The dark matter and the dark energy proceeding from elementary particles

Christian PIERRE

Abstract

As dark energy and dark matter are responsible for at least 96% of the structure of our universe, they have to be considered as fundamental blocks of the structure of all elementary particles.

Referring to the central thesis of algebraic quantum theory which envisages the structure of massive elementary particles in the three embedded shells ''space-time, middle-ground and mass'', the composition of the dark energy and of the dark matter is then proved to refer naturally respectively to the internal vacuum space-time substructure and to the vacuum space-time plus middle-ground bound substructures of elementary particles.

PACS: 02.10.De, 95.35.+d, 95.36.+x

Contents

1 Introduction 1

2 Dark energy structure of elementary particles 2

3 Dark matter and visible matter structure of elementary particles 10

4 The dark energy and the dark matter structures of elementary particles interacting

with the visible matter structure of other elementary particles 19

References 27

1

1 Introduction The most recent observations of cosmologists reveal a universe made up of 73% dark energy, 23% dark matter and only 4% ordinary (visible) matter. The exact nature of dark energy and dark matter is presently speculative. The first observational evidence for dark energy came from observations of accelerated expansion of supernovae [Per], [Rie]. The term ''dark energy'' was proposed by M. Turner [H-T] in 1998. This dark energy fills uniformly empty space and is known to interact only through gravity. The gravitational effects of its space energy correspond to those of Einstein's cosmological constant [P-R] which has negative pressure [Lem] equal (at a sign) to its energy density and which is then responsible for the expansion of the universe. The possible connection between the vacuum energy density of quantum field theories and the cosmological constant was envisaged by Zel'dovich [Zel] to take into account the quantum fluctuations of the vacuum energy density. But, this connection between quantum field theories (QFT) and general relativity (GR) through the cosmological constant led to a profound crisis since the theoretical expectations of QFT for the cosmological constant exceed observational limits by more than 100 orders of magnitude as outlined by S. Weinberg [Wei]. This problem results clearly from the lack of unification between GR ad QFT which are two theories describing the physical phenomena at different scales. So, the outcome of this dead-end proceeds necessarily through the fusion of these two theories. This was attempted by the author in the so-called algebraic quantum theory [Pie1], [Pie2], proposing a new interpretation of the equations of GR which are then in one-to-one correspondence with the equations of the internal dynamics of the vacuum and mass structures of a set of interacting (bisemi)particles at the (sub)Planckian scale. In this new context, the cosmological constant would deal with the internal vacuum substructure of elementary particles while the classical observed cosmological constant of general relativity refers to the ''macroscopic'' domain where the vacuum energy density alters the geometry of the space-time by its gravitational effects in such a way that the curvature be on an average null [Pie2]. On the other hand, the evidence for the dark matter comes mostly from the observation of the rotational speeds of galaxies and the orbital velocities of galaxies in clusters. It was first inferred by F. Zwicky [Zwi] in 1933 in studying the motions of the Coma cluster of galaxies. The presence of dark matter can be deduced from its gravitational effects on visible matter. It was detected by its gravitational lensing in 2006 [Nas]. It behaves like non radioactive dust forming haloes around galaxies and it is important in the structure formation of galaxies. The composition of dark matter is at present unknown [S-W] since it has never been observed in laboratory. Baryonic and non baryonic dark matter have been postulated among which the neutrino, having a very small mass, could be a candidate of hot non baryonic dark matter since it travels with a relativistic velocity. As dark energy and dark matter are responsible for 96% of the structure of our universe, they cannot be ignored in a unified quantum theory of matter which would have to consider them as fundamental structural blocks of the internal structure of elementary particles, taking into account the importance of dark matter in the formation of galaxies. It is thus the aim of this paper to show that the composition of dark energy and dark matter refers respectively to the vacuum space-time substructure and to the vacuum space-time plus dark mass (called middle-ground in [Pie1]) bound substructures of elementary (bisemi)particles in the frame of algebraic quantum theory (AQT) introduced in [Pie1] and briefly recalled in the next chapter.

2

Chapter 2 deals also with the nature of the dark energy structure of elementary particles as resulting from the unification of quantum field theories with general relativity. The deformations of the states of the vacuum space-time fields of the dark energy structure of elementary particles are responsible for their dark matter and visible matter structures. This constitutes the content of chapter 3 proving that the dark matter structure and the visible matter structure of an elementary particle are respectively twofold and threefold universal singular mathematical structures resulting from the blowups of the versal deformations of degenerate singularities on the vacuum internal dark energy structure. Finally, in chapter 4, the bilinear interactions between the dark energy and the dark matter structures of a set of elementary particles and the visible matter structure of another set of particles is studied.

