# Current conservation for SU (N)

• Tian
In summary, the conversation discusses the equation D^{\mu}F_{\mu\nu} = - j_{\nu} and the process of differentiating it covariantly and anti-symmetrizing it to obtain \frac{1}{2}[D^{\mu}, D^{\nu}]F_{\mu\nu} = D^{\nu}j_{\nu}. It is then shown that using the definition of the covariant derivative in the adjoint representation, [D^{\mu} , D^{\nu}]M = [F^{\mu\nu} ,M], and for M = F_{\mu\nu}, the equation simplifies to D^{\nu}j_{

#### Tian

Homework Statement
In the "An introduction to Quantum Field Thoery" of Peskin and Schroeder, the equation(15.51) of the chapter 15.3 gives the classical equation of motion, so from this equation to derive the current conservation.
Relevant Equations
the classical equation of motion for SU(N), please see my picture
Here is my solution

$$D^{\mu}F_{\mu\nu} = - j_{\nu},$$ Differentiate this covariantly and anti-symmetrized to obtain $$\frac{1}{2}[D^{\mu}, D^{\nu}]F_{\mu\nu} = D^{\nu}j_{\nu}. \ \ \ \ (1)$$ Now, from the definition of the covariant derivative in the adjoint representation (acting on any matrix-valued field) $$D^{\mu}M \equiv \partial^{\mu}M + [A^{\mu} , M],$$ you can show that $$[D^{\mu} , D^{\nu}]M = [F^{\mu\nu} ,M]$$ Thus, for $M = F_{\mu\nu}$, eq(1) becomes $$D^{\nu}j_{\nu} = \frac{1}{2}[F^{\mu\nu} , F_{\mu\nu}] = 0.$$

Thank you veery much . It should be done in the adjoint repesentation.

## 1. What is SU(N) in current conservation?

SU(N) is a mathematical representation of the symmetry group of a system with N different states. In the context of current conservation, it refers to the conservation of a certain type of charge, such as electric charge or color charge, in a system with N different charges.

## 2. Why is current conservation important in SU(N)?

Current conservation is important in SU(N) because it is a fundamental principle of nature that dictates the behavior of particles and their interactions. Without current conservation, the laws of physics would not be consistent and predictable.

## 3. How does current conservation apply to the Standard Model?

The Standard Model of particle physics is based on the principle of current conservation. It describes the interactions between particles and their conservation laws, including electric charge, color charge, and other types of charges.

## 4. Can current conservation be violated in SU(N)?

There are some theories that suggest current conservation may be violated in certain extreme conditions, such as in high energy collisions or in the early universe. However, current conservation has been extensively tested and has not been observed to be violated in any experiments so far.

## 5. How is current conservation experimentally verified in SU(N)?

Current conservation is experimentally verified by measuring the total charge before and after a particle interaction. If the total charge remains the same, then current conservation is upheld. This has been confirmed in numerous experiments, including high energy colliders and particle decay processes.