Vacuum stability conditions and potential minima for a matrix representation in lightcone orbit space

The orbit space for a scalar field in a complex square matrix representation obtains a Minkowski space structure from the Cauchy–Schwarz inequality. It can be used to find vacuum stability conditions and minima of the scalar potential. The method is suitable for fields such as a bidoublet, an SU(2) triplet or SU(3) octet. We use the formalism to find the vacuum stability conditions for the left-right symmetric potential of a bidoublet and left and right Higgs doublets.