2 Dark energy structure of elementary particles

2.1 Nature of algebraic quantum theory The algebraic quantum theory is a theory of internal structure of elementary particles: this internal structure is a universal mathematical structure in the heart of the global program of Langlands developed by the author in [Pie3], [Pie4], and resulting from one-to-one correspondences between representations of Galois groups and their holomorphic representations (and cuspidal forms) affected by singularities. A QT is thus a new quantum field theory [F-R-S] based on algebraic geometry and number theory and not deduced from classical mechanics as QFT but reaching the same objectives as QFT and even more.

2.2 Quantization by means of algebraic quanta of structure Indeed, AQT is a structural algebraic quantum field theory whose spatial building blocks are space structure algebraic quanta defined as irreducible closed algebraic subsets of degree N . Remark that the energy quanta are shifted spatial algebraic quanta resulting from differential Galois theory generating Galois groupoids on manifolds as described in [Pie5]. Assume that these structure algebraic quanta have a universal representation space given by open balls of radii equivalent to the Planck constant in the sense that all the roots of the splitting field of an algebraic quantum are mapped in an open ball of radius of magnitude of . These algebraic quanta can then be considered as big points at the Planck scale. By a change of scale [Pie2], each algebraic quantum, i.e. a quantum big point, can correspond to a macroscopic point of classical mechanics or general relativity if all points of this algebraic quantum implode on the considered open ball center viewed as the corresponding macroscopic point. This implosion of all points of an algebraic quantum inside an open ball of magnitude ħ can be described mathematically by a projective map acP from all roots of the algebraic quantum

onto the center of the considered open ball. This projective map is associated with a metric (tensor) contraction if the considered closed algebraic subset of the algebraic quantum is compact.

3

It describes the transition from the quantum level to the classical level in the frame of algebraic geometry taking into account that a set of q neighbouring algebraic quanta imploding onto their centers gives rise to a continuum compact subset of q ''classical'' points. This projective map acP is the inverse analogue of the deformation quantization [F-L-S]

consisting in deforming the algebra A of classical observables over a ground field k into the

algebra A over [[ ]]k in such way that / A A A where A is topologically free [[ ]]k -module. This leads to the existence of a new product u v , for , u v A , expanded in formal series:

1

( , )

rr

r

u v u v c u v

where rc is a Hochschild 2-cochain [Ster], [Ger].

2.3 Fusion of quantum mechanics with special relativity by bisemistructures

The fusion of quantum mechanics with special relativity is still an open problem since ''no exactly solved nontrivial model of interacting quantum field theory on Minkowsky space is yet known'' as noticed by S. Doplicher in [Dop]. This question was envisaged in AQT as a crucial step by considering a completely bilinear version of QFT by means of bisemistructures introduced by the author in [Pie6]. A fundamental bisemistructure refers to a bisemigroup appearing in the triple , ,L R RLG G G

where: (a) LG (resp. RG ) is a left (resp. right) semigroup under the addition of its left (resp. right)

elements iLg (resp.

iRg ) restricted to (or referring to) the upper (resp. lower) half space.

(b) R LG is a bilinear semigroup whose bielements ( )

i iR Lg g are submitted to the cross binary

operation according to:

,

( ) ( ) ( ) ( ) ,i i j j i j i j

R L R L R L

R L R L R R L L

G G G

g g g g g g g g

leading to cross products ( )i jR Lg g and ( )

j iR Lg g .

Important bisemigroups are algebraic bilinear semigroups of matrices 2GL ( )R LF F interpreted as

(bilinear) representations of product, right by left, of global Weil (or Galois) groups in the frame of the global program of Langlands [Pie3[ and leading to a consistent algebraic definition of elementary particle quantum fields having a bilinear character and inducing a ''bi-semi'' particle structure as outlined in section 2.6.

An algebraic bilinear semigroup 2GL ( )R LF F then decomposes according to:

2 2 2GL ( ) ( ) ( )t

R L R LF F T F T F

where 2 ( )t

RT F (resp. 2 ( )LT F ) is a right (resp. left) linear semigroup of lower (resp. upper) triangular

matrices with entries in RF (resp. LF ) which are right (resp. left) algebraic symmetric extensions

of a number field of characteristic 0.

4

Let , ,{ }mv v mL L

(resp. , ,{ }mv v mL L

) denote the set of right (resp. left) real pseudoramified

completions homeomorphic to the corresponding extensions associated with RF (resp. LF ) and characterized by degrees (of the corresponding extensions)

, ,resp. [ : ] · , [ : ] ·( )

m mv vL k N L k N

which are integers modulo N , where is an integer inferior to N . Let ( ) ( )R v L vM L M L be the representation space 2Repsp(GL ( ))v vL L of the bilinear complete

(algebraic) semigroup 2GL ( )v vL L over products of pairs of real completions.

