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Application to the above:
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We have used the obvious property: the transpose of the transpose of a matrix is the original matrix.
Therefore, globally:
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If I take a particle with non-zero mass, I can always imagine that I picked it from the tree of knowledge of rest and non-rest states, with zero momentum.
I have seen that I can also arrange things so as to cancel the transition, by placing myself in a reference frame that "accompanies the particle in its motion."
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I cannot consider a particle with zero rest energy E₀. That would make no physical sense. But I also know—or am supposed to know—that a particle cannot have zero spin, even in a hypothetical rest state. Moreover, not only does this spin, or "spin vector" s, always exist, but its magnitude s is invariant—it is indeed a characteristic of the particle. It is a half-integer multiple of h/2π, the reduced Planck constant. This is also a consequence of the "geometric quantization" invented by Souriau.
Always from geometry...
These "attributes" are somewhat more puzzling than the non-relativistic attributes mentioned earlier.
But it should be noted that this "geometric quantization" also applies to the non-relativistic world (Bargmann group), quantifying spin, individual angular momentum, "vorticity," spin—whatever name one gives it—of the particle, the material point, the object, the thing governed by the group. Its direction may change, but:
Don't touch my magnitude s.
All this passes through an additional variable z, regarded by some theorists and mathematicians as "a computational intermediary."
That said, in this five-dimensional space:
z, x, y, z, t
we move, we transport ourselves.
There are things that cause no problem, such as:
x → -x
y → -y
z → -z
which corresponds to a P-symmetry. If we apply this not to a point object, but to a set of linked points, the structures are transformed into their enantiomorphs, into their mirror images. But for an isolated particle, this is merely another kind of motion.
Still staying within 5D, we have seen certain attributes emerge.
In non-relativistic physics:
- Mass m
- Energy E
In relativistic physics:
- E and m intertwined into a single entity.
These are simple scalars. The mathematician would say they can be chosen equally well as positive or negative. They are merely choices made within a particular space of momenta, forming the momentum space, depending on n parameters (n being equal to the dimension of the group). In the momentum associated with the Poincaré group (non-extended):
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Jₚ = {E, p, M}
the parameters can, in principle, take any possible values, positive or negative.
Let J be the set of parameters defining the momentum. J is the momentum space. In this space, we should then be able to distinguish two domains:
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The group "overhangs" this space and ensures various types of transport. It contains elements that allow transforming one motion into another. As Souriau puts it:
The momentum follows the motion like its shadow.
Back to the subsection "Groups and Physics"