Lagrangian density for a complex scalar field (classical)

[Trave11er]

Hi.
Let's say we have a complex scalar field \varphi and we separate it into the real and the imaginary parts:
\varphi = (\varphi1 + i\varphi2)
It's Lagrangian density L is given by:
L = L(\varphi1) + L(\varphi1)
Can you tell the argument behind the idea that in summing the densities of cpts. we treat the imaginary part on equal basis with the real.

 

[K^2]

Do you mean to say L(\varphi) = L(\varphi1) + L(\varphi2)? That's because L(\varphi) = L(\varphi1) + L(i\varphi2) due to superposition principle, and L(i\varphi2)=L(\varphi2) due to U(1) symmetry. Neither are absolutely generally true. Former requires a linear Lagrangian, later requires it to be symmetric under U(1) transformations. Both of these are true in Quantum Mechanics, but not necessarily in general field theory.

 

[dextercioby]

U(1) symmetry follows from the general requirements for a Lagrangian field theory. The action must be real under complex conjugation, hence the lagrangian density must contain matched products of phi and phi star and subsequent spacetime derivatives.

 

[K^2]

You are right, it does follow from L = L^*. I never really thought of it that way.

 

[Trave11er]

Thank you for the answers - they are very insightful.