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).

 

[astrokreng]

I would like to first clarify that the term "mass-less modes" refers to modes of oscillation with zero rest mass, such as electromagnetic waves. In this context, the terms "symmetric" and "anti-symmetric" most likely refer to the symmetry of the interaction potential, rather than the motion of the fields.

Based on the given information, it is true that when the fields R and S are symmetric (i.e. the interaction potential is symmetric), there will be mass-less modes. This is because in this case, the potential V = m(R-S)^2 will be equal to zero, and the wave equation will reduce to the standard wave equation, which describes mass-less modes.

On the other hand, when the fields R and S are anti-symmetric (i.e. the interaction potential is anti-symmetric), there will be massive modes. This is because the potential V = m(R-S)^2 will have a non-zero value, leading to the appearance of a mass term in the wave equation. This can be seen in the one-dimensional example of two superimposed strings with a potential V proportional to the area between the strings squared. In this case, the anti-symmetric potential will lead to the appearance of a mass term in the wave equation, resulting in massive modes of oscillation.

In summary, the symmetry of the interaction potential between the fields R and S does affect the type of modes that can be observed. When the potential is symmetric, mass-less modes will be present, while an anti-symmetric potential will lead to the appearance of massive modes. This is due to the different forms of the wave equation that result from the different potential symmetries.