Probing the Limits of Nuclear Existence

Yorick BLUMENFELD CNRS/IN2P3, IJCLab, University of Paris-Saclay, Orsay, France

Researchers have discovered the heaviest bound isotope of sodium and characterized other neutron-rich nuclides, offering important benchmarks for refining nuclear models.

The neutron dripline marks one of the boundaries of nuclear existence-indicating isotopes of a given element with a maximum number of neutrons. Adding a neutron to a dripline isotope will cause the isotope to become unbound and release some of its neutrons. Mapping the dripline is a major goal of modern-day nuclear physics, as it is a testing ground for nuclear models and has implications for our understanding of neutron-stars and of the synthesis of elements in stellar explosions. Now, studies by two groups extend our knowledge of the properties of nuclei close to the dripline [1]. Working at the Radioactive Isotope Beam Factory (RIBF) in Japan, Deuk Soon Ahn of RIKEN and colleagues have discovered sodium-39 ( 39 Na), which likely marks the dripline location for the heaviest element to date (Fig. 1). Meanwhile, In the first experiments performed at the recently inaugurated Facility for Rare Isotope Beams (FRIB) at Michigan State University, Heather Crawford of Lawrence Berkeley National Laboratory and coworkers studied neutron-rich isotopes of other elements close to sodium, delivering measurements of previously unknown isotope lifetimes [START_REF] Crawford | Crossing N=28 Toward the Neutron Drip-line: First Measurement of Halflives at FRIB[END_REF].

The atomic nucleus is composed of neutrons and protons whose cohesion is insured by one of the four fundamental forces: the strong, or nuclear, force. The number of protons (Z) in the nucleus defines the chemical element. Nuclei of a given element with different numbers of neutrons (N) are called isotopes. Nuclei and their isotopes can be represented in the two-dimensional Segré chart (fig. 1), which shows all bound isotopes, including stable ones-those making up most of the world around us-and radioactive isotopes-which decay towards the stable ones. For a given element, neutrons can be added up to the point where the nuclear force is no longer strong enough to bind the nucleus together, causing neutrons to "drip out". This limit on the neutron-rich side of the chart is the neutron dripline.

After more than a century of nuclear research, elements have been discovered all the way to Z=118 (Oganesson). It is thus remarkable, that the neutron dripline is only known up to neon (Z=10), where it is located at N=24. Two challenges complicate the extension of the dripline: the extremely small probability with which the most neutron-rich nuclei can be produced in the lab and-once a putative dripline isotope has been observed-the difficulty of ruling out that even more neutron-rich isotopes of the same element exist. To produce these elusive nuclei, researchers use accelerator facilities where intense, high-energy beams of stable but already neutron-rich nuclei are smashed onto solid targets, causing nuclear reactions with target nuclei that produce a slew of different nuclides.

To produce 39 Na, Ahn and colleagues start with a 16-GeV beam of 48 Ca nuclei, which impinge at a rate of 3 10 12 nuclei per second on a target of Be atoms. The nuclides emerging from the collisions then traverse a series of magnetic spectrometers tuned to maximize the transmission of 39 Na nuclei and minimize that of contaminants. At the output of the spectrometers, nuclear detectors determine the Z and N of each nucleus. The team reports nine detection events of 39 Na collected over a two-day experiment in which approximately 5 10 17 48 Ca nuclei impinged on the Be target-a detection more challenging than finding the proverbial needle in a haystack! The existence of 39 Na provides information about nuclear stability and structure, offering an important benchmark for nuclear models. A remarkable feature of the dripline, which is located at N=16 for oxygen (Z=8), is that it extends to N=22 for fluorine (Z=9), to 24 for neon (Z=10), and, as revealed by the new data, to at least 28 for sodium. This feature implies that as Z is increased, an enhanced binding energy allows nuclei to incorporate more neutrons while remaining bound. Previous studies suggest that such additional binding ability is likely the consequence of deformation of nuclear shapes: 24 O, for instance, is presumed to be spherical [START_REF] Hoffman | Evidence for a doubly magic 24 O[END_REF], while neutron-rich nuclei with Z ≥ 9 are thought to exhibit a prolate shape, resembling a rugby ball. This behavior can be explained in the framework of the nuclear shell model, first elaborated by Nobel laureates Maria Goeppert-Mayer [START_REF] Goeppert-Mayer | On Closed Shells in Nuclei. II[END_REF] and Hans Jensen [START_REF] Haxel | On the "Magic Numbers" in Nuclear Structure[END_REF] and perfected over last 70 years. In nuclear-shell-based models, protons and neutrons occupy different shells, and deformation is induced when the energy-ordering of these shells changes, which can happen when neutrons are added. The observation of 39 Na offers an important validation for these models. In particular, a sophisticated shell model proposed in 2020 [Tsu20] not only correctly predicts the position of the dripline for oxygen, fluorine and neon driplines but also the existence of 39 Na. Other cruder models, however, fail to explain the new finding. In particular, the finite range droplet model [START_REF] Möller | New Finite-Range Droplet Mass Model and Equation-of-State Parameters[END_REF], which treats the nucleus as a liquid drop, and the socalled Hartree-Fock Bogoliubov mass model [START_REF] Goriely | Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XIII. The 2012 atomic mass evaluation and the symmetry coefficient[END_REF] both predict 39 Na to be unbound. Note that the discovery of 39 Na does not not rigorously prove experimentally that 39 Na sits at the dripline, since the experiments do not rule out the possibility that neutron-richer Na isotopes exist. However, the shell model that correctly predicts 39 Na also indicates that it is the most neutron-rich bound isotope of Na [START_REF] Tsunoda | The impact of nuclear shape on the emergence of the neutron dripline[END_REF].

