Two complex fields + interaction, conserved current?

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SUMMARY

The discussion centers on the implications of global phase changes in complex fields f_1 and f_2, which are solutions to the D'Alembert operator in three-dimensional space. It establishes that subtracting the square of the magnitude of the difference between these fields from their Lagrangian maintains invariance under a global phase change of exp(i*theta). This invariance suggests the existence of a conserved current. Furthermore, the conversation explores the conditions under which local phase invariance can lead to interactions between these fields, particularly in the context of massless and massive wave solutions.

PREREQUISITES
  • Understanding of the D'Alembert operator and its applications in field theory.
  • Familiarity with Lagrangian mechanics and its role in physics.
  • Knowledge of phase invariance in quantum field theory.
  • Concepts of massless and massive wave solutions in the context of complex fields.
NEXT STEPS
  • Research the implications of global phase invariance in quantum field theory.
  • Study the derivation and applications of conserved currents in field theories.
  • Explore local phase invariance and its role in generating interactions in quantum fields.
  • Examine the characteristics of massless versus massive solutions in wave equations.
USEFUL FOR

The discussion is beneficial for theoretical physicists, particularly those specializing in quantum field theory, as well as students and researchers interested in the mathematical foundations of field interactions and conservation laws.

Spinnor
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Say we have two complex fields, f_1 and f_2 which each are solutions of the D'Alembert operator or the wave operator in three space dimensions. Say we subtract the square of the magnitude of the difference of f_1 and f_2 from the Lagrangian for the fields f_1 and f_2. In that case the Lagrangian is unchanged if both f_1 and f_2 each under go a global phase change of exp(i*theta)? If so does that imply a conserved current?

Assume solutions of the form f_1 = f_2 and f_1 = - f_2 , such waves are mass-less and massive respectively?

Can we demand local phase invariance and get some type of interaction?

Thanks for any help!
 
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