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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 :
(341)
The components of the momentum become :
(342)
...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 :
(344)
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) :
(346)
Jep=
{ c , M , P } = { c , Jp }
Jp represents
the calssical Poincaré's momentum. Then the coadjoint action of the
momentum simply becomes :
(347)
The calculation is not
complicated and is similar to the one presented above. One computes the anti-action
:
(348)
Then the duality is expressed
through the constancy of the following scalar :
(349)
...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 :
(350)
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 :
(352)
The reader will say "well,
why don't we add 57 new scalars ? "
Just add six and identify
these new scalars to
(353)
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 , z3 , z 4
, z 5 , z6 )
i.e : space-time plus
six additional dimensions.
(355)
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
:
(356)
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
:
(357)
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.