The discovery of an isotope is but a first step for the study of nuclear structure. Knowledge of further properties, including the isotope's half-life, mass, spin, and the features of its excited states, is necessary to better constrain nuclear models and evaluate their predictive ability. Kicked off in May of this year (see Research News: Rare Isotopes for the Choosing), FRIB-the world's leading accelerator in terms of exotic-isotope production capabilities-will allow researchers to produce and study thousands of nuclei far from stability, some of which will have been created for the first time on Earth.

Crawford and her colleagues report the first results from FRIB. Their setup also involves a 48 Ca beam, but is tuned to produce heavier neutron-rich isotopes of elements, with Z ranging between 12-15 (Ma, AL, Si and P), and to measure their lifetimes. The team obtains five hitherto unknown lifetimes for those isotopes. Most of the obtained values agree well with theorical predictions-offering another demonstration of the predictive power of modern shell-model calculations [START_REF] Yoshida | Systematic shell-model study of β-decay properties and Gamow-Teller strength distributions in A≈40 neutron-rich nuclei[END_REF]. However, the researchers observe an an unexpected drop in half-life for 38 Mg compared to 37 Mg and to lighter Mg isoptopes, a small discrepancy that calls for further refinements of shell model calculations.

FRIB's experiments could not yet probe sufficiently neutron-rich nuclei to reach the dripline for elements in the Z=12-15 region. The results, however, come from FRIB's first exploratory run, in which the facility was operated at less than 1% of its target ion-beam intensity. A progressive intensity ramp-up planned for the coming months will allow the facility to access a wider set of isotopes, potentially leading to further extensions of the dripline.

With the edge of nuclear existence now known up to Z=11 (assuming that the 39 Na is indeed located at the dripline), a "terra incognita" of neutron-rich nuclei with Z ≥ 12 remains to be explored. Some of them will be created by FRIB and other new accelerators coming online, such as the Facility for Antiproton and Ion Research in Germany. Others may forever remain out of the reach of our terrestrial laboratories. Their existence and properties will have to be inferred through nuclear models, whose predictive power will be boosted by the ever more stringent tests offered by experiments such as those discussed here.


Guy Barrand

Symmetric Lorentz-Minkowski, Antisymmetric Dirac-Majorana

We show a nice symmetric/antisymmetric relation between the four vector Lorentz transformation and the Dirac spinor one in the Majorana representation. From the spinor one, we exhibit the antisymmetric pending of the symmetric Minkowski metric. We then rewrite the Dirac equation in various ways exploiting group properties induced by these relations, and this without complex numbers. We show also a nice relation with a five dimensional metric. When done, we will see that the traditional complex electromagnetic coupling could be handled also without complex numbers by just considering two coupled real fields instead of one complex field. Finally, we will show that going toward six or ten dimensional spacetime would be more natural from a group point of view.

1 The coordinate Lorentz transformation, the symmetric Minkowski matrix

A coordinate Lorentz transformation on a tuple x µ=0,1,2,3 of four real numbers can be written:

L c (A|x) µ def = (e Aη ) µ ν x ν (1)
with the Minkowski η 4x4 real symmetric matrix being:

η def =     1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1    
and A being a 4x4 real antisymmetric matrix. Concerning the exponential, someone must note the pattern: antisymmetric matrix A, parameterising a Lorentz transformation, multiplied by the constant η 4x4 real symmetric matrix. A 4x4 real antisymmetric matrix having six free parameters, we recover the number of parameters (three rotations plus three boosts) of a Lorentz transformation.

Scalar, vector Lorentz transformations

On a field φ(x) (no index), a tuple V µ=0,1,2,3 (x) of fields, they are defined with:

L s (A|φ)(L c (A|x)) def = φ(x) L • (A|V ) µ (L c (A|x)) def = (e Aη ) µ ν V ν (x) (2)
We can also introduce the "down" vector transformation on a V µ=0,1,2,3 (x):

L • (A|V ) µ (L c (A|x)) def = (e -Aη ) ν µ V ν (x)
3 η as a metric

If the length of a V µ (x) is defined, in matrix notation, with:

l[η](V )(x) def = t V (x)ηV (x) def = ( t V (x)) µ η µ ν V ν (x)
( t for the transposition operation), it can be shown easily that it is a Lorentz invariant:

l[η](L • (A|V ))(L c (A|x)) dem = l[η](V )(x)
due to the fact that:

t (e Aη )ηe Aη dem = η

It is a general property that if A is an antisymmetric square matrix and S a symmetric square matrix of same dimension, we have:

t (e AS )Se AS dem = e -SA e SA S dem = S

t (e SA )Ae SA dem = e -AS e AS A dem = A

This second property will have some importance in the following. If S and A are invertible, we have also:

e AS S -1 t (e AS ) dem = e AS S -1 e -SA dem = e AS e -AS S -1 dem = S -1

(3)

e SA A -1 t (e SA ) dem = e SA A -1 e -AS dem = e SA e -SA A -1 dem = A -1

Dirac spinor Lorentz transformation

On a tuple ψ α=0,1,2,3 (x) of fields, we define the "spinor" Lorentz transformation with:

L • (A|ψ) α (L c (A|x)) def = (e Γ[A] ) α β ψ β (x) (4)
with Γ[A] being:

Γ[A] def = 1 8 [γ µ , γ ν ](ηAη) µ ν
with the γ µ=0,1,2,3 being four 4x4 complex matrices verifying:

{γ µ , γ ν } = 2(η -1 ) µ ν I

(5)

with I being the four dimensional identity matrix.

