# A Scalar Fields with the Same Mass

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1. Jul 8, 2017

### hilbert2

In the Peskin&Schröder's QFT book there's an exercise that's about a pair of scalar fields, $\phi_1$ and $\phi_2$, having the field equations

$\left(\partial^{\mu}\partial_{\mu}+m^2 \right)\phi_1 = 0$
$\left(\partial^{\mu}\partial_{\mu}+m^2 \right)\phi_2 = 0$

where the mass parameter is the same for both fields. Now I can make a "rotation",

$\phi_{1}^{'} = \cos \alpha \phi_1 - \sin \alpha \phi_2$
$\phi_{2}^{'} = \cos \alpha \phi_2 + \sin \alpha \phi_1$

which mixes the two fields. The generator of this rotation apparently has properties of angular momentum, but has nothing to do with spacetime. The symmetry can also remain if a four-point interaction is added in the form of terms proportional to $(\phi_1)^4$, $(\phi_2)^4$ and $\phi_1^2 \phi_2^2$ in the Lagrangian density.

From other sources I got the impression that this has something to do with the Higgs mechanism... How does this extend to more than 2 scalar fields, and can a pair of vector, tensor or spinor fields have a similar property? If the fields have almost the same mass, is there some useful way to treat the mass difference as some kind of a perturbation?

Last edited: Jul 8, 2017
2. Jul 8, 2017

This is just a scalar field theory with an SU(N) internal symmetry. The field can be represented as a vector. If the symmetry is promoted to a local symmetry it will become a nonabelian gauge theory coupled to a scalar field.

You can also have this for spinor fields

3. Jul 8, 2017