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---------- (39) ---------- (40) ---------- (41) ---------- (42) | |

the

For regions of space where charge and current are absent, the right-hand sides vanish in all the equations. These first order differential equations can be translated into second order equations more suitable for describing the electromagnetic wave in relativistic invariant form. By virtue of the mathematical identity (

By substituting Eq.(43) into Eq.(42) and using another mathematical identity ()=0, we can define

which combines with Eq.(39) yields:

Substituting Eqs.(43), (44) into (40) and using one more identity ( )

The U(1) symmetry on the spin of

then Eqs.(45), and (46) can be written in simpler form :

and

The vector potential

if ' = - (1/c)

and

This is known as the

- Other choice for involves setting
- Eq.(45) is reduced to the Laplace's equation for time-independent electrostatic potential in free space, that's why it is also called "Coulomb gauge".
- The longitudinal component is eliminated, i.e., there is no component along the direction of motion. This is ensured by the condition
**A**' = 0. - The two transversal components correspond to the two polarizations as observed in electromagnetic radiation.
- Since ' = 0 in free space without the presence of any source, the wave equation Eq.(46) become :

^{2}**A**' - (1/c^{2})_{}**A**' = 0 ---------- (49) - The magnetic field is related to
**A**by**B**=**A**' ---------- (50) - The electric field is related to
**A**by**E**= - (1/c)_{}**A**' ---------- (51) - From Eqs.(49), (51), the wave equation for
**E**can be derived :

^{2}**E**- (1/c^{2})_{}**E**= 0 ---------- (50a) - From Eqs.(49), (50), the transverse gauge, and the identity leading to Eq.(46), we obtain the wave equation for
**B**:

^{2}**B**- (1/c^{2})_{}**B**= 0 ---------- (51a) - It is now clear that the introduction of the vector potential
**A**is mainly to simplify computations. For the case of transverse electromagnetic wave two components of**A**is required to describe the wave, while the**E**and**B**fields together introduce four components. - The Coulomb gauge is not covariant under a Lorentz transformation.
- See more in "Quantum Electrodynamics".

The solution of Eq.(49) is :

k

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