- #31
- 22,823
- 14,864
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
Click For Summary
Discussion Overview
The discussion revolves around verifying a relation in Yang-Mills theory involving a scalar field and covariant derivatives. Participants explore the implications of the relation, the definitions of the covariant derivative and field strength, and the notation used in the context of gauge theory.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant presents the relation \([D_{\mu},D_{\nu}]\Phi=F_{\mu\nu}\Phi\) and expresses confusion about the last term in their calculations, questioning how it relates to the field strength.
- Another participant suggests consulting Baez's textbook for clarification on the relation.
- Some participants clarify the notation, indicating that \([X,\Phi] = X\Phi - \Phi X\) and discussing the implications of using the adjoint representation.
- There are suggestions to write \([A_\mu,.]=\operatorname{ad}(A_\mu)\) and to consider the properties of the adjoint representation in the context of the equations presented.
- Participants discuss the notation of the covariant derivative and its implications, with some suggesting that the notation should separate mapping and variable clearly.
- There is a debate about whether \(\operatorname{ad}([A_{\mu},A_{\nu}])\Phi\) is equivalent to \([A_{\mu},A_{\nu}]\Phi\), with some participants expressing uncertainty about the conditions under which these expressions hold.
- One participant raises a question about the commutation of derivatives with the scalar field, particularly in relation to the product rule and the implications for the covariant derivative.
- Another participant emphasizes the need to clarify what is meant by \(\Phi\), whether it refers to a matrix in the Lie algebra or a column vector in a representation space.
Areas of Agreement / Disagreement
Participants express differing views on the notation and the implications of the adjoint representation. There is no consensus on the equivalence of certain expressions or the conditions under which they hold, indicating that multiple competing views remain.
Contextual Notes
Participants highlight limitations in notation and assumptions, particularly regarding the separation of operators and variables, and the implications of using different representations of the scalar field.
Who May Find This Useful
This discussion may be of interest to those studying gauge theories, particularly in the context of Yang-Mills theory, as well as those exploring the mathematical foundations of quantum field theory.
Similar threads
- · Replies 1 ·
- · Replies 7 ·
- · Replies 3 ·
- · Replies 17 ·
- · Replies 3 ·
- · Replies 1 ·
- · Replies 38 ·
- · Replies 7 ·
Undergrad
Georgi-Glashow model of W bosons/photons
- · Replies 2 ·
- · Replies 9 ·