Two coupled real Klein-Gordon fields ($m_{\pm }^2=m^2\mp g$)

For interaction density $g{\varphi }_1{\varphi }_2$, the combinations ${\varphi }_{\pm }={\varphi }_1\pm {\varphi }_2$ diagonalize the equations into independent Klein-Gordon equations with the displayed mass squares. When $0<g<m^2$, all Fourier frequencies are real and nonzero. Initial data in one original field generally excite both normal combinations.

A single spatial Fourier pattern initially displaced in one field excites two different normal frequencies. Their sum and difference generate slowly varying amplitude envelopes in the original fields. Total coupled-field energy is conserved, but the separate uncoupled energies need not be, and zeros of one displacement envelope do not automatically imply exact complete energy transfer.