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Particles with spin.
...The Poincaré 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 with the energy E and the momentum 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 has its momentum space and coadjoint action on this momentum space. If, instead of first choosing the relativistic world (Poincaré group), we had chosen the non-relativistic world, we would have had to refer to the Bargmann group. For computational details, see my lectures on groups. The Bargmann group is a non-trivial extension of the Galilean 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 regarding additional variables will be treated further. On this site, the 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 group on its momentum space. Paradoxically, the calculation in the non-relativistic world is somewhat more complicated than that in the relativistic world. The result is the following: (273)
The physicist identifies some familiar objects, like the velocity: (274)
and kinetic energy: (275)
m v is the momentum. 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 particular point of view on something, on a particle. If we consider the group SO(3), the matrices a, it means "seen from another angle". If we consider the group O(3), the matrices a, it adds the possibility to observe the "thing" through a mirror.
The translation vector (276)
of the Euclidean 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 delay Dt can be combined into a space-time translation vector: (277)
Look at the formulas, from the Bargmann group, we see that:
m' = m
Whatever the point of view, the mass remains 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 that we pass from a steady situation to a non-steady situation, the initial conditions correspond to:
E = 0 (zero energy)
**p **= 0 (zero momentum, zero velocity)
"passage" f = 0
Then the coadjoint action gives: (279)
"To consider" must be read in its etymological meaning.
A process server says: - Drawing up a survey and inventory.
...A static (v = 0) vision of things corresponds to the Euclidean group. The process server observes things at distance c. He observes facts at the time they happen (Dt = 0). Eventually, he looks from a certain angle (a different from 1).
...A general, flying over a battlefield 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 the 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 through space.
...The "north pole" of the Earth is different from the "north pole" of the Sun, by 23°, and it changes over time (26,000 years), due to equinox precession. The north indicated by the Sun (its own rotation axis) is not the same as the north indicated by our galaxy, the Milky Way, which has its own spinning movement (a gap of 90°). Even a galaxy moves, at three hundred miles per hour. With respect to what? With respect to the others. That's all we can say. The group corresponds to two different points of view.
...If I consider that the object is stationary, fixed in space and time, and has no spinning movement, all that I can say is:
- If I move away at distance c.
- If I observe the thing, while cruising at velocity v.
- If the information, coming from this thing, reaches me with a time delay Dt.
*With respect to me *:
---> The mass of the object is not modified.
----> I assign to the object a momentum mv, considered as apparent.
-----> The object acquires a "passage" **f **= m [ c - v Dt ]
-----> It acquires 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 components of a vector: (283b)
...Although the vector 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 vector product. (284)
...The reversed v indicates the vector 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 indicate indifferently a matrix or a vector.
This vector begins to look like something familiar to the physicist: the kinetic momentum.