What is the interpretation / meaning of relative signs between interaction terms in a Lagrangian density? (If there is none, are they even physically reasonable?)

Example: Let $\phi$ be a scalar field, then I am referring to Lagrangian densities such as $$L=\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi\,+\,\frac{\lambda_3}{3!} \phi^3 \,-\,\frac{\lambda_4}{4!} \phi^4.$$ Note the relative signs between the cubic and quartic interactions!

Note: This article did unfortunately not help as it was not properly answered.

  • 1
    $\begingroup$ $L=T-V$ where $T$ is the kinetic energy (filed derivative squared) and $V$ is the potential energy. In this case $V=-\,\frac{\lambda_3}{3!} \phi^3 \,+\,\frac{\lambda_4}{4!} \phi^4$. $\endgroup$ May 8 at 16:04
  • $\begingroup$ I mean the relative sign between the potential terms (In the qusetion I stated: "relative sign between INTERACTION terms"!). Is there any interpretation and if not, why not? $\endgroup$ May 8 at 16:16
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    $\begingroup$ You might plot the potential and survey the locations of its minimum in the 4 distinct logical possibilities involved. $\endgroup$ May 8 at 16:24

1 Answer 1


They determine the shape of the potential and hence its minima; possible implications include spontaneous symmetry breaking.


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