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Non-trivial extension of the Special Galileo's group.
**The Bargmann's group **( 1960 )
The following matrixes ( see my lectures on groups )
(338)
form a group, discovered by Bargmann in 1960. Here again, it acts on a five-dimensional space. Its dimension is 11, due to the presence of the scalar f . It's a non-trivial extension of the Special Galileo's group.
(339)
If one compute the coadjoint action of the group on its momentum, one gets :
(340)
...We see that this coadjoint action is more refined and that the mass interacts with the other components of the moment. We have analyzed that above and shown how it brings the physical meaning of the momentum's components.
...A momentum is a movement of a given particle. The Bargmann's group describes non-relativist movements. We may consider a particle at rest, with no energy, no impulsion, no spin. Just a non-zero mass :
m
**p **= 0
E = 0
**f **= 0
**l **= 0
We use the following element of the Bargmann's group :
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The components of the momentum become :
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...In a system of coordinates linked to the particle the passage **f **is still zero. We have show than the spin matrix identifies to kinetic momentum.
...Here, what is important is to look at the trivial extension of the Special Galileo's group (why "special" ? This will be explained further). When one performs this trivial extension, it just brings an additional scalar to the momentum.
Let us extend the Poincaré's group :
Central extension of the Poincaré's group. (343)
"ep" means "extended Poincaré's group". Lo is the element of the orthochron sub-group Lo of the complete Lorentz group L. So that we may consider the above element as the orthochron sub-group Gepo of a complete extended Poincaré's group, whose element is :
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The two act of five dimensional space :
(345) ( t , x , y , z , z ).
On can show that this extension cannot stand non zero terms on the first line, instead 0 = ( 0 0 0) , between 1 and f .
...As shown by J.M.Souriau, the geometric quantification method (Kostant-Kirilov-Souriau method) brings the Schrödinger equation from the Bargmann's group and the Klein Gordon equation from the extended Poincaré's group ( Structure des Systèmes Dynamiques, Dunod Ed. 1972 ). In addition this central extension of the group adds an extra scalar to the momentum (as in the trivial extension of the Bargmann's group) :
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Jep= { c , M , P } = { c , Jp }
Jp represents the calssical Poincaré's momentum. Then the coadjoint action of the momentum simply becomes :
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The calculation is not complicated and is similar to the one presented above. One computes the anti-action :
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Then the duality is expressed through the constancy of the following scalar :
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...So that we get an additional scalar c , which is just conserved through the coadjoint action. Since now this scalar had received no physical interpretation. We are going to clear up all that in the following. Obviously we can extend the group as many time we want :
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Each time, it adds an additional scalar
(351) Jpe = { c 1 , c 2 , c 3 ....., M , P } Jpe = { c 1 , c 2 , c 3 ....., Jp } and the coadjoint action becomes :
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The reader will say "well, why don't we add 57 new scalars ? "
Just add six and identify these new scalars to
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c 1 = q (electric charge)
c 2 = cB (baryonic charge)
c 3 = cL (leptonic charge)
c 4 = cm (muonic charge)
c 5 = ct (tauonic charge)
c 6 = v (gyromagnetic coefficient)
The group acts on the following ten dimensional space :
(354) ( x , y , z , t , z 1 , z 2 , z 3 , z 4 , z 5 , z 6 )
i.e : space-time plus six additional dimensions.
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Recall that this group is built with the orthochron sub-group
Lo = Ln (neutral component) U Ls (achieving space-inversion)
of the complete Lorentz group L.
The momentum becomes :
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Jpe = { q , cB , cL , cm , ct , v , Jp }
Jp being the part of the moment corresponding to the Poincaré's group Gop (orthochron sub-group).
What is the physical meaning ?
...A momentum belongs to a space, which is a n-manifold. The Poincaré's group owns ten dimensions, so that the Poincaré's group momentum is composed by ten quantities.
Then we add six more dimensions to the group, corresponding to the additional phasis :
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f1 ,f2 ,f3 ,f3 ,f5 ,f5
The momentum becomes :
(358) Jpe = { J1, J2 J3, J4, J5, J6, J7, J8, J9, J10, J11, J12, J13, J14, J15, J16 }
We decide that amon te set of scalars
(359) Jp = { J7, J8, J9, J10, J11, J12, J13, J14, J15, J16 }
we identify the energy E, the momentum p, the passage **f **, the spin antisymmetric matrix l .
...E and** p** may take all possible values, but quantum arguments impose the constancy of the modulus s of the spin vector (in a system of coordonates linked to the particle), which is not justified here and corresponds to Souriau's work.
We have six more scalars :
(360) J1, J2 J3, J4, J5, J6
...We decide that, among an infinity of possible choices, some discrete choices correspond to real particles (and anti-particles). Then, in the 16-manifold corresponding to the momentum space we select discrete movements corresponding to particules, with defined quantum numbers
(361) { q , cB , cL , cm , ct , v }
...For the moment the coadjoint action of the group just ensures the conservation of these quantities, along given movements. There are "passive quantum numbers" as well as the mass appeared as a passive quantity, when arising from the trivial extension of the Special Galileo's group.