation for $\mathrm{Re}\phi$ enough to determine $\phi$?

Qmechanic
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Octavius
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1 Answers1

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  1. For what it is worth, the standard convention is to divide each term in the Lagrangian with its symmetry factor, e.g. $${\cal L}~=~\mp \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^2\phi^2-\sum_{n\geq 3}\frac{\lambda_n}{n!}\phi^n\qquad(\text{real scalar field})$$ $${\cal L}~=~ \mp\partial_{\mu}\phi^{\ast} \partial^{\mu}\phi-m^2 \phi^{\ast}\phi -\sum_{n\geq 2}\frac{\lambda_{2n}}{(n!)^2}(\phi^{\ast}\phi)^n\qquad(\text{complex scalar field})$$ $${\cal L}~=~\mp \frac{1}{2}{\rm Tr}(\partial_{\mu}\phi\partial^{\mu}\phi)-\frac{1}{2}m^2{\rm Tr}(\phi^2)-\sum_{n\geq 3}\frac{\lambda_n}{n}{\rm Tr}(\phi^n)\qquad(\text{$N\times N$ Hermitian matrix field})$$ with Minkowski sign convention $(\mp,\pm,\pm,\pm)$, respectively, cf. e.g. this & this Phys.SE post.

  2. We leave it to the reader to work out what this means (i) for the corresponding Euler-Lagrange (EL) equations, and (ii) for the conversion of a complex scalar field into a pair of real scalar fields.

Qmechanic
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