20
Different movements.
As we saw earlier, the complete Poincaré group had contracted a disease related to its close relationship with a four-component group, the Lorentz group. It is thus composed of two sets: the orthochronous subgroup Go and the antichronous set Gat (which is not a group by itself). Here is the complete playing field:
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In the space J of momenta corresponding to movements occurring in the momentum space with negative energies:
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These are precisely the movements that greatly trouble physicists. In the case of a collision between two particles in the same space, one with positive energy and the other with negative energy, the result is NOTHING.
Before tackling such thorny problems, couldn't we instead focus on "normal" particles, in Coluche's sense?
Agreed. Let's follow Souriau's approach:
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Amputate the group of its antichronous part and keep only the orthochronous subgroup.
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Remove from the momentum space this part referring to material points with negative energy and negative mass.
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Restricted playing field, but then: no more problem.
J+ is supposed to represent a momentum associated with a movement occurring at positive energy.
Conversely,
J− will represent a momentum associated with a movement occurring with energy E < 0.
I choose an element g in my orthochronous subgroup Go. It induces a change in motion. The representative point jumps within the momentum space. But this remains without issue.
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On the left, for example, I have two different movements of the same particle.
Particle species are "species of momenta." In this momentum space J, I can distinguish different regions corresponding to different species. Below, we have limited ourselves to two particle species, corresponding to this linear boundary cutting the half-disk into two parts:
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The representative points in the J+ subspace of momentum, associated with movements occurring with positive energies, I have placed two points corresponding to the same species. For example, these could represent two movements of the electron species.
I have drawn an arrow (coadjoint action) allowing a continuous transition from one of these movements to the other.
On the other hand, if my points were chosen in the momentum subspace, in regions corresponding to two different species (for example, electrons and protons), there would be no group element, hence no coadjoint action, allowing transition from one of these movements to the other. As mentioned earlier.
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Back to the sub-summary "Groups and Physics"