Questions about *classical* gauge field theory (Abelian and Non-Abelian)

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SUMMARY

This discussion addresses the existence and relevance of exact solutions in classical gauge field theory, specifically focusing on Abelian and Non-Abelian theories. It confirms that exact solutions such as instantons and merons exist for non-Abelian gauge theory without matter fields, but their practical application is limited due to non-linearity. For Abelian and Non-Abelian theories with matter fields, no exact solutions are confirmed, and the well-behaved nature of these solutions remains unproven. The conversation emphasizes that classical solutions have limited relevance, with the complexities of Quantum Chromodynamics (QCD) primarily residing in Quantum Field Theory (QFT).

PREREQUISITES
  • Understanding of classical gauge field theory concepts
  • Familiarity with Abelian and Non-Abelian gauge theories
  • Knowledge of instantons and merons as solutions
  • Basic principles of Quantum Chromodynamics (QCD)
NEXT STEPS
  • Research the role of instantons and merons in gauge theories
  • Explore the implications of non-linearity in gauge field theories
  • Study Quantum Field Theory (QFT) and its relation to classical solutions
  • Investigate lattice gauge theories and their differences from classical approaches
USEFUL FOR

The discussion is beneficial for theoretical physicists, researchers in quantum field theory, and students exploring the complexities of gauge field theories and their solutions.

petergreat
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I know little about these issues, so my questions may be dumb.
1. Do exact solutions exist for non-Abelian gauge theory without matter fields?
2. Do exact solutions exist for Abelian/Non-Abelian theory with matter fields (scalar or spinor)?
2. Are solutions well-behaved? I'm asking because no one can prove it for Navier-Stokes equation. Another reason is that Non-linear waves can develop singularities, and gluons fields are non-linear waves.
 
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regarding 1: yes, e.g. instantons and merons are exact solutions of the theory; but due to the non-linearity they are of limited relevance as they can't be used to construct all solutions

regarding 2: afaik no!

regarding 3: afaik no proof exists

general remark: classical solutions are of limited relevance; all the hard stuff in QCD is purely QFT; even instantons and merons are used in rare cases only; so it's really the QFT that needs to be solved and it seems that classical solutions do not really help. Maybe this is slightly different in lattice gauge theories ...
 

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