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Question

What are the light bullets (electromagnetic solitonss)? Do they contradict the Maxwell equations?
Asked by: Stefan Bucur

Answer

Light bullets are the quanta of the electromagnetic (EM) field and are known as photons. They arise due to the quantisation of the EM field and as such do not contradict the classical equations of Maxwell. Think of the solutions to Maxwell's equations as the classical limit of the quantum solutions, i.e. photons.

This way, semi-classically light comes in little "bullets" with energy given by Planck's law. These are particles of light, which is not a new idea. Newton considered light to be little particles. This idea was forgotten about when the wave nature of light became apparent via diffraction experiments.

As in quantum mechanics we have wave/particle duality; that is a particle is also a wave, we have field/particle duality in quantum field theory. (I say this loosely as it is not always clear exactly how to get a particle from a field.) This means that the field and particle description of light are both consistent.

Being very slack about the definition of a solitons, Maxwell's equations do not have solitons solutions. (Here by solitons I am really referring to a classical solution which is topologically stable and distinct from the vacuum). A slight modification of the equations does have a solitons called a magnetic monopole. These were first considered by Dirac and are the direct generalisation of the electric monopole. This monopole is a point of magnetic charge. The corresponding classical equations are Maxwell's with a magnetic source term.

Interestingly, if such monopoles exist then we must have quantisation of electric and magnetic charge, that is both must be integers (in some correct units). However, EM is consistent without magnetic monopoles and none have ever been observed in nature.

A generalisation if EM called Yang-Mills gauge theory does not posses static non-singular monopole solutions. There are however, solutions which connect topologically discrete classical vacua. These are known as instantons and manifest themselves as tunnelling events. These instantons are very important as they give us information about the vacuum.

Coupling Yang-Mills to a Higgs (scalar) field does have monopole solutions. These could be cosmological significant. Such monopoles must be formed when we have spontaneous symmetry breaking. By spontaneous symmetry breaking we mean that the vacuum state does not have the same symmetry as the rest of the system. This is the case with the standard model of particle physics and we expect suchmonopoleple to be seen in nature.

Any book on quantum field theory will talk about photons and the classical solutions I have outlined here.
Answered by: Andrew James Bruce, Grad student, UK



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