It decomposes into a set of conjugacy class representatives on the real pseudoramified completions

,mvL

and ,mvL

according to [Pie7]:

, , ,

( ) ( ) { }m m mR v L v v v vM L M L M M

where m labels the multiplicity of the -th conjugacy class representative, ,m mv vM M

.

Let

,( )

mR vM

(resp.,

( )mL vM

) be a complex-valued differentiable function over the ( , )m -th

conjugacy class representative of 2 ( )tvT L (resp. 2 ( )vT L ) and let

, ,( ) ( )

m mR v L vM M

denote the corresponding bifunction on , ,m mv vM M

.

Then, the set , ,{ ( )}mR v mM

(resp., ,{ ( )}mL v mM

) of -valued differentiable functions,

localized in the lower (resp. upper) half space, is the set ( ( ( )))R R vM L (resp. ( ( ( )))L L vM L ) of

right (resp. left) sections of the semisheaf of rings ( ( ))R R vM L (resp. ( ( ))L L vM L ).

And, the set , , ,{ ( ) ( )}m mR v L v mM M

of differentiable bifunctions constitutes the

set ( ( ( )) ( ( )))R R v L L vM L M L of bisections of the bisemisheaf of rings

( ( )) ( ( ))R R v L L vM L M L over 2GL ( )v vL L .

2.4 Proposition

The bisemisheaf of rings ( ( ) ( ( )R R v L L vM L M L is a physical bosonic quantum string field of

an elementary particle. Proof. Let

, ,{ }m

T Tv v mL L

(resp., ,{ }m

T Tv v mL L

) be the set of right (resp. left) real pseudo-

ramified toroidal completions obtained from vL (resp. vL ) by a toroidal compactification of the

corresponding completions ,mvL

(resp.

,mvL

) [Pie3].

The set ,

{ }m

Tv mL

(resp. ,

{ }m

Tv mL

) of -th completions are then semicircles covering a 2-

dimensional right (resp. left) semitorus 2 ( )RT (resp. 2 ( )LT ), localized in the lower (resp. upper)

half space. Assume that the degree of

,m

TvL

and of ,m

TvL

is given by their transcendence degree equal to the

Galois extension degree , ,

:[ ] [ ]:m m

T Tv vL Lk k N

of the associated extension.

5

Then,

,m

TvL

and ,m

TvL

are toroidal completions or semicircles at quanta, a quantum being an

irreducible completion of degree N . As we are essentially interested in circles, toroidal completions

2 , 2m

TvL

and

2 , 2m

TvL

characterized by

degrees:

2 , 2 ,2 2: :[ ] [ ] 2

m m

T Tv vL L k Nk

will be taken into account. By this way, the corresponding completions

2 , 2mvL

and 2 , 2mvL

are closed paths or closed strings.

Now, each product

2 , 2 ,2 2{ }

m m

T Tv vL L

of symmetric circles rotating in opposite senses according

to [Pie2] is the representation of an harmonic oscillator. This is also the case for the product

2 , 2 ,2 2{ }

m mv vL L

of corresponding completions homeomorphic to 2 , 2 ,2 2

{ }m m

T Tv vL L

and for the

product 2 , 2 ,2 2

{ ( ) ( )}m mR v L vM M

of -valued differentiable functions.

Consequently, the set of packets

2 , 2 , 22 22 ,{ ( ) ( )}

m mR v L v mM M of bifunctions on

2 , 2 , 22 22 ,{ }

m mv v mL L behaves like a set of packets of harmonic oscillators characterized by

increasing integers 2 , 1 . Thus, the bisemisheaf of rings

2 , 2 , 22 22 ,( ( )) ( ( )) { ( ) ( )}

m mR R v L L v R v L v mM L M L M M

is a physical string field. It is a quantum string field because the set of sections of the bisemisheaf

( ( )) ( ( ))R R v L L vM L M L is a tower of increasing bistrings, i.e. products of symmetric right and

left strings, behaving like harmonic oscillators and characterized by a number of increasing biquanta, 2 , corresponding to the normal modes of the string field. This quantum string field is a bosonic field because biquanta can be added to (i.e. created) or removed (i.e. annihilated) from these bistrings by Galois automorphisms or antiautomorphisms as developed in [Pie1] and because each bistring with degree 2 was interpreted as a ''bound bisemiphoton'' at 2 biquanta. ■

2.5 Corner-Stone of the unification of GR with QFT: vacuum space-time substructure The other great challenge of AQT is the unification of QFT with GR. The corner-stone of this possible unification is the connection between the vacuum state of QFT and the cosmological constant of GR as recalled in section 1. This vacuum of QFT was assumed in [Pie1] and in [Pie2] to be of expanding space-time type in order to correspond to the de Sitter solution of the equations of GR and to be shared out amongst the own vacua of the elementary particles. Under this hypothesis, the vacuum substructure of every elementary particle must be of expanding space-time nature and can be described by a time and a space quantum string field. The generative string field is the time string field considered as a reservoir of space having an inertial effect on the space string field localized in an orthogonal space.

