Quartic Interaction - The Lagrangian

The Lagrangian

The Lagrangian for a real scalar field takes the form

This Lagrangian has a global Z₂ symmetry mapping φ to −φ. For a complex scalar field the Lagrangian is,

With n real scalar fields, we can have a φ4 model with a global SO(N) symmetry

Expanding the complex field in real and imaginary parts shows that it is equivalent to the SO(2) model of real scalar fields.

In all of the models above, the coupling constant λ must have a non-negative real part, since otherwise, the potential is unbounded below, and there can be no stable vacuum. Also, the Feynman path integral discussed below would be ill-defined. In 4 dimensions, φ4 theories have a Landau pole. This means that without a cut-off on the high-energy scale, renormalization would render the theory trivial.