| 19 |
Particles with spin.
...The
Poincarés' group describes the relativistic movement of a point-like
object. Similarly the Bargmann group describes the non-relativistic movement.
The components of the moment arise as pure geometric quantities. It's a geometrization
of physics .
The
physicists are familiar to the energy Eand impulsion vector p. But
they can be a little bit puzzled by the two other objects : the passage f
and the spin vector l . The
form of the momentum's components depends on the choice of coordinates.
...Each
dynamic group owns its momentum space and coadjoint action on this momentum
space. If, instead chosing at first the relativist world (Poincaré's
group) we had chosen the non-relativistic world, we would have to refer to
the Bargmann's group. For computational detail see my lectures on groups.
The Bargmann group
is a non-trivial extension of the Galileo's group :
(272)
As the reader can see,
this group acts on a five dimensional space :
r : space
t : time
z : an additional variable.
...These
questions of additional variable or additional variables will be treated further.
In this site full calculation of the coadjoint action of the Poincaré
group has been given above. One could also derive the calculation of the coadjoint
action of the Bargmann's group on its momentum. Paradoxically, the calculation
in the non-relativist world is somewhat more complicated than the one in relavistice
world.
The result is the
following :
(273)
The physicist identifies
somes familiar objects, like the velocity :
(274)
and kinetic energy :
(275)
m v is the impulsion.
Velocity
with respect to what ?
A group changes
the parameters of the movement, gives to a particle a velocity v and
a kinetic energy E .
We can choose the
opposite interpretation, and consider that a group is a peculiar point of
view on something, on a particle.
If we consider
the group SO(3), the matrixes a, it means "seen along another angle
of view".
If we consider
the group O(3), matrixes a, it adds the possibility to observe the
"thing" through a looking glass.
The translation vector
(276)
of the Euclid's group
adds "seen from elsewhere".
In dynamic groups, the
presence of a velocity v in the group means that the observer moves.
The time-translation
e = Dt means that the observer sees the thing after
some delay.
The translation
vector Dr and the time delayDt
can be put together in a space-time translation vector :
(277)
Look at the formulas,
from Bargmann's group, we see that :
m' = m
Whatever is the point of view, the mass in unchanged.
Let us simplify a little
bit this "point of view", choosing a = 1.
The coadjoint action
becomes :
(278)
...The
coadjoint action indicates the change in the movement's parameters. If we
consider we pass from a steady situation to a non-steady situation, the inital
conditions correspond to :
E = 0 ( zero energy )
p = 0 ( zero impulsion,
zero velocity )
"passage" f =
0
The the coadjoint action gives :
(279)
"To consider" must be
read in its etymologic meaning.
A process server says
:
- Drawing up a survey cum inventory.
...A
static (v = 0) vision of things, corresponds to the Euclid's group.
The process server observes things at distance c . He observes facts
at the time they happen (Dt = 0). Eventuallly he looks from a certain angle
(a different from 1).
...A
general, flying over a battle field, in a plane, is some sort of a process
server, who observes things from a moving point of view (from a plane cruising
at velocity v).
...But
a general, in his headquarters, looking at a movie taken by a pilotless plane,
a drone, some hours before, says :
- Considering the target, as it was one hour before (non zero Dt), seen from
a moving point of observation (non zero v) , from an altitude of five
thousand feet (non zero c), cruising at velocity v and taking
picture through a certain angle (a being different from 1).
...A
target has no defined velocity, or position, or orientation, even if it is
supposed to be a "fixed" building. Everything is relative. Even the Earth,
the Sun, our galaxy move over space.
...The
"north pole" of the Earth is different from the "north pole of the Sun", from
23°, and it changes in time ( 26,000 years), due to equinox precession.
The north indicated by the sun ( its own rotation axis ) is not the north
indicated by our galaxy, the milky way, which has its own spining movement
( 90°'s gap ). Even a galaxy moves, at a trhree hundred miles per hour.
With respect to what ? To the other ones. That's all we can say.
The group corresponds to two different points of view.
...If
Iconsider that the object is steady, fixed in space and time, and owns no
spining movement, all that Ican say is :
- If Imove off at distance c .
- If Iobserve the thing, when cruising at velocity v .
- If the information, coming from this thing, joins me with a time-delay Dt.
With respect to me:
---> The mass of the
object is not modified.
----> Igive the object an impulsion mv, considered as an apparent
one.
-----> The object gets a "passage" f = m [ c - v Dt ]
----> It gets a spin
(279b)
Write it in a more explicit
way :
(280)
(281)
(282)
or :
(283)
One may consider the three independent components of the spin matrix l
as the component of a vector :
(283b)
...Although
the vectorial product has not been defined in our space, i.e : we did not
give space a right-left orientation, we can consider the last expression as
a vectorial product.
(284)
...The
reversed v indicates the vectorial product. We see that the last line of the
formulas giving the coadjoint action corresponds to :
(285)
l is a matrix,
not a vector. But, according to the chosen notation, bold letters indicates
indifferently a matrix or a vector.
This vector begins to
look like something familiar to the physicist : the kinetic momentum
.
Index Dynamic Groups Theory