Proof that the E.M Field is invariant under guage transformation.

In summary, the E.M. Field is a physical field that encompasses all electromagnetic interactions and is responsible for the behavior of charged particles. A gauge transformation is a mathematical concept used to describe the symmetries of a physical system, and it is important to prove that the E.M. Field is invariant under this transformation. This proof is achieved using mathematical equations and principles, with practical applications including accurate models for electromagnetic devices and advancements in modern physics theories.
  • #1
hob
6
0
To prove:

[tex]F[/tex] [tex]\overline{} \mu\nu[/tex] = [tex]\nabla [/tex][tex]\overline{} \mu[/tex][tex]A[/tex] [tex]\overline{} \nu[/tex] - [tex]\nabla [/tex][tex]\overline{} \nu[/tex][tex]A[/tex] [tex]\overline{} \mu[/tex]

is invariant under the gauge transformation:

[tex]A[/tex] [tex]\overline{} \mu[/tex] [tex]\rightarrow[/tex] [tex]A[/tex] [tex]\overline{} \mu[/tex] + [tex]\nabla [/tex][tex]\overline{} \mu[/tex][tex]\Lambda[/tex]I end up with:

[tex]F[/tex] [tex]\overline{} \mu\nu[/tex] = [tex]F[/tex] [tex]\overline{} \mu\nu[/tex] + [[tex]\nabla [/tex][tex]\overline{} \mu[/tex],[tex]\nabla [/tex][tex]\overline{} \nu[/tex]][tex]\Lambda[/tex]

Which I guess is invariant provided [tex]\nabla [/tex][tex]\overline{} \mu[/tex] & [tex]\nabla [/tex][tex]\overline{} \nu[/tex] commute?

Do they commute? and if so why?

Many thanks.
 
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  • #2
Yes, they commute. In the case of normal minkowski space time and the Abelian gauge group U(1) the differential operators reduce to ordinary derivatives.
 
  • #3


Yes, \nabla \overline{} \mu and \nabla \overline{} \nu commute. This is because the commutator of two vector fields is given by:

[\nabla \overline{} \mu, \nabla \overline{} \nu] = \nabla_{\overline{} \mu} \nabla_{\overline{} \nu} - \nabla_{\overline{} \nu} \nabla_{\overline{} \mu}

where \nabla_{\overline{} \mu} and \nabla_{\overline{} \nu} are the covariant derivatives with respect to the vector fields \overline{} \mu and \overline{} \nu, respectively.

Now, the covariant derivative of a vector field with respect to another vector field is given by:

\nabla_{\overline{} \mu} \overline{} \nu = \partial_{\overline{} \mu} \overline{} \nu + \Gamma^{\lambda}_{\mu \nu} \overline{} \lambda

where \partial_{\overline{} \mu} is the partial derivative with respect to the coordinate \overline{} \mu and \Gamma^{\lambda}_{\mu \nu} are the Christoffel symbols of the second kind. These symbols represent the connection coefficients of the metric tensor and are symmetric in their lower indices.

Since the Christoffel symbols are symmetric in their lower indices, we have:

\Gamma^{\lambda}_{\mu \nu} = \Gamma^{\lambda}_{\nu \mu}

This means that the covariant derivative of a vector field with respect to another vector field is symmetric, i.e.:

\nabla_{\overline{} \mu} \overline{} \nu = \nabla_{\overline{} \nu} \overline{} \mu

Therefore, the commutator of two covariant derivatives, and thus two vector fields, is zero, i.e.:

[\nabla \overline{} \mu, \nabla \overline{} \nu] = 0

This shows that \nabla \overline{} \mu and \nabla \overline{} \nu commute, and thus the E.M. field is invariant under gauge transformation.
 

1. What is the E.M Field?

The E.M. Field, also known as the electromagnetic field, is a physical field that encompasses all electromagnetic interactions, including electric and magnetic fields. It is responsible for the behavior of charged particles and is a fundamental concept in physics.

2. What is a gauge transformation?

A gauge transformation is a mathematical concept used to describe the symmetries of a physical system. It involves changing the mathematical description of a system without altering its physical properties. In the context of the E.M. Field, it involves transforming the electric and magnetic fields while keeping their physical effects unchanged.

3. Why is it important to prove that the E.M. Field is invariant under gauge transformation?

Proving the invariance of the E.M. Field under gauge transformation is important because it confirms the fundamental principles of electromagnetism and allows for the development of accurate mathematical models to describe the behavior of electromagnetic interactions. It also helps to simplify calculations and predictions in various fields, such as engineering and astrophysics.

4. How is the invariance of the E.M. Field under gauge transformation proven?

The invariance of the E.M. Field under gauge transformation is proven using mathematical equations and principles, such as Maxwell's equations and the Lorentz transformation. These equations are manipulated to show that the electric and magnetic fields remain unchanged after a gauge transformation is applied, thereby proving their invariance.

5. What are the practical applications of the invariance of the E.M. Field under gauge transformation?

The invariance of the E.M. Field under gauge transformation has many practical applications, including the development of accurate models for electromagnetic devices, such as antennas and circuits. It also plays a crucial role in the study of quantum field theory and the development of modern physics theories, such as the Standard Model.

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