# How to break the continuous particle exchange symmetry

keywords: e+e+ moller scattering, supersymmetry, scalar supercharge, high density positron plasma, dipositronium spin states, positron statistics, unstable antiworld.

The idea to exchange fields by rotations in 4N dimensional space is IMHO brand new,
and it is realized in the linked papers. Overall the project has taken nearly 15 years since 1999, when I 1st time heard prof. Berry talk about continuous particle exchange by the mathematical analog of the belt trick. Actually I have tons of related calculations; most of them are related to the embedding of the tensor product of SO(4) and O(N) to SO(4N).

After all, the theory arrives at the super-symmetric electro-dynamics with charge conjugation taking bosons to fermions, positive charge to negative charge, changing the sign of the parity and the spin. In other words, positron and electron have same mass and spin; positron and electron have opposite charge, parity and statistics.

I’m not that ambitious to state that this theory more than just mathematical exercise related to the foundations of the Pauli principle ( spin-statistics theorem ). Basically I say that one can go around the Pauli principle if the statistics can be changed by the charge conjugation.

The experimental data on two positron systems is very limited. As of September 4 2014 there is no experimental data on positron-positron scattering; no experimental data on positron-positronium scattering; no experimental data on positronium-positronium scattering; no experimental data on Positronium molecule (Ps2) ground state magnetization.

There is now lot of efforts to make positronium laser; it requires high density cold positronium plasma; possibly the condensed state of the positronium or positronium molecules. Joint Quantum Institute. “Stimulated mutual annihilation: How to make a gamma-ray laser with positronium.” ScienceDaily. ScienceDaily, 1 May 2014.

Medical application use positron emission tomography and annihilation radiation tomography. The opposite Parity of electron and positron put a constrain on radiation polarization. However the annihilation events are not coherent. All symmetries of the positronium are listed in the text book. Fresh reference.

The positronium Hyper-fine structure is analysed many years, now at \alpha^7 order by Adkins&Fell . The Rich’s review gives all the data 1930-1980. The L=1 state of the positronium molecule was predicted by K. Varga, J. Usukura, and Y. Suzuki PRL(1998), and L=1 to L=0 transition (251nm) was measured recently by D. B. Cassidy et all PRL(2012).

=========================================
I hope that the Positronium molecule (Ps2) life-time can bring some evidences in favor of the theory. Here is my 2nd paper on the dipositronium magnetic moment. The linear Zeeman effect is predicted for some of $S=1, S=2$ states; if the above theory is correct, then $S=1$ becomes the ground state; it can be unambiguously identified by the linear Zeeman split.

The N=1, L=0, S=0 state of the positronium is the short living (125ps); the N=1, L=0, S=1 state of the positronium is the long living (142ns). The posironium molecule (Ps2) lifetime is in wide range 0 – 150ns with both L=0 and L=1 at S=0, see D. B. Cassidy & A. P. Mills publications in Nature(2007) and PRL(2012). This is very much puzzling; basically they did not consider any direct annihilation inside the Ps2 moleciule. They assume Ps2 molecule made from two positronium “atoms” with S=1 M=1 and S=1 M=-1. Therefore the Ps2 molecule lifetime is determined by dissociation Ps2 → o-Ps + o-Ps, then flip o-Ps → p-Ps, then annihilation p-Ps→2γ

I propose that the positronium molecule is made from ortho- and para-positronium “atoms”; but annihilation is limited once they are coupled. So the process become Ps2 → o-Ps + p-Ps, then annihilation p-Ps→2γ; in this model the Ps2 lifetime is just the dissociation time. I hope to continue the parity analysis for all processes as my next project.

## 28 thoughts on “How to break the continuous particle exchange symmetry”

1. For The Symmetry of Positronium (Ps2) Molecule see following papers. The numeric calculations of the Ps2 binding energies are popular with ~50 authors in the list.

http://journals.aps.org/pra/abstract/10.1103/PhysRevA.58.1918
J. Usukura, K. Varga, and Y. Suzuki, Signature of the existence of the positronium molecule, Phys. Rev. A 58, 1918 ( 1998 )

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.92.043401
D. M. Schrader, Symmetry of Dipositronium Ps2, Phys. Rev. Lett. 92, 043401 (2004)

http://iopscience.iop.org/1751-8121/40/39/014/
Alexei M Frolov and Farrukh A Chishtie, J. Phys. A: Math. Theor. 40 11923 (2007), Annihilation of the electron–positron pairs in polyelectrons

