| 22 |
Permanent magnets.
...If
we put a piece of iron in a strong magnetic field, when this inducting magnetic
field is cut, this metal will keep a permanent magnetization. Why ?
...The
magnetic field acts on the spins of electrons, which behave like mini-magnetic
dipoles, mini-magnets. But why do they keep the orientation imposed by the
inducting field, after this one is cut ?
...Because
electrons are like Panurge's sheeps. Each one follows the field due to its
neighbours. Then they all keep their parallelism. This order can be destroyed
if one warms or hammers it the metal.
The magnetic momentum
of anti-matter.
...The
charge conjugation reverses the gyromagnetic coefficient, in the Dirac's antimatter.
Then the spin s being unchanged but it reverses the magnetic momentum of the
particle.
Notice this matter
anti-matter symmetry does not change the energyE , nor the particle's
impulsion p .
The four components
of the Lorentz group.
Above, we presented that
we called the "PT-group", a four components ruling the P, Tand PT symmetries.
(300)
Then was presented the
Galileo group "space and time oriented".
(301)
Then was presented the
complete four components Galileo group.
(302)
with P, Tand PT symmetries.
The Lorentz group element
(4,4) L obeys the axiomatic definition :
(303)
(304)
L acts on space
time :
(305)
Like the Galileo's complete
group the complete Lorentz group has four components :
Ln
elements : keep space and time orientation unchanged.
Ls
elements : achieve space inversion ( P-symmetry )
Lt
elements : achieve time-inversion (T-symmetry )
Lst
elements : achieve space and time inversion (PT-symmetry)
Give an example of matrixes
which belong to the four components :
(306)
An = 1 ( neutral element) 
Ln
keeps space and time unchanged
As
Ls inverses space.
At
Lt inverses time.
Ast
Lst inverses space and time.
The neutral component
is a sub-group of the complete Lorentz group.
Remark :
(307)
At
= - As
Ast
= - An
Two component form a
sub-group :
(308)
Lo = Ln
U Ls
whose elements
do not reverse time. Souriau calls it the orthochron sub-group Lo of
the complete Lorentz group L . The rest of the group, the set of matrixes
which belong to the third and fourth components :
Lac = Lt
U Lst
do not form a group but
a set of matrix, that Souriau calls antichron set. Then
the complete Lorentz group is (
U for "union")
(309)
L = Lo U
Lac
But, writing
(310)
m
Lo , avec m
= ± 1
we get the complete
group.
The four components
ofthe Poincaré's group.
From the Lorentz group
one builds the Poincaré's group :
(311)
C being the space-time
translation vector :
(312)
...The
complete Poincaré's group owns four components, due to the four components
structure of the Lorentz Group. In classical physics the Poincarés'
group is limited to its neutral component.
...We
have built in preceding sections the coadjoint action of the group on its
momentum's space, which works "in general", whatever the chosen component
is. In the following we examine the action for the different components. This
was done earlier by J.M.Souriau :
Souriau, Structure des Systèmes Dynamiques, Dunod 1973, in french,
andBirkhauser Ed. 1997, in english, chapiter III, page 197, in a section entitled
: Inversions of space and time.
Index Dynamic Groups Theory