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

In summary, the conversation discusses whether exact solutions exist for non-Abelian gauge theory with or without matter fields, and if these solutions are well-behaved. It is mentioned that instantons and merons are exact solutions, but their relevance is limited due to non-linearity. There is no proof for the well-behaved nature of these solutions, and classical solutions are not very helpful in solving QFT.
  • #1
petergreat
267
4
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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  • #2
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 ...
 

1. What is gauge field theory?

Gauge field theory is a mathematical framework used to describe the interactions between particles in physics. It is based on the idea of symmetries, specifically gauge symmetries, which dictate how particles behave and interact with each other.

2. What is the difference between Abelian and Non-Abelian gauge field theory?

In Abelian gauge field theory, the gauge transformation is commutative, meaning the order in which the transformations are applied does not matter. In Non-Abelian gauge field theory, the gauge transformation is non-commutative, and the order in which the transformations are applied can affect the final result.

3. How is gauge field theory related to electromagnetism?

Gauge field theory is the mathematical foundation of electromagnetism. The gauge field for electromagnetism is the electromagnetic potential, and the gauge symmetry is the conservation of electric charge.

4. What is the role of gauge fields in the Standard Model of particle physics?

Gauge fields play a crucial role in the Standard Model of particle physics, which is the most widely accepted theory of how particles and forces interact in the universe. The Standard Model includes three non-Abelian gauge field theories: quantum chromodynamics, electroweak theory, and the Higgs mechanism.

5. What are some applications of gauge field theory?

Gauge field theory has many practical applications in physics, including in the development of new technologies such as particle accelerators and lasers. It also plays a crucial role in understanding and predicting the behavior of matter and forces at the subatomic level, as well as in cosmology and the study of the early universe.

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