• CAF123
In summary, the adjoint transformation of gluon is a mathematical operation used to describe how the properties of a gluon change when it is transformed between reference frames. It is closely related to color charge and is important for understanding the strong force and developing models for subatomic particle behavior. The transformation is calculated using equations from group theory and has applications in fields such as theoretical and particle physics.
CAF123
Gold Member
It is commonly written in the literature that due to it transforming in the adjoint representation of the gauge group, a gauge field is lie algebra valued and may be decomposed as ##A_{\mu} = A_{\mu}^a T^a##. For SU(3) the adjoint representation is 8 dimensional so objects transforming under the adjoint representation are 8x1 real Cartesian vectors and 3x3 traceless hermitean matrices via the lie group adjoint map. The latter motivates writing ##A_{\mu}## in terms of generators, ##A_{\mu} = A_{\mu}^a T^a##.

My first question is, this equation is said to be valid independent of the representation of ##T^a## - but how can this be true? In some representation other than the fundamental representation, the ##T^a## will not be 3x3 hermitean traceless matrices and thus will not contain 8 real parameters needed for transformation under the adjoint rep. But we know the gluon field transforms under the adjoint representation so does this line of reasoning not constrain the ##T^a## to be the Gell mann matrices?

Consider the following small computation:
$$A_{\mu}^a \rightarrow A_{\mu}^a D_b^{\,\,a} \Rightarrow A_{\mu}^a t^a \rightarrow A_{\mu}^b D_b^{\,\,a}t^a$$ Now, since ##Ut_bU^{-1} = D_b^{\,\,a}t^a## we have ##A_{\mu}^a t^a \rightarrow A_{\mu}^b (U t^b U^{-1}) = U A_{\mu}^b t^b U^{-1}##.
The transformation law for the ##A_{\mu}^a## is in fact ##A_{\mu} \rightarrow UA_{\mu}U^{-1} - i/g (\partial_{\mu} U) U^{-1}##.
1) What is the error that amounts to these two formulae not being reconciled?
2) The latter equation doesn't seem to express the fact that the gluon field transforms in the adjoint representation. I was thinking under SU(3) colour, since this is a global transformation, U will be independent of spacetime so the derivative term goes to zero but is there a more general argument?

Thanks!

Hello,

Thank you for your forum post. The equation ##A_{\mu} = A_{\mu}^a T^a## is valid independent of the representation of ##T^a## because it is a general form of writing a gauge field in terms of its generators. The generators ##T^a## can take on different representations, such as the fundamental representation or the adjoint representation, but the general form of the equation remains the same. In other words, the specific form of ##T^a## does not affect the validity of the equation.

Regarding your computation, the error may lie in the fact that you are using different transformation laws for the gauge field and the generators. The correct transformation law for the gauge field is ##A_{\mu} \rightarrow U A_{\mu} U^{-1} + \frac{i}{g}(\partial_{\mu} U) U^{-1}##, which is consistent with the transformation law for the generators. This is known as the adjoint representation of the gauge field.

To answer your second question, the fact that the gluon field transforms in the adjoint representation is captured in the transformation law for the gauge field, as shown above. As you correctly stated, since the transformation is global, the derivative term goes to zero and we are left with the adjoint representation of the gauge field. This is a more general argument and does not depend on the specific representation of the generators.

I hope this helps clarify your questions. Please let me know if you have any further inquiries.

## 1. What is the adjoint transformation of gluon?

The adjoint transformation of gluon is a mathematical operation that describes how the properties of a gluon change when it is transformed from one reference frame to another. It is an important concept in quantum chromodynamics (QCD), the theory that describes the behavior of the strong nuclear force.

## 2. How is the adjoint transformation of gluon related to color charge?

The adjoint transformation of gluon is closely related to color charge, which is a property of quarks and gluons that describes their interaction with the strong force. The transformation changes the color charge of the gluon in a specific way, depending on the reference frame.

## 3. What is the purpose of studying the adjoint transformation of gluon?

Studying the adjoint transformation of gluon is important for understanding the behavior of the strong force and how it affects the behavior of quarks and gluons. It also helps in developing mathematical models and theories that can accurately describe and predict the behavior of subatomic particles.

## 4. How is the adjoint transformation of gluon calculated?

The adjoint transformation of gluon is calculated using mathematical equations and principles from the field of group theory. These equations take into account the properties and interactions of gluons and their transformation between different reference frames.

## 5. Are there any real-world applications of the adjoint transformation of gluon?

While the adjoint transformation of gluon is primarily studied in theoretical physics, it has applications in fields such as particle physics, nuclear physics, and cosmology. It helps in understanding the behavior of particles and forces at a microscopic level, which can have implications for technologies such as nuclear power and particle accelerators.

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