# A Relevant interactions in quantum field theory

1. Sep 18, 2016

### spaghetti3451

For a $\phi^{3}$ quantum field theory, the interaction term is $\displaystyle{\frac{g}{3!}\phi^{3}}$, where $g$ is the coupling constant.

The mass dimension of the coupling constant $g$ is $1$, which means that $\displaystyle{\frac{g}{E}}$ is dimensionless.

Therefore, $\displaystyle{\frac{g}{3!}\phi^{3}}$ is a small pertubation at high energies $E \gg g$, but a large perturbation at low energies $E \ll g$.

Terms with this behavior are called relevant because they’re most relevant at low energies.

However, I do not understand why the interaction term is called relevant if we cannot use perturbation theory at low energies (where the term $\displaystyle{\frac{g}{3!}\phi^{3}}$ is a large pertubation). Is it because quantum field theory is only applicable in the relativistic limit, where $E \gg g$ and the perturbation is small ?

2. Sep 19, 2016

### Demystifier

The philosophy behind this terminology is that all field theories are only effective theories, which means that they are valid only at large distances (low energies). If a contribution is relevant, it means that it is not negligible. But it does not mean that it is not sufficiently small for application of perturbation theory. For instance, $\alpha=1/137$ is small but not negligible.