Em field 6 degrees of freedoms & 4-potential

In summary, the electromagnetic field strength includes six degrees of freedom, with three for the electric field strength and three for the magnetic field strength, while the four-potential only includes three degrees of freedom, with two for the photon and one for the Higgs boson. This is because not all combinations of electric and magnetic field components are allowed by Maxwell's equations, which restrict the number of degrees of freedom to three. Adjusting the four-potential to encompass six degrees of freedom would result in combinations that do not satisfy Maxwell's equations.
  • #1
kye
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What does it mean that the electromagnetic field strength includes six degrees of freedom (three for the electric field strength, three for the magnetic field strength), whereas the four-potential includes only three degrees of freedom (two for the photon, one for the Higgs boson)? What is the four-potential? Why isn't it adjusted to be able to encompass 6 degrees of freedom too?
 
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  • #2
You don't really have 6 degrees of freedom... You can't just arbitrarily pick all the electric and magnetic field components and expect them to satisfy Maxwell's equations. As a simple example, suppose [itex]E_x(x,y,z,t) = cos(\omega t + k z) [/itex] and set everything else to zero for all space and time ([itex]E_y = E_z = B_x = B_y = B_z = 0[/itex]). Those particular electric and magnetic fields are not allowed by Maxwell's equations. Specifically, if you plug them into Faraday's law
[tex]\vec{\nabla}\times\vec{E} = -\frac{\partial \vec{B}}{\partial t}[/tex]
you'll find that the left hand side is non-zero while the right hand side is zero, which is obviously a contradiction. There's no charge density/current density combination I can pick which will give me those particular fields.

So you don't really have 6 degrees of freedom because not all combinations of [itex](E_x, E_y, E_z, B_x, B_y, B_z)[/itex] are allowed. They need to satisfy Maxwell's equations.
 

1. What are the 6 degrees of freedom in an electromagnetic field?

The 6 degrees of freedom in an electromagnetic field refer to the 6 independent components of the electric and magnetic fields. These include the three components of the electric field (Ex, Ey, and Ez) and the three components of the magnetic field (Bx, By, and Bz).

2. What is the significance of the 4-potential in electromagnetism?

The 4-potential, also known as the electromagnetic potential, is a mathematical representation of the electric and magnetic fields in space and time. It allows for a more elegant and concise formulation of Maxwell's equations, which describe the behavior of electromagnetic fields.

3. How are the 6 degrees of freedom related to the 4-potential?

The electric and magnetic fields, represented by the 6 degrees of freedom, are derived from the 4-potential through a set of mathematical equations known as Maxwell's equations. The 4-potential is the fundamental quantity from which all electromagnetic phenomena can be derived.

4. Can the 6 degrees of freedom and 4-potential be used to describe all types of electromagnetic fields?

Yes, the 6 degrees of freedom and 4-potential can be used to describe all types of electromagnetic fields, including static and dynamic fields, as well as those generated by moving charges or changing magnetic fields.

5. How is the concept of gauge invariance related to the 4-potential?

Gauge invariance refers to the fact that the 4-potential can be shifted by a certain amount without changing the physical properties of the electromagnetic field. This allows for different mathematical representations of the same physical phenomenon, and is an important concept in the theory of electromagnetism.

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