6

To the time string field, noted in condensed notation ( )R L

T TST STM M , corresponds an operator-

valued time string field ( )p p

R L

T T

ST STM M obtained from ( )R L

T TST STM M by the action of the

differential bioperator:

; ; : ( ) ( )

p p

R L R L

T T T TST ST ST STR ST L STT T M M M M

in such a way that

; ; 0 0; 0 ; 0

R L

ST STR ST L ST

t r ST t r ST

T T i s i sc t c t

acts on each bisection or bistring

2 , 2 ,2 2( ) ( )

R L m mR v L vM M

where ST and t rc are respectively the Planck constant and the velocity of light at this (space)-

time internal level, i.e. the level of the dark energy.

;R STT and ;L STT are directional derivatives in the time variable 0t in the directions 0Rs

and 0Ls

respectively. The left strings rotate in opposite sense with respect to the corresponding symmetric right strings

and have two senses of rotation: this gives to the time string field ( )p p

R L

T T

ST STM M a structure of spin

1/ 2 field.

The operator-valued space string field, noted ( )p p

R L

S S

ST STM M , can be generated from the

(operator-valued) time string field ( )p p

R L

T T

ST STM M by the (composition of) morphisms of its

right and left semifields:

( )

( )(resp :.

:

)

p p p

R R R

p p p

L L L

T T r S

ST ST STt r R

T T r S

ST ST STt r L

E M M M

E M M M

where RE (resp. LE ) is an endomorphism [Pie1] based on Galois antiautomorphisms and

transforming the semifield p

R

T

STM (resp. p

L

T

STM ) into a disconnected time residue semifield ( )p

R

T r

STM

(resp. ( )p

L

T r

STM ) and into a complementary time semifield ( )p

R

T i

STM (resp. ( )p

L

T i

STM ) which, projected

through the origin into the orthogonal three-dimensional space by the morphism t r , gives rise to

the space semifield p

R

S

STM (resp. p

L

S

STM ).

The (operator-valued) space-time fields ( ) ( )( ) ( )

p p p p

R R L L

T r S T r S

ST ST ST STM M M M then describe the

dynamics of the internal vacuum structure of a lepton of the first family, i.e. essentially the electron and its neutrino

e . Referring to the introduction, these space-time fields are

responsible for the dark energy structure of the electron (and of its neutrino). They decompose as a bisemistructure [Pie6] into the following disconnected space-time (i.e. dark energy) subfields

7

( ) ( )

magn elec

p p p p p p p p

R L R R L L

p p p p p p p p p p

R L R L R L

T S T S T S T S

ST ST ST ST ST ST

T S T S S S S T T S

ST ST ST ST ST STD

M M M M M M

M M M M M M

where:

(a) ( ) p p p p

R L

T S T S

ST STDM M

is a four-dimensional bilinear diagonal subfield;

(b) magn( )

p p

R L

S S

ST STM M is a three-dimensional off-diagonal magnetic subfield generated from the

exchange of magnetic biquanta between the right and left space subsemifields p

R

S

STM and

p

L

S

STM as described in [Pie7];

(c) ( ) ( )

elec )p p p p

R L

S T T S

ST STM M is a three-dimensional off-diagonal electric subfield generated from

the exchange of electric biquanta between the right space (resp. right time) and left time (resp. left space) subsemifields [Pie1].

From a bigger time string field (noted p p(Bar);T (Bar);T

R LST STM M ) (i.e. a time baryonic string field with

much more packets of bistrings), another endomorphism Rt

E (resp. Lt

E ) can be envisaged in

such a way that Rt

E (resp. Lt

E ) participates in the decomposition of the right (resp. left)

baryonic time semifield p(Bar);T

RSTM (resp. p(Bar);T

LSTM ) into a residue time semifield p(Bar);T (r)

RSTM (resp.

p(Bar);T (r)

LSTM ) and into three connected complementary semiquark time semifields ; 31{ }

i p

R

q T

ST iM

(resp. ; 3

1{ }i p

L

q T

ST iM ) according to:

p p

p p

3(Bar);T (Bar);T (r) ;

1

3(Bar);T (Bar);T (r) ;

1

(resp.

:

: )

i p

R R Ri R

i p

L L Li L

q T

ST ST STt t ti

q T

ST ST STt t ti

E M M M

E M M M

where:

RtE (resp.

