Wave eq., two real fields with interaction, wave speed.

[Spinnor]

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.

 

[turin]

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?

I believe that is "correct". However, mass is a result of quantization. If they're just classical fields, then I believe you should replace "mass" with "attenuation (constant)". Oh, and "the wave equation" should be relativistically invariant (or covariant if R and S have internal structure).