Cosmological Physics of Antimatter

**..**When Souriau explicitly describes the action of the different elements of the Poincaré group, he finds:

gp ( Ln , C) : I E --> E ; p --> p ; f ---> f ; l ----> l

**..**The elements of this orthochronous (neutral) component of the group conserve energy, momentum, passage and spin.

gp ( Ls , C) : I E --> E ; p --> - p ; f ---> - f ; l ----> l

**..**This element of the second component of the orthochronous subset of the Poincaré group's matrices conserves energy and spin, but reverses passage and momentum.

gp ( Lt , C) : I E --> - E ; p --> p ; f ---> - f ; l ----> l

**..**This element of the third component of the group, which belongs to the antichronous subset (according to Souriau's definition), reverses energy and passage, but conserves momentum and spin.

gp ( Lst , C) : I E --> - E ; p --> - p ; f ---> f ; l ----> l

**..**This fourth element, which belongs to the antichronous subset of the Poincaré group, conserves passage and spin, but reverses energy and momentum.

In all four cases, the spin remains unchanged.

The elements of the two antichronous components of the Poincaré group reverse energy.

**..**This is a very important result, discovered by Souriau in 1972, which can be found in his book, chapter III, page 197 (in the French edition), devoted to the inversions of space and time.

Quantum characteristics come from the so-called "extended Poincaré group":

**....**The dimension of the group then becomes 11.

**....**f is a phase.

...A group acts on its associated space (here, spacetime plus an additional dimension z, the "Kaluza dimension"). But it acts on its momentum space through coadjoint action. The number of components of the momentum J is equal to the number of dimensions of the group. For the non-extended Poincaré group, the components of the momentum are:

**....**Classically, these components are grouped:

Jep = { c, M , P } = { c, M, p, E }

where p is the momentum:

p = { px , py , pz }

while E is the energy. P is the four-vector:

M is an antisymmetric matrix, as defined by Souriau:

**....**If we consider the extended Poincaré group, we obtain an additional scalar component in the momentum, classically identified with the electric charge:

**....**The action of the extended Poincaré group on its momentum space gives:

**....**That we "read" as: conservation of the electric charge c. It is now possible to extend this group by adding new extra dimensions, similar to Kaluza's. In the following, Lo represents the orthochronous subgroup of the Poincaré group. Note that:

• Lo gives the antichronous subset for:

• Ln = Lst

• Ls = Lt

**....**Here, we have limited the Lorentz group to its neutral component Lo, which will be explained later. The subsequent action of this extended group on its momentum space becomes:

**....**The first lines only show the conservation of quantum numbers, the electric charge being one of them.

Geometric definition of Dirac's antimatter.

**....**Introduce the following vector f and matrix l:

**....**Now, introduce the new group:

**....**It is a two-component group. Clearly, from above, the l = -1 component reverses the quantum charges ci. Note that it also reverses the zi dimensions. We suggest that this is the general geometric definition of antimatter, a (z-symmetry): inversion of the extra-dimensions zi.

Geometric definition of Feynman antimatter.

Now write the group:

**....**It becomes a four-component group. (m = 1) elements achieve PT-symmetry. The corresponding action on the momentum space becomes:

**....**Take (l = +1) and (m = -1). We get a PT-symmetry. Quantum charges remain unchanged, but extra-dimensions are reversed. According to our geometric definition of antimatter, this corresponds to Feynman's antimatter.

Group acting on a two-points bundle space.

..Introduce a bundle index b and write the action of a new group:

..The action on the momentum space is identical. A dynamical group governs the movements of mass points. Given a movement, an element of the group may define another one, and we have seen that antimatter is nothing but a different movement of the particle, along the reversed additional dimensions zi. The Poincaré group raises a physical problem by introducing antichronous movements, corresponding to T-symmetry. Similarly, the so-called Feynman's antimatter raises the same problem, for the considered movement was also T-symmetric. Here, the problem is solved, for antichronous movements take place in the twin space, in the b = -1 sheet of the bundle.

m = 1 causes a T-symmetry and what we will call a B-symmetry (bundle symmetry).

..Now, positive energy and negative energy particles cannot meet and completely annihilate, as they live in distinct twin sheets.

Geometric interpretation of the CPT theorem.

..In the above group, choose:

l = -1 ; m = -1

..We achieve a CPT-symmetry:

• spacetime is reversed

• quantum numbers ci are reversed

but the additional dimensions zi remain unchanged, so that this corresponds to a particle of matter. The CPT-symmetric of a particle of matter is a particle of matter, except it has negative mass and energy and lives in the twin sheet.

The CPT-symmetric of matter in the matter of the twin sheet, whose contribution to the gravitational field is negative.

..Similarly, if we choose:

l = +1 ; m = -1

we get the PT-symmetric of the particle. If we take a particle of matter, its PT-symmetric is antimatter, for we have a z-symmetry. It lives in the twin sheet, due to the subsequent B-symmetry.

Matter-antimatter duality holds in the twin universe.

..All particles of the twin universe have apparent negative energy (including photons, neutrinos, etc.). All massive particles have an apparent negative mass. Quod erat demonstrandum.

References :

[1] A.Sakharov : "CP violation and baryonic asymmetry of the Universe". ZhETF Pis'ma 5 : 32-35 (1967) : Translation JETP Lett. 5 : 24-27 (1967) [2] A.Sakharov : "A multisheet cosmological model" Preprint Institute of Applied Mathematics, Moscow 1970 [3] A.Sakharov : "Cosmological model of the Universe with a time-vector inversion". ZhETF 79 : 689-693 (1980) : Translation in Sov. Phys. JETP 52 : 349-351 (1980) [4] A.Sakharov : "Topological structure of elementary particles and CPT asymmetry" in "Problems in theoretical physics", dedicated to the memory of I.E.Tamm, Nauka, Moscow 1972 pp. 243-247 [5] J.P.Petit : "Enantiomorphic universes with opposite time arrows", CRAS of May 8, 1977, t.285 pp. 1217-1221 [6] J.P.Petit : "Universes in interaction with their image in the mirror of time". CRAS of June 6, 1977, t. 284, series A, pp. 1413-1416 [7] J.P.Petit : The missing mass effect. Il Nuovo Cimento, B, vol. 109, July 1994, pp. 697-710 [8] J.P.Petit, Twin Universe Cosmology. Astrophysics and Space Science. Astr. And Sp. Sc. 226 : 273-307, 1995 [9] J.P.Petit, "We have lost half of the universe", Ed. Albin Michel, France, 1997. [10] - J.P.Petit : An interpretation of cosmological model with variable light velocity. Modern Physics Letters A, Vol. 3, n°16, nov 1988, p.1527 [11] - J.P.Petit : Cosmological model with variable light velocity: the interpretation of red shifts. Modern Physics Letters A, Vol.3 , n° 18, dec. 1988, p.1733 [12] - J.P.Petit & Maurice Viton : Gauge cosmological model with variable light velocity. Comparison with QSO observational data. Modern Physics Letters A Vol.4 , n°23 (1989) pp. 2201-2210 [13] - R.Adler, M.Bazin and M.Schiffer : Introduction to general relativity, Mc Graw Hill Book Cie. 1975, chapter 14, "TOV equation". [14] - Oppenheimer J.R. and H. Snyder (1939) : On continued gravitational contraction, Phys. Rev. 55 : 455 [15] J.M.Souriau : Structure des Systèmes Dynamiques, Dunod-France Ed. 1972 and Birkhauser Ed. 1997. [16] Interview of Fort, Ciel et Espace Jr. June 2000.