Gauge transformation of Yang-Mills field strength

In summary, a gauge invariant Lagrangian is important in non-abelian theories and the Yang-Mills gauge boson free field term can be transformed under gauge transformation in the same way as the field strength. The actual Lagrangian term is the trace of the field strength matrix, which is equivalent to decomposing it as a linear combination of the generator matrices. This trace is essential and not just a compact notation.
  • #1
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Hi. I'm reading about non-abelian theories and have thus far an understanding that a gauge invariant Lagrangian is something to strive for. I previously thought that the Yang-Mills gauge boson free field term ##-1/4 F^2 ## was gauge invariant, but now after realizing that the field strength transform homogeneously under gauge transformation U(x) s.t

$$ F \to UFU^\dagger$$

it seems like this term also transform like

$$F^2 \to UF^2U^\dagger.$$

Am I right there? If so isn't this a problem for the physics? Why, why not?
 
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  • #2
The actual lagrangian term is the trace of ##F^2##:

##{\rm tr} (F^{\mu \nu} F_{\mu \nu})##

This is gauge invariant. Sometimes this gets written as

##F^{a \mu \nu} F^a_{\mu \nu}##

where ##F^a_{\mu \nu}## is defined by ##F_{\mu \nu} \equiv F^a_{\mu \nu} T^a##. Here ##a## is a gauge group adjoint representation index, the ##T^a## are the generator matrices of the adjoint representation and ##F^a_{\mu \nu}## is a set of numbers (not matrices). This is just decomposing the field strength matrix as a linear combination of the generators.

The second form of the Lagrangian is equivalent because ##{\rm tr} (T^a T^b) = \delta^{a b}##. So if you plug the decomposition of ##F^{\mu \nu}## into the first form of the Lagrangian, you get the second form.
 
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  • #3
Ah, that's true. So the trace is actually essential. I've heard that it was nothing but compact notation :)
 
  • #4
It should most probably be seen like this after gauge transformation
[itex] UFU^\dagger UFU^\dagger= UF^2U^\dagger [/itex]
 

1. What is gauge transformation in the context of Yang-Mills field strength?

Gauge transformation is a mathematical concept in the field of theoretical physics, specifically in the study of Yang-Mills field strength. It refers to the process of changing the mathematical description of a physical system without altering its physical properties. In the context of Yang-Mills field strength, gauge transformation involves changing the gauge fields, which describe the interactions between elementary particles, without changing their physical behavior.

2. Why is gauge transformation important in the study of Yang-Mills field strength?

Gauge transformation is important in the study of Yang-Mills field strength because it allows us to simplify the mathematical equations used to describe the behavior of particles. It also helps us understand the symmetries and properties of the system, which can lead to deeper insights into the fundamental forces of nature.

3. How does gauge transformation affect the physical behavior of particles?

Gauge transformation does not affect the physical behavior of particles. It only changes the mathematical representation of the system. This means that the observable properties and interactions of particles remain the same, even after a gauge transformation is applied.

4. Can gauge transformation be applied to all types of physical systems?

No, gauge transformation is specific to certain types of physical systems, particularly those described by the Yang-Mills field strength. It is a fundamental concept in the study of quantum field theory and is not applicable to classical mechanics or other branches of physics.

5. Are there any experimental evidence or applications of gauge transformation in the real world?

While gauge transformation is a purely mathematical concept, it has been successfully applied to various physical systems in the real world. For example, it is used in the Standard Model of particle physics to describe the strong and weak nuclear forces, as well as in the study of quantum chromodynamics. It also has applications in condensed matter physics, such as in the theory of superconductivity.

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