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A Scalar Fields with the Same Mass

  1. Jul 8, 2017 #1

    hilbert2

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    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
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  3. Jul 8, 2017 #2

    radium

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    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
     
  4. Jul 8, 2017 #3

    hilbert2

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    Thanks. I'll still need to read more about this to understand the case where the symmetry is only approximate.
     
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