14
Central extension of the Poincaré group.
A reference to such an extension can be found in J.M. Souriau's book, Structures des Systèmes Dynamiques. His method of geometric quantization allows one, starting from the group, to recover the equations of Quantum Mechanics. For example, the Bargmann group, describing a non-relativistic particle, leads to the Schrödinger equation, which is also non-relativistic.
The starting point is the Galilei group.
It is a (5,5) matrix constructed in this way:
(152)
The rotation matrix depends on three parameters, the Euler angles. Thus, the dimension of the group is ten.
Using the notations:
(153)
(154)
associated with spacetime:
(155)
Strangely enough, the construction of the coadjoint action of the group on its momentum space does not make the mass m appear as a geometric object. This can only be achieved through a non-trivial extension of the group, namely the Bargmann group (1960).
(156)
The presence of the scalar
f
increases the dimension of the group by one: eleven.
This group acts on a five-dimensional space, spacetime, plus one additional dimension
z
, through the action:
(157)
The coadjoint action of the Bargmann group on its momentum was given above. One sees that adding the scalar
f
, by increasing the group's dimension, adds a component to the momentum, which then identifies with the mass m (which is conserved in the process: m' = m).
Starting from the Bargmann group and using its method of geometric quantization, Souriau can then construct the non-relativistic Schrödinger equation.
The relativistic quantum equation is the Klein-Gordon equation. It was therefore logical to seek which group could generate it. This is the central extension:
(158)
"pe" for "extended Poincaré". Here we have constructed the Poincaré group starting from the orthochronous subgroup of the Lorentz group Lo.
The space associated with this group is also five-dimensional:
(159)
(t, x, y, z, z).
This extension is simpler than Bargmann's, but in fact things are always easier in the relativistic case. We can additionally prove that between the 1 and the
f
in the first row, only the row matrix
0
= (0 0 0) can appear: all zeros.
The method of geometric quantization then leads to the Klein-Gordon equation. Regarding the group's action on its momentum space, we obtain:
(160)
J_pe = {c, M, P} = {c, J_p}
The calculation is not complicated. It fully mirrors the calculation of the coadjoint action of the Poincaré group on its momentum.
We compute the anti-action:
(160b)
then express the invariance of the scalar product (duality):
(160c)
If you manage to get through this calculation, it will be a very good sign. It will mean you're beginning to enter this complex domain.
A scalar
c
then appears, whose sole function is to be conserved. What does it mean? No explanation is given. It is simply "something that is conserved." One could, for example, assign it the status of electric charge.
The first idea that comes to mind is to perform this type of extension multiple times:
(161)
It will be shown later that this operation can be performed indefinitely, each time adding an extra scalar:
(162)
J_pe = {c₁, c₂, c₃, ..., M, P}
J_pe = {c₁, c₂, c₃, ..., J_p}
with the following coadjoint action:
(163)
We will then consider certain discrete values of the momentum components as the particle's charges.
Well, the reader might say, indeed, we could add, for example, six additional rows. We would then obtain invariance of scalars that could be identified as:
(164)
c₁ = e (electric charge)
c₂ = cB (baryon charge)
c₃ = cL (lepton charge)
c₄ = cₘ (muon charge)
c₅ = cₜ (tau charge)
c₆ = v (gyromagnetic ratio)
It would suffice to consider the group, with its corresponding action, associated with a ten-dimensional space:
(165)
(x, y, z, t, z₁, z₂, z₃, z₄, z₅, z₆)
(166)
Once again, we construct the group around the orthochronous subgroup Lo of the Lorentz group:
Lo = Ln (neutral component)
» Ln (space inversion).
This two-component group then simply produces six scalars that accompany the particle without interacting with anything else. The momentum becomes:
(167)
J_pe = {q, cB, cL, cₘ, cₜ, v, J_p}
J_p being the "Poincaré part." But this remains of limited interest.
Back to the sub-summary "Groups and Physics"