2. http://www.sciencedirect.com/science/article/pii/S0168583X04003945 , Positrons in the Universe, Jacques Paula, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, Volume 221, July 2004, Pages 215–224, [The systematic paper on cosmic annihilation radiation. ]
http://www.quantumdiaries.org/2013/07/08/cosmology-and-dark-matter/ , [The polarized 511keV radiation coming from universe raise the issue of the positrons produced from the dark matter. Once I claim that the positrons are bosons, they can be in condensed state in the depth of cosmos, then the 511keV cosmic rays are evidences of the interaction of the dark matter with our world.]

3. The scalar super-charge pass conditions of the Coleman-Mandula theorem (1967), but obviously violates the Pauli principle; it predicts multiplet states of same spin and mass, but opposite charge, parity and statistics. This series of papers explains how to bypass the Pauli principle. It turns out that one flavor of anticommuting fields is enough to get positively defined energy and correct expressions for the current and charge.

Nice explanation on the Coleman-Mandula theorem is given by
M.~F.~Sohnius, {\it Introducing Supersymmetry}, Phys. Rep., {\bf 128}, 39 (1985)
http://www-fp.usc.es/~edels/SUSY/Sohnius_Introducing%20SUSY.pdf

4. I’ve just read a couple of review papers on progress in antimatter, none mentioned molecular antihydrogen; The difficulty is in relatively low density of all three species: positrons, antiprotons and antihydrogen. Still; if the antihydrogen is formed, then the moleculas should be formed right away;
the binding energy of the molecule is 1/3 of the binding energy of the atom.

links on hydrogen-antihydrogen; so far I don’t understand the point; are these complexes too stable?
http://journals.aps.org/pra/cited-by/10.1103/PhysRevA.73.052503
http://arxiv.org/abs/0803.2445
http://arxiv.org/abs/physics/0506160

5. The formation of the molecular antihydrogen is problematic (or even not possible) if positrons commute. unfortunately experimental data on low-density plasma is not available yet. Even formation of antihydrogen from positrons and anti-protons is not easy and requires substantial time.

The dissociation of molecular hydrogen to atomic hydrogen is complicated plasma problem; it requires lot of energy (binding energy for the molecule is ~ 5eV ). However formation of molecules is not easy too because it requires relatively high density of atomic hydrogen.

(see particularly chapter VI)
http://www.nrgnair.com/MPT/02AIR/atm.h2a.htm
practically there are few papers in Phys. Rev. with dissociation by ~20-60eV UV light.

6. The SO(6) can be used to rotate (exchange) two SO(3) spinors; SO(6) has 4-dimensional [0,0,1] representation which branches to [1]x[1] of SO(3) x SO(3) and it is the best to study spinor interchange rotations. Maximum one can use 8-dimensional [0,1,0]+[0,0,1] in order to use the covering algebra of gamma-matrices.

Berry in his paper wrote that 10 dimensions are required for exchange of two SO(3) spinors; there are 10-dimensional reps [0,0,2] and [0,2,0] in SO(6), but they branch to [0][0]+[2][2] of SO(3) and these are spin S=1 states. I don’t think that this was Berry intention.

D3 = WeylCharacterRing(“D3″, style=”coroots”)
D3(1,0,0).degree()
D3(0,1,0).degree()
D3(0,0,1).degree()
D3(2,0,0).degree()
D3(0,2,0).degree()
D3(0,0,2).degree()
D3(0,1,1).degree()
︡6 4 4 20 10 10 15

D3 = WeylCharacterRing(“D3″, style=”coroots”)
B1xB1 = WeylCharacterRing(“B1xB1″, style=”coroots”)
b=branching_rule(“D3″,”B1xB1″,rule=”orthogonal_sum”)
b

b.branch(D3(1,0,0))
b.branch(D3(0,1,0))
b.branch(D3(0,0,1))
b.branch(D3(2,0,0))
b.branch(D3(0,2,0))
b.branch(D3(0,0,2))
b.branch(D3(0,1,1))

