now the dispersion relation Finally, for the case of a symmetry breaking one has being and the following dispersion relation holds These wave solutions are interesting as, notwithstanding we started with an equation with a wrong mass sign, the dispersion relation has the right one. Besides, Jacobi function has no real zeros and so the field is never zero but moves around a given constant value that is initially chosen describing a spontaneous breaking of symmetry. A proof of uniqueness can be provided if we note that the solution can be sought in the form being . Then, the partial differential equation becomes an ordinary differential equation that is the one defining the Jacobi elliptic function with satisfying the proper dispersion relation. See also Scalar field theory Quantum triviality Landau pole Renormalization Higgs mechanism Goldstone boson Coleman–Weinberg potential References Further reading 't Hooft, G., "The Conceptual Basis of Quantum Field Theory" (online version). Quantum field theory Subatomic particles with spin 0