Redundant degrees of freedom in EM fields?

In summary, there are a total of 6 degrees of freedom for the electric and magnetic fields. However, these are related to each other through Maxwell's equations, allowing for the potential to be found and reducing the degrees of freedom to 4. Of these, 3 are in the vector potential and 1 is in the scalar potential. This leaves 2 redundant degrees of freedom, which are related to the polarization of the electromagnetic wave and the gauge symmetry. To fix the gauge completely, a constraint is needed, such as setting the scalar potential to 0 and having a transverse vector potential, resulting in only 2 polarization degrees of freedom.
  • #1
DOTDO
7
0
If we consider E and B individually, there are 6 total degrees of freedom.

But they are actually related to each other by Maxwell's equations.

So we can find potentials and reduce dof to 4; 3 in vector potential and 1 in scalar potential.
Thus, there remain 2 redundant dof.

This is the problem...

I think one of them gives the polarization of EM wave and the other gives the gauge symmetry.

For example, the vector potential has gauge symmetry, A' = A + del (f)

f is scalar and has one dof.

Am I understanding right ?
 
Last edited:
Physics news on Phys.org
  • #2
There are 6 physical degrees of freedom, ##\vec{E}## and ##\vec{B}##. Indeed, the homogeneous Maxwell equations,
$$\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}$$
lead us to introduce the scalar and vector potentials ##\Phi## and ##\vec{A}##. The relation to the physical field is
$$\vec{E}=-\frac{1}{c} \partial_t \vec{A}-\vec{\nabla} \Phi, \quad \vec{B}=\vec{\nabla} \times \vec{A}.$$
The gauge symmetry makes everything invariant under the gauge transformations,
$$\vec{A}'=\vec{A} - \vec{\nabla} \chi, \quad \Phi'=\Phi+\frac{1}{c} \partial_t \chi$$
with an arbitrary scalar field ##\chi##. Thus we can pose one constraint in addition to the inhomogeneous Maxwell equations. If we choose the Lorenz-gauge condition,
$$\frac{1}{c} \partial_t \Phi + \vec{\nabla} \cdot \vec{A}=0,$$
the inhomogeneous Maxwell equations become separated
$$\Box \Phi=\rho, \quad \Box \vec{A}=\frac{1}{c} \vec{j}, \quad \frac{1}{c^2} \partial_t^2 - \Delta.$$
The entire system of Maxwell equations are only consistent, if electric charge is conserved,
$$\partial_t \rho+\vec{\nabla} \cdot \vec{j}=0.$$
It turns out that this fixing of the gauge with the Lorenz gauge makes the solutions to the wave equations unique if charge and current densities are present.

For the free Maxwell field that's not the case, but even with the Lorenz-gauge constraint you still have the freedom to change the gauge via a field obeying the source-free wave equation. Thus you have to use another constraint to fix the gauge completely. One convenient possibility is to demand
$$\Phi=0$$
in addition to the Lorenz condition. This means you have
$$\Phi=0, \quad \vec{\nabla} \cdot \vec{A}=0,$$
i.e., you have only two transverse field-degrees of freedom left. There are only 2 polarization degrees of freedom (e.g., helicity +1 and helicity -1 states, which is the same as left- and right-circular polarized waves).
 

1. What are redundant degrees of freedom in EM fields?

Redundant degrees of freedom in EM fields refer to the extra parameters that describe the behavior of electromagnetic waves beyond the necessary ones for describing the wave. These extra parameters do not add any new information and can be expressed in terms of the necessary parameters.

2. Why are redundant degrees of freedom important to consider?

Considering redundant degrees of freedom is important because it helps in simplifying the mathematical description of electromagnetic waves. It also helps in identifying the necessary parameters that are needed to fully describe the behavior of the wave and eliminates the need to consider unnecessary parameters.

3. How do redundant degrees of freedom affect the polarization of EM waves?

Redundant degrees of freedom can affect the polarization of EM waves by allowing for multiple ways to describe the same polarization state. This can lead to different representations of the same polarization state, which can be problematic when trying to analyze and interpret experimental data.

4. What is the relationship between redundant degrees of freedom and gauge invariance?

Redundant degrees of freedom are closely related to gauge invariance. Gauge invariance refers to the ability to choose different mathematical descriptions of the same physical system without affecting the physical results. In electromagnetism, redundant degrees of freedom allow for different gauge choices, but the physical results remain the same.

5. How can we eliminate redundant degrees of freedom in EM fields?

There are various mathematical techniques that can be used to eliminate redundant degrees of freedom in EM fields. One approach is to use gauge transformations to remove the unnecessary parameters. Another approach is to use gauge-invariant quantities to describe the behavior of the wave, which automatically eliminates the redundant degrees of freedom.

Similar threads

Replies
8
Views
716
Replies
9
Views
762
  • Classical Physics
Replies
31
Views
2K
Replies
1
Views
527
  • Advanced Physics Homework Help
Replies
15
Views
1K
  • Classical Physics
Replies
7
Views
729
  • Classical Physics
Replies
7
Views
1K
Replies
6
Views
839
  • Special and General Relativity
Replies
7
Views
2K
  • Other Physics Topics
Replies
1
Views
2K
Back
Top