Presented in this form, the e Γ [A] appearing in formula (4) is less appealing than the e Aη appearing in (2); it looks a little bit exotic and much more complicated, but we are going to show that in fact it can be presented with the same similar simpler structure than in e Aη . About the definition and properties of Γ[A], have a look to Appendix B with S = η. In particular we have the two important properties:

e Γ[A] γ µ e -Γ[A] dem = (e -Aη ) µ ν γ ν (6) e Γ[A] γ µ e -Γ[A] dem = (e Aη ) ν µ γ ν (7)
with the down γ µ being:

γ µ def = η ν µ γ ν
Since the e Γ[A] matrix looks complex, up so far, the ψ α (x) have to be complex numbers too.

The Dirac equation

Over a ψ α (x) tuple, it reads:

i(γ µ ) α β ∂ µ ψ β (x) = mc ψ α (x)
or in a more compact matrix notation:

iγ µ ∂ µ ψ(x) = mc ψ(x) (8)
By using (1), (4) and exploiting (6) we can show quite easily that it is a Lorentz invariant.

Terminology, notation

In this article, the words "coordinate", "scalar", "vector", "tensor", "spinor", etc are used as a qualifier for a transformation and not to define a tuple/matrix of numbers or functions/fields as for x µ , φ(x), V µ (x), ψ µ (x) or later g µν (x).

On an advanced theory, the same tuple/matrix of numbers/functions may be subjected to various transformations, and then using such qualifiers on them can lead to confusion.

(For example, we can see in Appendix D that the tuple of functions ψ α (x) is subjected to a scalar transformation for R but to a spinor one for the L I transformation).

It is only in a context where only one transformation is around that we can displace such qualifier toward a tuple/matrix, and then speak for example of the A µ (x) "vector" electromagnetic potential or the ψ(x) "spinor". (In this text, we stick to the tradition of noting a tuple subjected to a vector Lorentz transformation L • with an uppercase latin letter, as V (x), and of using the lowercase greek letter ψ(x) for a tuple submitted to a spinor Lorentz transformation L • ).

For the transformations, we use a notation that may look heavy, but which is in general complete in the sense that it carries all the needed informations:

T qualif ier (parameters|whatever) indices (arguments)

with "qualifier" ' that could be upward or downward, "parameters" being a set of tuples/matrices of numbers/functions and "whatever" being a tuple/matrix of numbers or functions submitted to the transformation. The result of a transformation being also a tuple/matrix of numbers/functions, it has in general upward/downward indices and ending brackets with arguments.

For functions/fields depending of parameters, we use:

name[parameters](arguments)

We use also def = in case an equality is a definition (left side is defined by the right side), and dem = when an equality comes from a demonstration (left side is demonstrated to be the right side). We have also cas = in case an equality is demonstrated by using a computer algebra system (CAS).

We definitely avoid the practice of setting = c = 1 which complicates the reading of the dimensionality of quantities.

Let us go now to the core of this article.

7 The A a basis, Γ a matrices and θ a parameterisation Let us define the antisymmetric A a=1,2,3,4,5,6 six real matrices with:

A 1 def =     0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0     A 2 def =     0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0     A 3 def =     0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0     A 4 def =     0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0     A 5 def =     0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0     A 6 def =     0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0    
These matrices form a basis for any 4x4 real antisymmetric matrix (see Appendix B for some of their properties).

Instead of parameterising a Lorentz transformation with a A antisymmetric matrix, we can do it by using six real numbers θ a so that:

A = θ a A a ⇔ θ a dem = - 1 2 T r(A a A)
With this choice for the A a matrices, the tuple (θ 1,2,3 , 0, 0, 0) parameterises a spatial rotation, whilst (0,0,0,θ 4,5,6 ) parameterises a boost.

By using also the Γ a matrices defined with:

Γ a def = 1 8 [γ µ , γ ν ](A a ) µ ν
(see also Appendix B with S = η), we can write now:

L c (θ|x) µ def = (e θ a A a η ) µ ν x ν L • (θ|V ) µ (L c (θ|x)) def = (e θ a A a η ) µ ν V ν (x) L • (θ|ψ) α (L c (θ|x)) def = (e θ a Γ a ) α β ψ β (x)
8 Hidden Dirac-Majorana antisymmetric ξ matrix

In the Majorana representation, the γ µ=0,1,2,3 matrices are (see Appendix A):

γ 0 def = i     0 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 0     γ 1 def = i     1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1     γ 2 def = i     0 0 0 1 0 0 -1 0 0 -1 0 0 1 0 0 0     γ 3 def = i     0 -1 0 0 -1 0 0 0 0 0 0 -1 0 0 -1 0    
If now defining the real antisymmetric matrix:

ξ def =     0 0 0 1 0 0 -1 0 0 1 0 0 -1 0 0 0     dem = iγ 0
we can show the remarkable fact that:

Γ a dem = S a ξ with: S 1 def = 1 2     0 -1 0 0 -1 0 0 0 0 0 0 1 0 0 1 0     S 2 def = 1 2     0 0 -1 0 0 0 0 -1 -1 0 0 0 0 -1 0 0     S 3 def = 1 2     -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1     S 4 def = 1 2     1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1     S 5 def = 1 2     0 0 0 1 0 0 -1 0 0 -1 0 0 1 0 0 0     S 6 def = 1 2     0 -1 0 0 -1 0 0 0 0 0 0 -1 0 0 -1 0    
being all real symmetric matrices!

We have the remarkable fact that the coordinate, vector and spinor Lorentz transformations with the θ a as parameters, now look like:

L c (θ|x) µ def = (e θ a A a η ) µ ν x ν L • (θ|V ) µ (L c (θ|x)) def = (e θ a A a η ) µ ν V ν (x) (9) L • (θ|ψ) α (L c (θ|x)) def = (e θ a S a ξ ) α β ψ β (x) (10)
We note the remarkable and nice interchange of symmetric/antisymmetric real matrices between ( 9) and (10) with:

(A a , η) ⇔ (S a , ξ) (antisyms, sym) ⇔ (syms, antisym)

To spot this interchange, we can write the vector/spinor group commutators together:

[A a η, A b η] dem = l ab c A c η (11) [S a ξ, S b ξ] dem = l ab c S c ξ
with the same l ab c real constants (see Appendices B and C).

We note also that since e θ a Saξ is real, the ψ α (x) tuple does not have to be complex and can stay a priori real.

Four symmetric matrices are lacking

The set S a , despite of being linearly independent, is not a complete set to form a basis of the set of symmetric 4x4 real matrices S (4, R). We should have ten matrices instead of six; four are lacking! Four? but we have four γ µ , and if . . . yes! It can be shown that:

γ µ dem = -i Sµ ξ (12) with: S0 def = I def =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     S1 def =     0 0 0 -1 0 0 -1 0 0 -1 0 0 -1 0 0 0     S2 def =     1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1     S3 def =     0 0 -1 0 0 0 0 1 -1 0 0 0 0 1 0 0     being all symmetric too!
We note that (7) can be written:

e θ a S a ξ ( Sµ ξ) e -θ b S b ξ dem = (e θ c A c η ) ν µ ( Sν ξ) (13)
We note also that we have:

{ Sµ ξ, Sν ξ} dem = -2η µ ν I
10 The Sα ξ Dirac equation

We can rewrite now the Dirac equation as:

(η -1 ) µ α Sα ξ ∂ µ ψ(x) = mc ψ(x) (14)
Since all matrices are real, the ψ α (x) tuple is a priori real.

For reasons that will appear clear later, we are going to rewrite it as:

h µα (x) Sα ξ ∂ µ ψ(x) = mc ψ(x) (15)
with the h µα (x) real fields being constant and defined as:

h µα (x) def = h µα def = (η -1 ) µ α dem = η µ α
If the h µα (x) fields are Lorentz transformed as:

L •• (θ|h) µα (L c (θ|x)) def = (e θ a A a η ) µ ν (e θ b A b η ) α β h νβ (x)
we have, due to (3):

L •• (θ|h) µα (x) dem = (η -1 ) µ α
and we can show, with the help of ( 1), ( 10), (13), that ( 15) is Lorentz invariant. The introduction of the h(x) fields may look cumbersome, but we do this to be able to compare with other Dirac like equations coming in further sections.

11 The S a ξ more natural Dirac like equation So it would be much more natural to introduce the equation:

h µa (x) S a ξ ∂ µ ψ(x) = mc ψ(x) (16) ≡ h µa (x)Γ a ∂ µ ψ(x) = mc ψ(x)
This equation would be Lorentz invariant if the, a priori real, h µa (x) fields be Lorentz transformed as:

L • (θ|h) µa (L c (θ|x)) def = (e θ c A c η ) µ ν (e θ d L d ) a b h νb (x) (17)
It is a remarkable trick from Dirac that the lonely relation (5), leading to (6) avoided to consider the more natural equation ( 16). Once having written (6), the Dirac equation (8), crowded with complex numbers, looks an ad hoc construction from the group point of view. Someone may argue that (16) introduces the extra fields h µa (x) that may look cumbersome, but it is not new to see the introduction of coworking real fields related to the Dirac equation; this had been introduced already, through "vierbein", by people wanting to handle the Dirac equation within the framework of general relativity (see Appendix D). We see here that coworking fields emerged naturally lonely from a group point of view.

12 The A a η Dirac like equation At this point, it is worth noting that we could introduce also the equation:

h µa (x)(A a η) ν λ ∂ µ V λ (x) = mc V ν (x) h µa (x) A a η ∂ µ V (x) = mc V (x) (18)
which would be also Lorentz invariant with the same (17) transformation for the h(x) fields, the (2) one for the V µ (x) fields, and by exploiting:

e θ c A c η A a η e -θ d A d η dem = (e θ e L e ) b a A b η
induced by (11). We are not aware of any attempt to do physics with such version of the Dirac equation which is invariant by vector Lorentz transformation only.

The ten Σ A matrices

We can show that the set of ten matrices Σ A such that:

Σ A=1,10 def = 1 2 S0,1,2,3 , S a=1,2,3,4,5,6
forms a basis of S (4, R) with:

{Σ A ξ} = { 1 2 iγ µ , Γ a }
Concerning group constants we have now:

[Σ A ξ, Σ B ξ] dem = C AB C Σ C ξ (19)
with the real C AB C that can be computed with the same T r() technique as in (34). If we define the C A adjoint ten 10x10 real matrices with:

(C A ) C B def = C AB C
by using ten Θ A parameters, we have the exponential relation:

e Θ C Σ C ξ (Σ A ξ) e -Θ D Σ D ξ dem = (e Θ E C E ) B A (Σ B ξ)
13.1 The Σ A ξ Dirac like equation

If considering that the {Σ A } set is more complete than the {S a } set as a basis of 4x4 real symmetric matrices, we can also write:

h µA (x) Σ A ξ ∂ µ ψ(x) = mc ψ(x) (20)
But here, since the associated group involves ten Θ A parameters, we can't define a coordinate Lorentz transformation using η. Instead, we can continue to speak of an internal transformation T that let the x µ invariant but transforms h µA (x) and ψ α (x) such that:

T c (Θ|x) µ def = x µ = δ µ ν x ν T • (Θ|ψ) α (x) def = (e Θ D Σ D ξ ) α β ψ β (x) T • (Θ|h) µA (x) def = (e Θ D C D ) A B h µB (x) = δ µ ν (e Θ D C D ) A B h νB (x)
14 Connecting to the fifth dimension

We have a remarkable connection between η, the ten Σ A ξ 4x4 real matrices and a similar logic around a five dimensional 5x5 real matrix metric defined with:

η def = η 0 0 1 def =       1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1      
and a basis ÃA=1,10 of the ten 5x5 real antisymmetric matrices defined with: = 10, it is not surprising to find the same number of parameters, but it is more surprising to see that both set of ten 4x4 matrices {Σ A ξ} and the upper ten 5x5 { ÃA η} matrices give exactly the same group constants.

Ã1 def =       0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0       Ã2 def =       0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0       Ã3 def =       0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0       Ã4 def =       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0       Ã4+a(=1,

The five dimensional Dirac like equation

If, in the below, the index µ and ν=0,1,2,3 are taken for A=1,2,3,4 and a,b,c=1,2,3,4,5,6 for A=5,6,7,8,9,10, we can check that:

[ Ãa η Ãb η] dem = l ab c Ãc η [ Ãa η, õ η] dem = (A a η) ν µ Ãν η
We have then, with 5x5 real matrices:

e θ a Ãa η ( õ η) e -θ b Ãb η dem = (e θ c A c η ) ν µ ( Ãν η) similar to (13), so that we can write an equation similar to the Dirac one but on a Ψ α=0,1,2,3,5 (x) tuple:

(η -1 ) µ α Ãα η ∂ µ Ψ(x) = mc Ψ(x) (21)
with Ψ(x) transforming in a Lorentz transformation with:

L 5• (θ|Ψ) α (L c (θ|x)) def = (e θ a Ãa η) α β Ψ β (x)
In this equation there is no exotic complex gamma matrices and the spinor transformation in four dimensions becomes a more natural vector transformation in five dimensions!

A word on electromagnetism

Thanks to the Majorana representation of the gamma matrices, we have been able to get rid of complex numbers in our various equations, but traditionally the sticky "i" appears also when handling electromagnetism by writing:

iγ µ {∂ µ ψ(x) + i q c Φ µ (x)ψ(x)} = mc ψ(x) (22)
with q being the electromagnetic charge and Φ µ (x) being the electromagnetic potential related to the three dimensional Maxwell real U(t, x) and A(t, x) with:

Φ µ (x def = (ct, x)) def = (U(t, x), -A(t, x))
The Φ µ (x) would be Lorentz transformed with:

L • (A|Φ) µ (L c (A|x)) def = (e -Aη ) ν µ Φ ν (x)
By using our Sα ξ representation, it becomes:

h µα Sα ξ {∂ µ ψ(x) + i q c Φ µ (x)ψ(x)} = mc ψ(x) (23)
which exhibits the fact that "i" appears now only in the electromagnetic coupling to Φ µ (x).

The complex coupling induces that ψ(x) has to be complex, but if writing:

ψ[V , W ](x) def = V (x) + iW (x)
we see that (23) can be written as two equations without complex numbers on two real coupled fields V (x) and W (x):

h µα Sα ξ {∂ µ V (x) - q c Φ µ (x)W (x)} = mc V (x) h µα Sα ξ {∂ µ W (x) + q c Φ µ (x)V (x)} = mc W (x)
or:

[h µα Sα ξ ∂ µ - mc I]V (x) = [ q c Φ µ (x) h µα Sα ξ]W (x) (24) [h µα Sα ξ ∂ µ - mc I]W (x) = -[ q c Φ µ (x) h µα Sα ξ]V (x) (25)
We see now that complex numbers appear for electromagnetism in (22) mainly to write in a more compact form two coupled real quantities.