B1xB1(2,0) + B1xB1(0,2)
B1xB1(1,1)
B1xB1(1,1)
B1xB1(0,0) + B1xB1(2,2) + B1xB1(4,0) + B1xB1(0,4)
B1xB1(0,0) + B1xB1(2,2)
B1xB1(0,0) + B1xB1(2,2)
B1xB1(2,0) + B1xB1(2,2) + B1xB1(0,2)

7. There is bunch of papers on the double beta decay; it can be neutrinoless if the neutrino field is real (Majorano); in this case neutrino and antineutrino are the same thing and and can be virtual particle in the double beta decay. Klapdor Kleingrothaus claimed that he has seen the neutrinoless double-beta decay; A. S. Barabash in his “Double Beta Decay: Historical Review of 75 Years of Research” say that “The Moscow part of the Collaboration disagreed with this statement [69].”

My theory predicts opposite statistics for left and right fields, therefore for neutrino and antineutrino.
The detection of neutrino-less double beta decay would imply that neutrino and antineutrino is the same thing and fail of present theory. Actually, I need to check if the scalar supercharge can be introduced into the Majorano Lagrangian. So far things are not bad, because SO(10) grandunification does have separate representation for the antineutrino.

8. The fraction of positrons in primary cosmic beams is now big deal and it was measured with high accuracy by Alpha Magnetic Spectrometer on the International Space Station. The data is discussed widely on the internet. There are bunch of models for excess of positrons at both low and high energies.

Let me suggest that bosonic statistics of positrons would explain the excess of positrons at low energies.
The idea is that the particles are in thermal equilibrium somewhere far in space; then the distribution
seen at the earth orbit is average of fermi/bose statistics over perpendicular to the beam momenta:

N_e µ ò dp_x dp_y 1 / eE_p/T +1 = \ln (1+e-E_p/T)

N_e+ µ ò dp_x dp_y 1 / eE_p/T -1 = \ln 1/(1-e-E_p/T)

N_e+ / N_e µ \ln 1/(1-e-E_p/T) / \ln (1+e-E_p/T)

last equation again:



fit to the experiment at low energies (1-8 GeV) with T=1.5GeV

The original Data from linked paper:

9. Positron thermalization is popular topic; quite a few people working on it since ~1955; another keyword is the Positron Annihilation Lifetime Spectroscopy.
http://positrons.physics.lsa.umich.edu/current/home.html
http://positron.physik.uni-halle.de/talks/
http://journals.aps.org/pr/abstract/10.1103/PhysRev.97.1557

I’m not sure if the diffusion of positron in metal can relax the positron energy; people say that it moves and get trapped, probably emits the phonon.

10. Resubmitted all three papers to PRA, take me some time,
still have some doubts about definition of the products like
(b^\dagger Q f) (f^\dagger Q b) vs (f^\dagger Q b)(b^\dagger Q f)
do they commute or anticommute? should commute, because they are current

11. Astrophysics Papers on positronium:
http://journals.aps.org/pra/references/10.1103/PhysRevA.89.062704
N. Guessoum, P. Jean, and W. Gillard, Astronomy & Astrophysics 436, 171 (2005).
N. Prantzos et al., Rev. Mod. Phys. 83, 1001 (2011).
P. A. Milne, New Astron. Rev. 50, 548 (2006).
B. L. Brown and M. Leventhal, Astrophys. J. 319, 637 (1987).
S. C. Ellis and J. Bland-Hawthorn, Astrophys. J. 707, 457 (2009).
C. J. Crannell, G. Joyce, R. Ramatay, and C. Werntz, Astrophys. J. 210, 582 (1976).

12. Jut checked the chiral anomaly diagram for the supersymmetric QED; Need to repeat the calculation thoroughly; for the 1st glance the sign of the right loop get changed. In regular QED signs of the derivatives of the left and right chiral currents are opposite; in this theory signs are the same. I don’t know if there is any great physical consequence out of it.

13. The order parameter for SO(4N) to SO(4) spontaneous symmetry lowering in my previous paper is the symmetric tensor $h_{\mu\nu}$. Once one of its diagonal elements acquire the observation value the SO(4N) get lowered to SO(4N-1).

Just got two insights.

1st. Both metrics tensor $g_{\mu\nu}$ and graviton field $h_{\mu\nu}$ are good candidates to break the 4N-Lorentz symmetry and low the dimension. Just need to find solutions to the free Einstein equation with $g$ variation from subspace to subspace.