LtE ) is the baryonic endomorphism based on Galois antiautomorphisms

decomposing p(Bar);T

RSTM (resp. p(Bar);T

LSTM ) into a core residue time semifield p(Bar);T (r)

RSTM (resp.

p(Bar);T (r)

LSTM ) and into a connected complementary time semifield p(Bar);T (i)

RSTM (resp.

p(Bar);T (i)

LSTM );

it t is the morphism sending p(Bar);T (i)

RSTM (resp. p(Bar);T (i)

LSTM ) into a three-dimensional

connected complementary time semispace and transforming it into the three time semifields ; 3

1{ }i p

R

q T

ST iM (resp. ; 3

1{ }i p

L

q T

ST iM ) of the semiquarks.

The time string semifields of the three right (resp. left) semiquarks are then connected to the residue core time semifield of the right (resp. left) semibaryon leading to the confinement of these.

8

The space semifield ;i p

R

q S

STM (resp. ;i p

L

q S

STM ) of the three right (resp. left) semiquarks can be

generated by ( )i i RR R

t r iE (resp. ( )i i LL L

t r iE ) morphisms from their respective time semifields as

developed for semileptons. So, the space-time internal vacuum of a right (resp. left) semibaryon is given by:

3(Bar); - (Bar); ;

1

3(Bar); - (Bar); ;

1

(resp

. ).

p p p i p p

R R R

p p p i p p

L L L

T S T q T S

ST ST STi

T S T q T S

ST ST STi

M M M

M M M

The (operator-valued) space-time string field p p p p(Bar);T -S (Bar);T -S( )

R LST STM M then describes the

dynamics of the internal vacuum structure of a baryon and corresponds to the dark energy structure of this baryon. It decomposes according to:

1

(Bar); - (Bar); - (Bar); ( ) (Bar); ( )

3 3; ; ; ;

1

3 3(Bar); ( ) ; ; (Bar); ( )

1 1

p p p p p p

R L R L

i p p i p p i p p j p p

R L R L

p i p p i p p p

R L R L

T S T S T r T r

ST ST ST ST

q T S q T S q T S q T S

ST ST STi j

i

STi

T r q T S q T S T r

ST ST ST STi

j

i

M M M M

M M M M

M M M M

where:

(Bar); ( ) (Bar); ( )( )

p p

R L

T r T r

ST STM M is the core central time string field of the baryon, i.e. its core dark

energy structure;

; ;( )

i p p i p p

R L

q T S q T S

ST STM M

is the internal vacuum string field of the i -th quark, i.e. the dark

energy structure of this quark. It decomposes into diagonal space and time string fields, magnetic and electric fields as envisaged before for the lepton;

; ;( )

i p p j p p

R L

q T S q T S

ST STM M

are the mixed fields of interaction between the i -th right semiquark

and the j -th left semiquark. They decompose similarly into diagonal time and space fields of interaction, which are of gravitational nature as described in [Pie7], and into off diagonal magnetic and electric fields of interaction. These gravitational, electric and magnetic fields are ''dark energy interaction fields'' between quarks confined inside a baryon;

(Bar); ( ) ; ; (Bar); ( )( ) ( )

p i p p i p p p

R L R L

T r q T S q T S T r

ST ST ST STM M M M

are mixed ''strong'' string fields of

interaction between core central time string semifields of semibaryons and space-time string semifields of semiquarks. They are responsible for the generation of ''meson dark energy'' after compactification of these interacting fields as developed [Pie8].

2.6 Bisemiparticle dark energy structure The dark energy structure of an elementary particle then refers to the space-time fields of its internal vacuum structure, called space-time (ST). As these fields are bisemifields, i.e. products of right semifields by their symmetric left correspondents, an elementary particle at this dark energy level is a bisemiparticle composed of the

9

product of a left semiparticle, localized in the upper half space, by its symmetric right (co)semiparticle, localized in the lower half space in such a way that:

the space-time fields of its internal vacuum lead to a working interaction bisemispace responsible for the electric charge (i.e. the electric internal field), the magnetic moment (i.e. the magnetic internal space field), and an internal gravitational field in the case of a baryon;

the right semiparticle, dual of the left semiparticle, is projected on the latter and is unobservable unless the bisemiparticle be split into a pair of ''particle-antiparticle'' at this dark energy level.

The leptons are then called bisemileptons and the baryons bisemibaryons. Note that the bisemifermions of the second and third families have similar but much more complex structures than those of the first family. They are generated from bisemifermions of the first family as described in [ Pie1] and in [Pie8].

2.7 Proposition The dark energy structure of an elementary particle is a universal mathematical structure. Proof. Indeed, the dark energy structure of an elementary particle is composed of the direct sum of a time field and of a space field localized in orthogonal (bisemi)spaces, or orthogonal to each other. On the other hand, each time or space dark energy string field is a bisemisheaf of rings as developed in proposition 2.4. This bisemisheaf of rings is the analytic representation of the corresponding algebraic bilinear semigroup which is the representation of the product, right by left, of global Weil groups in the frame of the global program of Langlands as proved in [ Pie4] and recalled in section 2.3. In the same context, this bisemisheaf gives rise to the product, right by left, of holomorphic functions (and/or cuspidal forms) by taking into account their power series development as developed in [ Pie3]. Thus, the dark energy structure of an elementary particle, originating from algebraic biquanta generated by the product, right by left, of global Weil (or Galois) groups, can be represented by the sum of a time and a space holomorphic bifunction (and/or cuspidal biform). This explains the universal mathematical character of the given dark energy structure of an elementary particle. ■

2.8 Does the dark energy of elementary particles correspond to a new kind of aether?

The dark energy structure of elementary particles, being of space-time nature, is expansive since it is not confined by an external boundary shell. On the other hand, as this dark energy corresponds to the internal and generative primary structure of elementary particles, it fills uniformly the ''empty space''. Thus, this dark energy may correspond to the aether [Kra] of the physicists of the nineteenth century! … leading to a paradox since the aether wind was ruled out by the Michelson-Morley experiment. This was the starting point of the special relativity of Einstein who postulated the invariance of the light velocity in the vacuum. What is the solution of this apparent paradox? The aether envisaged by the physicists of the nineteenth century was supposed to correspond to ''visible'' matter (the only defined matter at that time) which is composed, in the present context of algebraic quantum theory, of three embedded structure shells ST M G M (ST space-time, MG middle ground, M mass) defining the structure of (visible) massive elementary particles as

10

it will be seen in the following chapter. This is also the case of the light of which photons have a structure in three embedded shells ST M G M . Now, the dark energy, viewed as an aether, has an elementary particle composition made only of the ''ST '' substructure. Consequently, the only relevant interaction between the visible light and the ''dark energy aether'' seems to be at first sight between the ''ST '' level of the dark energy and the boundary exterior shell '' M '' of the visible light as it will be discussed in chapter 4. But, as the rotational velocities, i.e. the frequencies of the shells ''ST '' and '' M '' are very different, no real quanta of interaction between these two shells can be effectively exchanged and thus, no real interaction exists between these two shells. On the other hand, interactions still exist between the ''ST '' structure of the dark energy and the ''ST '' substructure of the light. But, as the ''ST '' vacuum substructure of the light is bounded by the '' MG '' and '' M '' exterior shells, no real quanta can be added to this one in order to increase its frequency. Thus, it can be said that the dark energy could be interpreted as a new kind of aether differing from the aether of the nineteenth century physicists by the facts that:

(a) the dark energy does not really affect the velocity of the visible light; (b) the dark energy envisaged as an aether is not an absolute reference system.

3 Dark matter and visible matter structure of elementary particles

3.1 Singular universal mathematical structures

As introduced in section 2.7, the dark energy structure of an elementary fermion (i.e. a (bisemi)-lepton e , , or a (bisemi)quark u , d , s , c , b , t ) but, also of a (bisemi)baryon (as resulting from section 2.5), is a universal mathematical structure in the context of the global program of Langlands. It will then be seen in this chapter that the dark matter and the visible matter structures of an elementary fermion correspond to ''singular'' universal mathematical structures obtained from its dark energy vacuum substructure by versal deformations and blowups of these. These dark and visible matters of elementary particles are then deformations of states of the vacuum space time fields of their dark energy structure as developed in [Pie1].

3.2 One- and two-dimensional sections of the vacuum space-time fields

Let ST ST( )

p p p p

R L

T S T SM M

denote the ''vacuum'' space-time fields of the dark energy structure of an

elementary bisemifermion as developed in section 2.5. As these vacuum space-time fields have a spatial extension of the order of the Planck length, they are submitted to strong fluctuations which generate singularities on their sections.

Let 2 , 2 ,2 2

,{ ( ) ( )}p p

m m

S S

R v L v mM M denote the set of bisections of the bisemisheaf of rings

( ( )) ( ( ))p pS S

R R v L L vM L M L being a space vacuum string field as introduced in proposition 2.4 and

noted there ST ST

p p

R L

S SM M .

11

Then, 2 , 2

,{ ( )}p

m

S

L v mM (resp.

2 , 2,{ ( )}p

m

S

R v mM ) will be the set ( ( ( )))pS

L L vM L (resp.

( ( ( )))pS

R R vM L ) of left (resp. right) one-dimensional sections of the left (resp. right) semisheaf

( ( ))pS

L L vM L (resp. ( ( ))pS

R R vM L ) [Har].

Assume that the '' m '' one-dimensional sections 2 , 2

{ ( )}p

m

S

L v mM

(resp. 2 , 2

{ ( )}p

m

S

R v mM

) of each

packet '' '' are glued together under a compactification map '' c '' in order to generate a surface

2 ( )( )pS

L v cM

(resp. 2 ( )( )pS

R v cM

) as described in [Pie3] and in [Pie8].

Then, 2 ( ){ ( )}pS

L v cM (resp.

2 ( ){ ( )}pS

R v cM ) will denote the set ( )( ( ( )))pS

L L v cM L (resp.

( )( ( ( )))pS

R R v cM L ) of left (resp. right) two-dimensional sections of the left (resp. right) semisheaf

( )( ( ))pS

L L v cM L (resp. ( )( ( ))pS

R R v cM L ).

3.3 Degenerate singularities on the sections of the vacuum ''ST'' fields

Under external perturbations due to the strong fluctuations at the Planck scale, degenerate singularities are produced on the left and right sections of the above mentioned semisheaves. As an elementary particle has a spatial extension of the order of the Planck length, it is assumed that a same kind of degenerate singularity is generated on each left (resp. right) section which is a left (resp. right) differentiable function (resp. cofunction). On one- or two-dimensional sections, the simple germs 1( ) kf x x , 1 3k , which are singular points of corank 1 and multiplicity '' 1k '', can be produced in a 3-dimensional (semi)space by singularization morphisms which are defined as sets of contracting surjective morphisms of normal crossing divisors as developed in [Pie4]. On a two-dimensional section, the possible degenerate singular points are essentially the following germs of corank 2 and multiplicity inferior or equal to 3 [Tho], [Arn]: 3 2 3 3( , ) 3 , ( , ) .f x y x y x f x y x y

They are also produced by a set of contracting surjective morphisms of normal crossing divisors.

3.4 Versal deformations of degenerate singularities

Under the same kind of external perturbations, these degenerate singular germs are submitted to versal deformations interpreted as extensions of the contracting surjective morphisms of singularizations as proved in [Pie4]. The versal deformation or unfolding of a germ 1( ) kf x x of corank 1 and multiplicity '' 1k '' is given, in the frame of the Malgrange preparation theorem, by:

1

1

( , ( , )) ( ) ( , ) .k

ii

i

F x a y z f x a y z x

12

1

1

( , ( , )) ( , )k

ii i

i

R x a y z a y z x

is the polynomial of the quotient algebra of the versal unfolding of the

degenerate germ ( )f x with:

1 1{ , , , , }i kx x x being the basis of this quotient algebra of dimension ( 1)k , which is thus finitely generated;

( , )ia y z being a function of two variables on which is mapped the respective generator ix

as developed in [Tho] and [Mal]. This function ( , )ia y z belonging to a section of the space vacuum semifield is two-dimensional

because the set of sections of the left (resp. right) semisheaf ( ( ))pS

L L vM L (resp. ( ( ))pS

R R vM L ) is

assumed to be compactified in a three-dimensional spatial semispace as developed in [Pie2] and justified in [Pie1]. More concretely, the versal unfolding of the degenerate germs 1( ) kf x x in codimension inferior or equal to thee are (to a translation):

1) the fold: 3( )f x x of which versal unfolding in codimension 1 is 3 1

1 1( , ) ;F x a x a x

2) the cusp: 4( )f x x of which versal unfolding in codimension 2 is 4 1 2

1 2 1 2( , , ) ;F x a a x a x a x

3) the swallowtail: 5( )f x x of which versal unfolding in codimension 3 is 5 1 2 3

1 2 3 1 2 3( , , , ) ;F x a a a x a x a x a x

where ia is a contracted notation for ( , )ia y z , 1 3i .

The degenerate germs of corank 2 and multiplicity inferior of equal to 3 are [Tho] (to a translation):

1) the elliptic umbilic: 3 2( , ) 3f x y x xy of which versal unfolding in codimension 3 is 3 2 2 2

1 2 1 2( , , , ) 3 ( ) ;F x y b b x xy b x y b y

2) the hyperbolic umbilic: 3 3( , )f x y x y of which versal unfolding in codimension 3 is 3 3

2 3 4 2 3 4( , , , , )F x y b b b x y b y b x b xy

where 2b and 3b are two variable functions and 1b and 4b are one variable functions on a

section of the space vacuum semifield.

More generally, let ( )pS

L LM (resp. ( )pS

R RM ) denote the left (resp. right) semisheaf of space

vacuum field of which sections: 1) are one- or two-dimensional; 2) are affected by the same kind of degenerate singularities of corank 1 or 2 as developed in

sections 3.2 and 3.3.