Charge conjugation transformation C

With (22), charge conjugation transformation would consist to find a transformation C (ψ) such that:

iγ µ {∂ µ -i q c Φ µ (x)}C (ψ)(x) = mc C (ψ)(x) (26)
By using the Dirac (or Chiral) representation of the gamma matrices, we get that:

C (ψ)(x) dem = iγ 2 ψ * (x) (27)
but by working with (24, 25), we see that:

C (V , W )(x) dem = (W , V )(x) (28)
that is to say the charge conjugation is reduced to just a swapping of the two coupled fields! We find this much more appealing than the much more algebraic (27). (On ψ(x), (28) would be written C (ψ)(x) dem = iψ * (x)).

C transformation and our other Dirac like equations

The same conclusion would be reached with our other equations ( 16), ( 18), (20) and even with the five dimensional (21) one. Here too, electromagnetism could be handled by two coupled real fields.

About Maxwell equations

When writing Maxwell equations in a four dimensional form, it appears equations like:

∂ µ F µν (x) = J ν (x)
where F µν (x) is antisymmetric. This equation is Lorentz invariant with:

L • (θ|J) µ (L c (θ|x)) def = (e θ a A a η ) µ ν J ν (x) L •• (θ|F ) µν (L c (θ|x)) def = (e θ a A a η ) µ α (e θ b A b η ) ν β F αβ (x) (29)
Since F µν is antisymmetric, we can introduce the six F a (x) fields such that:

F µν (x) def = (A a ) µ ν F a (x)
With this, the upper equation becomes:

(A a ) µ ν ∂ µ F a (x) = J ν (x) (30)
which reminds the (18) equation.

Since (seee Appendix B with S = η) we have:

e θ c A c η A a t (e θ d A d η ) dem = (e θ e L e ) b a A b
We can show that (29) induces:

L (θ|F ) a (L c (θ|x)) dem = (e θ c L c ) a b F b (x)
which let (30) Lorentz invariant.

16 Toward six or ten dimensional "adjoint" spacetime?

In the case of ( 14) and (21), h(x) is a constant and a square (4x4) real matrix, but in case of ( 16) , ( 18) and (20) it is not a square matrix (4x6 or 4x10) and a priori not a constant field. Being not square is not handy, but we can recover that by just going to a space-time with six or ten dimensions! Indeed, if instead of (x µ=0,1,2,3 ) coordinates we go toward (X a=1→6 ) or (X A=1→10 ), then ( 16) or (20) become:

h ab (X) S b ξ ∂ a ψ(X) = mc ψ(X) h AB (X) Σ B ξ ∂ A ψ(X) = mc ψ(X)
They will be invariant with the transformations:

T c (θ|X) a def = (e θ d L d ) a b X b T • (θ|ψ) α (T c (θ|X)) def = (e θ a S a ξ ) α β ψ β (X) T (θ|h) ab (T c (θ|X)) def = (e θ c L c ) a e (e θ d L d ) b f h ef (X)
or:

T c (Θ|X) A def = (e Θ D C D ) A B X B T • (Θ|ψ) α (T c (Θ|X)) def = (e Θ A Σ A ξ ) α β ψ β (X) T (Θ|h) AB (T c (Θ|X)) def = (e Θ C C C ) A E (e Θ D C D ) B F h EF (X)
We can even show that we can have constant h(X) fields. Indeed, if taking:

h ab (X) def = G a b
with the G 6x6 real matrix defined by:

G def = G 6 def = I 3x3 0 0 -I 3x3 I 3x3 def =   1 0 0 0 1 0 0 0 1  
we have:

T (θ|h) ab (T c (θ|X)) dem = h ab
and the same for h AB (X) with the G 10 10x10 real matrix defined by:

h AB (X) def = (G 10 ) A B G 10 def = η 0 0 G 6 14
All the formulas used in this article had been checked numerically by computer. On the author GitHub gbarrand, the repository "papers" contains the open source "SMAD" C++ unitary test program that verifies the formulas found here.

Conclusions

Thanks to E.Majorana, it is interesting to see that the four dimensional vector and spinor Lorentz transformations show a similar "AS/SA" real matrix pattern (S for Symmetric, A for Antisymmetric), and that the "SA" real pattern for the spinor transformation is connected to an "AS" real pattern in fifth dimension. With this in head, we see now the Dirac spinor Lorentz transformation differently; instead of being related to some exotic mathematical trick through complex Dirac matrices introduced through (5), we see it now deeply related to a real antisymmetric ξ matrix which is the pending of the real symmetric η matrix for the four vector transformation. By pushing to the fifth dimension, we can even see it as a five vector transformation. Moreover, following the logic around this ξ matrix, we see that the traditional Dirac equation looks rather incomplete and could be extended in a more natural way to something as ( 16) or (20) that we write again here:

h µa (x) S a ξ ∂ µ ψ(x) = mc ψ(x) h µA (x) Σ A ξ ∂ µ ψ(x) = mc ψ(x)
with the ψ(x) tuple not needed to be complex at this point, and the coworking h(x) real fields coming naturally from a group point of view. We saw also that electromagnetism could be handled without complex numbers in all our Dirac like equations by just introducing two coupled real fields. Finally, we showed that going toward six or ten dimensional spacetime would be more natural from a group point of view.

with the standard Pauli 2x2 complex matrices being defined as:

σ 1 def = 0 1 1 0 σ 2 def = 0 -i i 0 σ 3 def = 1 0 0 -1
With less complex numbers around, we can write them directly as:

