Global SU(N) Symmetry and Complex Scalar Invariance

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SUMMARY

A complex scalar, such as a complex Lagrangian, is invariant under a global SU(N) symmetry for N greater than 1. This invariance implies that both the real and imaginary parts of the complex scalar maintain their invariance independently with respect to the SU(N) symmetry. The invariance of the complex conjugate scalar also follows from this definition, confirming that scalars cannot be transformed into other scalars through matrix operations.

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  • Understanding of global SU(N) symmetry
  • Knowledge of complex scalars and their properties
  • Familiarity with Lagrangian mechanics
  • Basic concepts of invariance in physics
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The discussion is beneficial for theoretical physicists, students of quantum field theory, and anyone interested in the mathematical foundations of symmetry in particle physics.

isospin
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If a complex scalar (e.g. a complex Lagrangian) is invariant under a global SU(N) symmetry, the complex conjugate of this scalar is also invariant, right?
That is, the real and imaginary part of the complex scalar are both invariances with respect to the SU(N) independently.
I am not quite sure if it, is the statement correct?

Thanks.
 
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A scalar, by definition, would be invariant under a global SU(N) symmetry for N>1. There is no way to act sensibly on a scalar with a matrix to get a scalar back. The invariance of the conjugate follows.
 

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