a4126
| 26 |
|---|
Zoos of particles and antiparticles.
...Particles constitute species, but there also exist particular movements and particular species in momentum space. We can construct the following two zoos:
(362)
From these two zoos, we can write the corresponding moments:
(363) Jpe = { q , cB , cL , cm , ct , v , Jp }
Jj = { 0 , 0 , 0 , 0 , 0 , 0 , Jp } : photon
Jp = { 1 , 1 , 0 , 0 , 0 , vp , Jp } : proton
Jn = { 0 , 1 , 0 , 0 , 0 , vn , Jp } : neutron
Je = { -1 , 0 , 1 , 0 , 0 , ve , Jp } : electron
Jne = { 0 , 0 , 1 , 0 , 0 , vne , Jp } : electron neutrino
Jnm = { 0 , 0 , 0 , 1 , 0 , vnm , Jp } : mu neutrino
Jnt = { 0 , 0 , 0 , 0 , 1 , vnt , Jp } : tau neutrino
...By proceeding in this way, we have a priori created these two distinct zoos: species of matter and species of antimatter. No group action allows transforming a particle into an antiparticle.
All of this rests on the following dynamical group:
(364)
What is momentum?
...Recall that, when constructing the Poincaré group, we began with the Lorentz group element L, which was axiomatically defined using a "mirror" matrix G:
(365)
(366)
This is related to a quadratic form: the Minkowski metric.
(367)
...A Minkowski metric applies to empty space. Our group describes isolated particles, not systems composed of several interacting particles. The motion of a particle is a geodesic in Minkowski spacetime: a straight line. If the particle has zero mass, this corresponds to a geodesic of zero length, but it is not incorrect to represent particle motions as straight lines in spacetime.
(365b)
...The set of points forming momentum space represents all possible movements of all possible species of particles. A group action (coadjoint action), based on a given element g of the dynamical group G, transforms one movement into another.
(366b)
(367b)
...On the figure above, we see how an element of the group can transform a given electron movement into another movement of the same species. However, using coadjoint action and group elements, we cannot transform an electron's movement into that of a neutron or a photon. The space of movements is divided into subsets, each corresponding to all possible movements of a given species.
...As shown above, the full Poincaré group leads to particles with negative energy. Therefore, if we now choose not to exclude them, we must consider two distinct subspaces: