Sign up ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free.

We consider a scalar theory in a $1+D$ dimensional flat Minkowski space-time, with a general self-interaction potential, whose action can be written as \begin{equation} A=\int dt\, d^D\! x \left[\frac12(\partial_t\phi)^2-\frac12(\partial_i\phi)^2 -U(\phi)\right] \,, \end{equation} where $\phi$ is a real scalar field, $\partial_t=\partial/\partial t$, $\partial_i=\partial/\partial x^i$ and $i=1,2,\ldots,D$. The equation of motion following from \eqref{action} is a non-linear wave equation (NLWE) which is given as $$ -\phi_{,tt} + \Delta \phi = U'(\phi)=\phi +\sum\limits_{k=2}^{\infty}g_k\phi^k \tag{1}$$ where $$\quad{\Delta}=\sum_{i=1}^{D}\frac{\partial^2}{\partial x_i^2}\,. $$ I just want to know the logic, that how they transformed the equation (1)

share|cite|improve this question
Related post by OP: – Qmechanic Mar 1 '13 at 12:10

1 Answer 1

They just expanded the "potential" derivative $U'(\phi)$ in a Taylor series in powers of $\phi$. Kind of polynomial self-interaction.

The coefficient $g_1$ can always be made to be unity with help of redefinition of $t$ and $x$, so they wrote $g_1=1$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.