Connection between four current and Noether current.

Additionally, the Noether current can be canceled out by including the electromagnetic interaction term -eJ^{\mu}A_{\mu} and performing a gauge transformation, demonstrating the close relationship between the two terms.
  • #1
thepopasmurf
76
0
Hi,
I was wondering if someone could explain how the Noether current and four-current are connected.

For context, I'm going through a derivation to show that local phase symmetry requires the electromagnetic field. I'm at a stage where the I have a Noether current of a complex field (for Klein-Gordon Lagrangian), the next bit of the derivation then says that the Noether current [itex]J^{\mu}=-i(\varphi\partial\varphi^{*}-\varphi^{*}\partial\varphi)[/itex] can be canceled out if one includes the electromagnetic interaction term [itex]-eJ^{\mu}A_{\mu}[/itex] and then does a gauge transformation.

How are the two J terms related, because it seems to say that the four-current is equal to the Noether current.

Thanks,
 
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  • #2
The Noether current and the four-current are related in that they are both generated by the same symmetry transformation. The Noether current is generated by a specific global symmetry transformation, while the four-current is generated by a local gauge symmetry transformation. Specifically, the four-current is generated by the U(1) gauge transformation associated with electromagnetism. In this way, the Noether current can be thought of as the source of the four-current.
 

1. What is the Noether current and how is it related to four current?

The Noether current is a mathematical concept in physics that represents the conserved quantities associated with symmetries of a physical system. The four current, also known as the electromagnetic current, is a quantity in electromagnetism that describes the flow of electric charge and current in a system. The Noether current is related to the four current through a mathematical relationship known as Noether's theorem.

2. How does Noether's theorem explain the connection between four current and Noether current?

Noether's theorem states that for every continuous symmetry in a physical system, there exists a corresponding conserved quantity. In the case of the four current and Noether current, the continuous symmetry is the invariance of the laws of physics under a change in the reference frame. This symmetry gives rise to the conserved quantities, which are the components of the four current, and the Noether current is the mathematical representation of these conserved quantities.

3. What are the implications of the connection between four current and Noether current?

The connection between four current and Noether current has several implications in physics. It provides a deeper understanding of the relationship between symmetries and conserved quantities in physical systems. It also allows for the development of new mathematical tools and techniques for analyzing and solving problems in electromagnetism and other areas of physics.

4. Can the connection between four current and Noether current be observed in real-world phenomena?

Yes, the connection between four current and Noether current can be observed in various real-world phenomena. For example, the conservation of electric charge and current in electromagnetic systems is a direct result of the Noether current. Additionally, the laws of physics that remain the same in all reference frames, such as the laws of Maxwell's equations, are evidence of the connection between four current and Noether current.

5. How is the connection between four current and Noether current relevant in modern physics?

The connection between four current and Noether current is essential in modern physics as it provides a framework for understanding the fundamental principles of conservation and symmetry in physical systems. It is also relevant in various areas of research, such as quantum field theory and particle physics, where the concept of symmetry plays a crucial role in understanding the behavior of particles and their interactions.

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