γ 0 def = i     0 0 0 -1 0 0 1 0 0 -1 0 0 1 0 0 0     γ 1 def = i     1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1     γ 2 def = i     0 0 0 1 0 0 -1 0 0 -1 0 0 1 0 0 0     γ 3 def = i     0 -1 0 0 -1 0 0 0 0 0 0 -1 0 0 -1 0     Appendix B γ µ , Γ[A] matrices
Let us have γ µ and a S real symmetric square matrices such that:

{γ µ , γ ν } def = 2S µ ν I
I being the identity matrix. Note that we do not specify the dimensions of them! The γ µ can be real or complex. With a little bit of algebra, we can show that it induces:

[[γ µ , γ ν ], γ α ] dem = 4S α ν γ µ -4S α µ γ ν (31)
If A being a real antisymmetric square matrix of same dimension as S, let us define Γ[A]:

Γ[A] def = 1 8 [γ µ , γ ν ](A) µ ν
The relation (31) induces:

[Γ[A], γ µ ] dem = (AS) ν µ γ ν
It is a general property (see [START_REF] Itzykson | Quantum Field Theory[END_REF] p.70) that for square matrices M, N a of same dimension:

[M, N a ] def = C b a N b ⇒ e M N a e -M dem = (e C ) b a N b (32)
Then we have the important property:

e Γ[A] γ µ e -Γ[A] dem = (e AS ) ν µ γ ν B.1 γ µ "up" matrices
If defining γ µ with:

γ µ def = = (S -1 ) µ ν γ ν
we have:

{γ µ , γ ν } dem = 2(S -1 ) µ ν I Γ[A] dem = 1 8 [γ µ , γ ν ](SAS) µ ν [Γ[A], γ µ ] dem = (-AS) µ ν γ ν
It is a general property (see [START_REF] Itzykson | Quantum Field Theory[END_REF] p.70) that for square matrices M, N a of same dimension:

[M, N a ] def = C a b N b ⇒ e M N a e -M dem = (e C ) a b N b
Then we have the important property:

e Γ[A] γ µ e -Γ[A] dem = (e -AS ) µ ν γ ν B.2 The A a basis
Let us define the antisymmetric A a=1,2,3,4,5,6 six matrices with:

A 1 def =     0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0     A 2 def =     0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0     A 3 def =     0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0     A 4 def =     0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0     A 5 def =     0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0     A 6 def =     0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0    
These matrices form a basis for any 4x4 real antisymmetric matrix. We can show that:

T r(A a A b ) dem = -2δ ab B.3 A a S,

adjoint matrices

It appears that:

A a SA b -A b SA a
is antisymmetric, and then we can develop it on the A a basis and write:

A a SA b -A b SA a dem = C ab c A c ⇒ [A a S, A b S] dem = C ab c A c S (33)
The C ab c can be computed by using the matrix trace T r() with:

T e X e Y = e Z with:

Z[X, Y ] dem = X + Y + 1 2 [X, Y ] + 1 12 [X, [X, Y ]] - 1 12 [Y, [X, Y ]] + ...
ǫ 1 def =   0 0 0 0 0 1 0 -1 0   ǫ 2 def =   0 0 -1 0 0 0 1 0 0   ǫ 3 def =   0 1 0 -1 0 0 0 0 0  
The L a have all the properties shown in the Appendix B with S = η, In particular we have:

e θ c A c η A a η e -θ d A d η dem = (e θ e L e ) b a A b η e θ c Γ c Γ a e -θ d Γ d dem = (e θ e L e ) b a Γ b e θ c L c L a e -θ d L d dem = (e θ e L e ) b a L b Appendix D R transformation, Vierbein D.1 General frame transformation R
One feature of general relativity is the invariance of formulas according to a general frame transformation. The idea behind this being that laws of physics should look the same in any reference frame. With the same kind of notation used in the first paragraph (1), a general coordinate transformation on the tuple x µ=0,1,2,3 can be written:

R c (r|x) µ def = r µ (x)
the r µ (x) being four "well behaved" functions (of the x µ ) that are seen as the parameters of the transformation R (similar as an antisymmetric matrix A is seen as the parameter of a Lorentz transformation). If we introduce the notation:

R[r] µ ν (x) def = ∂ ν {r µ }(x) R[r] µ ν (x) def = ∂ ν {(r -1 ) µ }(r(x))
we have:

Rµ α (x)R α ν (x) dem = ∂ ν {(r -1 • r) µ (x)}(x) dem = ∂ ν {x µ } dem = δ µ ν R µ α (x) Rα ν (x) dem = ∂ ν {(r • r -1 ) µ (x)}(r(x)) dem = δ µ ν
With this, a general frame transformation for a φ(x), a V µ (x), a T µν (x), a T µν (x), etc, reads:

R s (r|φ)(R c (r|x)) def = φ(x) R • (r|V ) µ (R c (r|x)) def = R µ ν (x)V µ (x) (35) R •• (r|T ) µν (R c (r|x)) def = R µ α (x)R ν β (x)T αβ (x) R •• (r|T ) µν (R c (r|x)) def = Rα µ (x) Rβ ν (x)T αβ (x)
For all the upper definitions, we have the important composition group property:

R(r 1 |R(r 2 |whatever)) dem = R(r 1 • r 2 |whatever)
We can note that for these, a Lorentz transformation L is a particular case of a R if we take:

r µ (x) def = (e Aη ) µ ν x ν
If, by using a g µν (x), we define the length of of a V µ (x) with:

l[g](V )(x) def = g µν (x)V µ (x)V ν (x)
with the upper definitions (35), it is easy to show that it is invariant by a general frame transformation:

l[R •• (r|g)](R • (r|V ))(R c (r|x)) dem = l[g](V )(x)

