9
Spin particles.
The Poincaré group describes
the relativistic motion of a point-like object. Similarly, the Bargmann group, whose expression will be given later, describes
the non-relativistic motion of a point-like object, which is then called "mass point".
Therefore, this technique, the calculation
of the coadjoint action of the group on the space of moments, has allowed
hidden elements, the attributes
of the object to emerge: the
components of the momentum
.
What is remarkable is that this approach, due to Souriau, makes appear the
key objects of the physicist
as purely geometric objects
. He therefore achieved an unprecedented work of
geometrization of physics
.
Apart from energy and momentum,
other components, the "twisting" and the "passing" greatly confuse
the physicist. What are these?
The expression of the components of the momentum
obviously depends on the coordinate system
chosen.
The simplest is probably to go back briefly to the non-relativistic, otherwise to the expression of the coadjoint action,
as it would have emerged from the analysis of the Bargmann group.
(111)
Mysterious formula. What is it for?
How does it work?
In the box above, the physicist will have recognized some familiar objects:
(112)
are just two expressions of the velocity vector {
vx , vy , vz }, the first in the form of a column matrix and the
second in the form of a row matrix. The product of the two matrices
is a scalar:
(113)
something that begins to resemble
kinetic energy.
m v
is a momentum.
The traditional physicist, when dealing with the dynamics
of a point mass, only knows three things:
-
The mass m
-
The momentum m
v
- The kinetic energy 1
/ 2
mv2
Yes, but velocity
relative to what
?
A group is also a
point of view
on the
things. One can then either consider that one transports, using the group,
objects (as we saw with the Euclidean group),
with respect to a fixed observer, or, the object being
fixed, that one observes it differently.
If one retains this displacement, this transport
of objects, regarding the
dynamic groups
, those of physics
(in contrast to the Euclidean group where time does not appear), one must also say that one
animates
the objects, by giving them
velocity
v
and energy E .
If one adopts the inverse point of view: considering
that the object is fixed and considering oneself to move, what sense
can be given to the groups?
The Euclidean group would then mean:
"Seen elsewhere and from another angle".
"The elsewhere" is the
translation vector
:
(114)
"The seen from another angle" is the rotation matrix
a
, a rotation in space (which could be explicitly expressed with
Euler angles, which we will not do).
Regarding dynamic groups, this view, this
point of view on the "things" must be enriched. Staying within the
context of the Bargmann group, the introduction of this velocity
v
means that, in addition, the observer, who observes this mass point from elsewhere
(translation vector
c
), from another angle (rotation matrix
a
), is also animated, with respect to this mass point
assumed to be stationary, with a velocity
v
.
And, to be complete, to make it more complex,
it does not evolve in the same time as the particle, the observed mass point. It is shifted relative to
it by a time interval
Dt.
In other words; it observes it elsewhere, but it is an spatio-temporal elsewhere,
corresponding to the spatio-temporal translation vector:
(115)
By taking such a "distance", with respect to
this mass point, what do I notice? First that: m' = m
This does not affect its mass.
I can make my life easier by canceling the rotation.
It is already complicated enough to observe a mass point
from elsewhere, seen from another time, shifted, perched on
a skateboard moving at a velocity
v
. Is it absolutely
necessary to twist my neck?
No. Let's do
a = 1
.
but this detail is generally omitted
in calculations. The coadjoint action, thus specified, becomes
:
(117)
"Consider" should be taken here in its etymological sense. What do I do when I consider a situation,
the sky, a battlefield, the film taken by a spy plane?
A bailiff will write:
- Considering the state of the place.....
Static vision, corresponding to the Euclidean group. The bailiff
observes the objects at a distance
c
, at the same moment (
D t
= 0), in principle stationary (
v
= 0). In case, under
a particular angle, under "a certain angle".
A general walking in a reconnaissance plane is a kind of bailiff who moves (
v
0).
But a staff officer watching the film taken by a
spy plane, a "drone", is facing a situation shifted
in time. He is forced to say:
- Let us consider the target, seen from such a point, in a banked turn,
at such a speed, and moreover as it appeared
two hours earlier...
The target does not have a particular proper velocity.
One cannot consider it as fixed, even if it is "a fixed installation".
Even the Earth moves, the Sun also, the galaxy, etc.
Back to the sub-section "groups
and physics"