Gauge fixing in 5D vs 4D, and photon polarization

[kuon]

I'm sorry if what I say is not right, or I haven't understood it right,

- In 3+1 D we have the photon with spin 1 => it has two polarizations.

Our Gauge field $$A_\mu$$ has 4 components => We have two extra degrees of freedom. => We need to get rid of the extra 2 fixing the gauge. 1. The Lorentz gauge is not enough to fix them. What else can we do?- In 4+1 D we have the photon with spin 1=> it has two polarizations.

Our Gauge field $$A_\mu$$ has 5 components => We have THREE extra degrees of freedom. => We need to get rid of the extra 3 fixing the gauge.

2. The Lorentz gauge is not enough to fix them. What else can we do?

Thank you very much for your attention :)

 

[Bob_for_short]

I'm sorry if what I say is not right, or I haven't understood it right,

- In 3+1 D we have the photon with spin 1 => it has two polarizations.

Our Gauge field $$A_\mu$$ has 4 components => We have two extra degrees of freedom. => We need to get rid of the extra 2 fixing the gauge.


1. The Lorentz gauge is not enough to fix them. What else can we do?

In the Lorentz gauge there are two extra degrees of freedom which do not contribute even without getting rid of them. They cannot exchange the energy-momentum with physical degrees of freedom.

 

[Entropia]

Thank you for your question. You are correct in saying that in 3+1 dimensions, the photon has two polarizations and the gauge field has four components, leaving two extra degrees of freedom that need to be fixed. The Lorentz gauge is one way to fix these extra degrees of freedom, but it is not enough. This is because the Lorentz gauge only fixes three of the four components of the gauge field, leaving one extra degree of freedom.

In 4+1 dimensions, the photon still has two polarizations, but the gauge field now has five components, leaving three extra degrees of freedom that need to be fixed. Again, the Lorentz gauge is not enough to fix all of these extra degrees of freedom.

To fix these extra degrees of freedom in both 3+1 and 4+1 dimensions, we can use a technique called gauge fixing. This involves choosing a specific gauge condition that will fix the extra degrees of freedom in the gauge field. One common gauge condition used is the Coulomb gauge, which fixes all of the extra degrees of freedom in both 3+1 and 4+1 dimensions.

In summary, in both 3+1 and 4+1 dimensions, the Lorentz gauge is not enough to fix all of the extra degrees of freedom in the gauge field. Gauge fixing is necessary to fully constrain the gauge field and accurately describe the behavior of the photon. I hope this helps clarify the concept of gauge fixing in higher dimensions.