Say we have two real fields, R(x,t) and S(x,t), which satisfy the 3-dimensional wave equation. Now let there be an interaction potential between the fields R and S of the form, V = m(R-S)^2. Suppose the "motion" of the fields is either symmetric or anti-symmetric, that is R(x,t) = + or - S(x,t). Then is it true we will have mass-less modes when R and S are symmetric and massive modes when R and S are anti-symmetric? A one-dimensional example, two superimposed strings with a potential V proportional to the area between the strings squared? Thank you for any help.