D.2 Spinor transformation? Vierbein

Concerning a spinor transformation R • , we have a slight problem since we have no obvious extension of the Lorentz transformation L • (4) for them. To handle the Dirac equation within general relativity, people do a (not so appealing) compound construction by introducing "vierbein" fields e α µ (x) bearing two indices of different kind concerning transformations. The vierbein fields are considered to be more fundamental than the metric g µν (x), and in fact define the metric with: Concerning transformations for e α µ (x), the R one is applied for the µ down index, but an "internal" or "local" Lorentz one L I is applied for the α upper one. The L I is defined as:

L c I (A|x) µ def = x µ
(then no effect on x µ ), and on a ψ(x):

L • I (A|ψ) α (x) def = (e Γ[A] ) α β ψ β (x)
We note then the important difference with a L ; the x µ are not changed with a L I transformation.

D.4 Dirac equation with vierbein

The Dirac equation on "curved spacetime" is now written:

ẽµ α (x)iγ α ∂ µ ψ(x) = mc ψ(x) (37)
Someone can check that this equation is invariant by both the R and L I transformations with:

R s (r|ψ) α (R c (r|x)) The compound construction (37) is probably related to the fact that, put all together, it makes no sense to attempt to mix the Dirac equation, related to the "quantum world", with maths related to the macroscopic world modelled with general relativity, and this without having unified in first place the ideas related to these two worlds. (A "quantum gravity" theory will probably be based on only one single "crystal clear" transformation).

By using the Majorana representation, the equation (37) can be written:

h µα (x) Sα ξ ∂ µ ψ(x) = mc ψ(x) h µα (x) def = ẽµ β (x)(η -1 ) β α
which is similar to (15) but with the h µα (x) fields being now not constant and transforming in a mixed way for each index (with R for the first one and L I for the second one). We can reach, here too, the conclusion that this equation is not so natural from a group point of view and something as (16) or (20) would be better.

  Indeed, from the group point of view, since: [S a ξ, S b ξ] dem = l ab c S c ξ and by defining the L a "adjoint" six 6x6 real matrices: (L a ) b c def = l ac b we have the exponential relation: e θ c S c ξ S a ξ e -θ d S d ξ dem = (e θ e L e ) b a S b ξ

  group constants as in (19): nicely strange! Since the number of parameters in a (n=4)x4 symmetric matrix is n(n+1) 2 = 10 and the number of parameters for a (d=n+1=5)x5 antisymmetric one is d(d-1) 2 = (n+1)n 2

  r[A d (A a SA b -A b SA a )] = C ab c T r(A d A c ) = C ab c (-2δ dc ) commutators: [C a , C b ] dem = C ab c C cand by exploiting (32) we have:e θ c A c S A a S e -θ d A d S dem = (e θ e C e ) b a A b S e θ c C c C a e -θ d C d dem = (e θ e C e ) b a C bIf S is invertible, from the first upper relation we can deduce that we have also:e θ c A c S A a t (e θ d A d S ) dem = (e θ e C e ) b a A b B.4 Γ a matricesBy using the upper A a real matrices, we define: , γ ν ](A a ) µ νWe can demonstrate that the commutators (33) induce:[Γ a , Γ b ] dem = C ab c Γ cwith the same real group constants C ab c . By exploiting (32) we have also:e θ c Γ c Γ a e -θ d Γ d dem = (e θ e C e ) b a Γ bB.5 The Baker-Campbell-Hausdorff developmentIf X and Y are matrices of same dimension, we have:

g

  µν (x) def = e α µ (x)e β ν (x)η αβ(36)Are introduced also co-vierbein fields ẽµ α (x) such that:e α µ (x)ẽ µ β (x) def = δ α β ẽµ α (x)e α ν (x) def = δ µ ν D.3 Local (or internal) Lorentz transformation L I

def=

  ψ α (x)L • I (A|ψ) α (x) def = (e Γ[A] ) α β ψ β (x) R • (r|e α ) µ (R c (r|x)) def = Rν µ (x)e α ν (x) L • I (A|e µ ) α (x) def = (e Aη ) α β e β µ (x) R s (r|e µ ) α (R c (r|x)) def = e α µ (x) L s I (A|e α ) µ (x) def = e α µ (x) R • (r|ẽ α ) µ (R c (r|x)) def = R µ ν (x)ẽ ν α (x) L I• (A|ẽ µ ) α (x) def = (e -Aη ) β α ẽµ β (x) R s (r|ẽ µ ) α (R c (r|x)) def = e µ α (x) L Is (A|ẽ α ) µ (x) def = ẽµ α (x)To play with the upper transformations over the vierbein and co-vierbein fields, we can show that with the metric defined by (36) we have:R •• (r|g) µν (x) dem = R • (r|e α ) µ (x)R • (r|e β ) ν (x)η αβ and: L Is (A|g) µν (x) def = g µν (x) dem = L • I (A|e µ ) α (x)L • I (A|e ν ) β (x)η αβ

Appendix A γ µ in Majorana representation

The γ µ=0,1,2,3 in the Majorana representation are presented in general as (see [START_REF] Itzykson | Quantum Field Theory[END_REF] p.694):