13
The four components of the Poincaré group.
Starting from the Lorentz group, one constructs the Poincaré group, already mentioned:
(142)
C is the "spacetime translation" vector.
(143)
This Poincaré group will also have four components, each associated with the corresponding component of the Lorentz group.
Above, the action of the group on its space of motions. But what is interesting are the actions of the four components on the momentum. See: Souriau, Structure des Systèmes Dynamiques, Dunod 1973 (and Birkhäuser 1997, in English), Chapter III, page 197, section titled:
Space and time inversions.
Recall the components of the momentum associated with the Poincaré group:
E: energy
p: momentum
f: passage
l: rotation
To stay close to Souriau's notation, let us denote:
- L_n the neutral component of the Lorentz group.
- L_s the component that inverts space.
- L_t the component that inverts time.
- L_st the component that inverts both.
Since C is a spacetime translation, the four components of the Poincaré group are:
g_p(L_n, C): neutral component
g_p(L_s, C): space inversion
g_p(L_t, C): time inversion
g_p(L_st, C): simultaneous space and time inversion.
Let us now examine their effects on the momentum components. We must consider the formulas giving the group's action on its momentum space:
(144)
P is the four-vector:
(145)
We can write the matrices to analyze:
(146)
with l = ±1 and m = ±1.
L_n = L(l = 1; m = 1)
L_s = L(l = -1; m = 1)
L_t = L(l = 1; m = -1)
L_st = L(l = -1; m = -1)
(147)
(148)
Now consider the action on rotation and passage.
(149)
But in what interests us, C = 0
(150)
thus l' = l and f' = l m f
We deduce:
(151)
g_p(L_n, C):
E → E; p → p; f → f; l → l
g_p(L_s, C):
E → E; p → -p; f → -f; l → l
g_p(L_t, C):
E → -E; p → p; f → -f; l → l
g_p(L_st, C):
E → -E; p → -p; f → f; l → l
Inversions never change the rotation l.
On the other hand, time inversion and energy inversion E → -E are synonymous.
Rotation is synonymous with spin when quantized. No inversion alters it.
Spin (as the magnitude of the particle's rotation vector) is merely a number.
The energy of a particle at rest is mc².
Time inversion is synonymous with mass inversion m.
Spatial inversion does not invert mass.
The first two components of the group are named by Souriau "orthochronous," and the last two "antichronous."
He notes that all this raises the problem of negative masses, which physicists generally dislike. Indeed, what of the result of a collision between two particles with energies +mc² and -mc²? Complete annihilation occurs. This is not merely matter-antimatter annihilation, which yields photons. Rather, it would be a phenomenon producing pure nothingness.
To avoid the problem of negative masses, Souriau considers two solutions. The first is simply to decide that particles with negative mass do not exist. The second is to exclude antichronous transformations.
Paraphrasing, we might say:
- God, in His infinite wisdom...
Let us continue building elements that will serve as a foundation for our own work.
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