Suppose I have a self interacting real scalar field ##\phi## with equation of motion(adsbygoogle = window.adsbygoogle || []).push({});

##\partial^i \partial_i \phi + m^2 \phi = -A \phi^2 - B\phi^3##,

and I attempt to find constant solutions ##\phi (x,t) = C## for it. The trivial solution is the zero solution ##\phi (x,t) = 0##, but there can also be two more constant solutions depending on the values of ##A##, ##B## and ##m##. Obviously ##B## needs to be positive for the system's energy to be bounded from below, but ##A## seems to be arbitrary. Higher order interaction terms would probably make this non-renormalizable.

Do the two nonzero constant solutions have any physical significance? Does this equation allow situations where the static solutions are unstable, becoming something else very quickly if perturbed even a little?

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# A Constant Solutions of Real Scalar Field

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