2nd. Above idea is wrong. The metrics of the space with the weak gravity is $g_00 = 1 + (2/c^2)U(r)$, $g_11=g_22=g_33=-1$, but the Lorentz invariance isn’t broken. In other words we can always rotate the space even if it is curved.

3rd. I was surprised to find papers with gravitons breaking the Lorentz invariance (eg http://arxiv.org/abs/0905.0955v1)

14. Meet Yakov Itin two weeks ago. He brought my attention to few his papers on the Lorenz invariance violations and related literature. I’ve decided to write the reference review.

15. More fresh papers on laser produced positrons:
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.114.215001
Hui Chen, F. Fiuza, A. Link, A. Hazi, M. Hill, D. Hoarty, S. James, S. Kerr, D. D. Meyerhofer, J. Myatt, J. Park, Y. Sentoku, and G. J. Williams Phys. Rev. Lett. 114, 215001

http://journals.aps.org/pra/abstract/10.1103/PhysRevA.89.012708
Estimating positronium formation for plasma applications T. C. Naginey, B. B. Pollock, Eric W. Stacy, H. R. J. Walters, and Colm T. Whelan Phys. Rev. A 89, 012708

16. Supersymmetry breaking at finite temperature: The Goldstone fermion
Daniel Boyanovsky
Phys. Rev. D 29, 743 – Published 15 February 1984
http://journals.aps.org/prd/abstract/10.1103/PhysRevD.29.743

The supersymmetry Lagrangian cannot be written at final temperature; he cited few old papers discussing the the poles of the green function on complex \omega plane. there is no way to impose the boundary conditions on mixed fermion-boson field.

other way around is to introduce fermi/bose particle number operators into some `effective’ thermal potential. not sure if this is valid procedure. any way the ward identities give the broken super-symmetry.

17. The positron / electron ratio just mean that the chemical potential is on electron side.
Take electrons as fermionic antiparticles, positron as fermionic or bosonic particles we get
n_p/n_e = { e^{(E-|\mu|)/T} +1 \over e^{(E+|\mu|)/T} \pm 1 }
the ratio of 0.05 = e^(-2|\mu|/T) => \mu/T = 1.5, take \mu = .5 .. 1GeV
the density of relativistic particles at this energy is n_e ~ (1/6\pi^2) (\mu/\hbar c )^3
\hbar c = 1e-6 eV cm; \mu / hbar c = 1e15 cm^-1; n_e ~ 1e45 cm^-3
pretty dense plasma somewhere in the universe

18. The outline of the talk about antimatter asymmetry and nonexistance of periodic table for antimatter
1. Intro to the antimatter (\bar p, \bar n, \bar d, \bar \alpha, e+), synthesis
2. Energy to matter conversion (photon+photon -> p+\bar p or n+\bar n), the barion number (#p+#n – #\bar p – #\bar n) is zero at birth of the universe (matter=antimatter).
3. Evidence for lack of antimatter today, none on earth, none in the galaxy or in cosmic rays(less than 1/10,000), none in interstellar space (Faraday rotation observed), no much annihilation products seen, no evidence for anti-stars etc; Today barion number is 10^100; Puzzle.
4. Neutron oscillation, Andrey D Sakharov 1967 paper (2391 citations)
5. The new idea: Antimatter is bosonic => unstable => all antimatter collapsed to antineutron stars
6. Stability of matter as I need it (S. Chandrasekhar 1931)
a. For the ideal gas (and bosons) P=nkT
b. For ideal fermions P=k’ n^{5/3}
c. For general case P=n^x
d. Gravitation (or any 1/r potential) \int (n/r) r^2 dr = k n^x
e. r dr = k n^{x-2} dn
f. for Fermions: R = k n^{2/3} (here n can be taken in the center of the star or average)
g. for Bosons: R = ln n => means collapse
7. Same is true for atoms (Thomas-Fermi); multi-electron atoms will be unstable.
8. Extensions of the periodic table to antimatter sector will not be possible
9. The paper shows how to bypass the Pauli theorem assuming anticommutation between CPT transformation and exchange.
a. result: positive “frequency” particles are bosonic; negative “frequency” particles are fermionic; need to switch definition.
b. CPT converts matter to antimatter; it is allowed to convert fermions to bosons?
c. supersymmetric QED is possible; energy, charge and green function are CPT invariant.
d. The Wick’s theorem is not CPT invariant.