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Also, just a notational issue: The notation should be ##F_{\mu\nu}##, not ##F_{\mu,\nu}##. It is quite common to use ##,\mu## as additional subscripts instead of writing out ##\partial_\mu## in front.
Verifying the Relation in Yang-Mills Theory with a Scalar Field
- Context: Graduate
- Thread starter Othin
- Start date
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SUMMARY
This discussion focuses on verifying the relation in Yang-Mills theory involving a scalar field, specifically the equation \([D_{\mu},D_{\nu}]\Phi=F_{\mu\nu}\Phi\). The covariant derivative is defined as \(D_{\mu}=\partial_{\mu} + [A_{\mu},\Phi]\), and the field strength tensor is given by \(F_{\mu,\nu}=\partial_{\mu} A_{\nu} - \partial_{\nu}A_{\mu} + [A_{\mu}, A_{\nu}]\). Participants highlight issues with notation and the interpretation of the scalar field \(\Phi\), emphasizing the importance of using the adjoint representation and clarifying the roles of operators acting on \(\Phi\). The discussion references Baez's textbook "Gauge Fields, Knots and Gravity" for further insights on the topic.
PREREQUISITES- Understanding of Yang-Mills theory and its mathematical framework.
- Familiarity with Lie algebra and adjoint representation.
- Knowledge of covariant derivatives and field strength tensors.
- Experience with commutators and their properties in quantum field theory.
- Study the adjoint representation in detail, particularly in the context of Yang-Mills theory.
- Review the derivation and properties of the field strength tensor \(F_{\mu\nu}\) in gauge theories.
- Explore the implications of Jacobi's identity in the context of Lie algebras.
- Read Baez's "Gauge Fields, Knots and Gravity" for a comprehensive understanding of gauge theories.
This discussion is beneficial for theoretical physicists, mathematicians specializing in gauge theories, and advanced students studying quantum field theory and Yang-Mills theory.
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