13

Then, the versal deformation of the semisheaf ( )pS

L LM (resp. ( )pS

R RM ) of differentiable

functions (resp. cofunctions) endowed with singular germs of corank 1 or 2 is given by the contracting fiber bundle:

: ( ) ( )

(resp. : ( ) ( ))

p p

L L

p p

R R

S S

S L L S L L

S S

S R R S R R

D M M

D M M

in such a way that the fiber 11{ , ( ), }

L

i kS L ix

(resp. 11{ , ( ), }

R

i kS R ix

) is given by the set

of ( 1)k semisheaves of the base LS (resp. RS ) of the considered versal deformation where

( )iL x (resp. ( )i

R x ) denotes the semisheaf of monomials '' ix '' with respect to all the sections of

( )pS

L LM (resp. ( )pS

R RM ).

These semisheaves ( )iL x (resp. ( )i

R x ) of monomials are projected on the respective coefficient

semisheaves ( )L ia (resp. ( )R ia ) or ( )L ib (resp. ( )R ib ) of which sections are respectively the

coefficient functions ia or ib .

As the semisheaf ( )pS

L LM (resp. ( )pS

R RM ) is defined on the algebraic semigroup LM (resp.

RM ) according to section 2.3, the semisheaves of monomials ( )iL x (resp. ( )i

R x ) and the

semisheaves of coefficients are also algebraic and characterized by a same set of increasing ranks, being algebraic dimensions defined from global residue degrees as developed in [Pie4].

3.5 Blowups of the versal deformations

A blowup of the versal deformation

LSD (resp. RSD ) can be envisaged: it consists in the

extension of the quotient algebra of the versal deformation and corresponds to the inverse versal deformation 1

LSD (resp. 1

RSD ). It is based on the following smooth endomorphism:

[ ( )] ( ) ( ) , 1 1,

(resp. [ ( )] ( ) ( ) ) ,

i

i

i i iL L r L Ix

i i iR R r R Ix

E x x x i k

E x x x

based on Galois antiautomorphisms [Pie1], [Pie4], where: ( )i

L rx (resp. ( )iR rx ) is the residual monomial semisheaf on the respective coefficient

semisheaf; ( )i

L Ix (resp. ( )iR Ix ) is the complementary monomial semisheaf disconnected from the

respective coefficient semisheaf on which it was projected. Let

LS (resp. RS ) denote the set

1 -11 1{ [ ( )]} (resp. { [ ( )]} )i i

i k i kL i R ix x

E x E x

of smooth endomorphisms disconnecting totally the monomial semisheaves ( )iL x (resp. ( )i

R x )

from the respective coefficient semisheaves. Let { , , }

x iL xV VT T (resp. { , , }

x iR xV VT T ) denote the set of tangent bundles obtained by

projecting all the disconnected monomial base semisheaves ( )iL Ix (resp. ( )i

R Ix ) in the vertical

tangent space.

14

Then, the extension of the quotient algebra of the versal deformation of the singular semisheaf

( )pS

L LM (resp. ( )pS

R RM ), having an isolated degenerate singularity on each section, is realized

by the spreading out isomorphism ( ) (resp. ( )) .

x L x RL RL V S R V SS T T S T T

as developed in [Pie1] and [Pie4]. Let ( )i

L x (resp. ( )iR x ) and ( )j

L x (resp. ( )jR x ) be two monomial base semisheaves of the

base LS of the versal deformation.

Then, ( )i jL LDx (resp. ( )i j

R RDx ) will denote the gluing up of these two monomial semisheaves

on a connected domain i jLDx (resp. i j

RDx ).

Let (1)p

L

S

S T (resp. (1)p

R

S

S T ) be the set LS (resp.

RS ) of the base monomial semisheaves totally

disconnected from pS

L (resp. pS

R ) by the blowup 1

LSD (resp. 1

RSD of the versal deformation) in

such a way that these monomial semisheaves: a) are glued together section by section;

b) cover partially the residue singular semisheaf pS

L (resp. pS

R ), in the sense that each

section of pS

L (resp. pS

R ) is totally or partially covered by the corresponding section of

(1)p

L

S

S T (resp. (1)p

R

S

S T ) obtained by gluing up the base monomials of the versal deformation.

Remark that the blowup 1

LSD (resp. 1

RSD ) of the versal deformation has been envisaged as being

maximal, i.e. when the base monomial semisheaves are totally disconnected from the singular

semisheaf pS

L (resp. pS

R ). The intermediate cases, i.e. when the monomial semisheaves are

partially disconnected from pS

L (resp. pS

R ), are studied in [Pie1] and in [Pie4].

3.6 Sequence of blowups of versal deformations

Referring to section 3.4, it appears that the blowup of the versal deformation of the swallowtail

5 1 2 31 2 3 1 2 3( , , , )F x a a a x a x a x a x generates a singular base monomial 3( )f x x .

Consequently, the blowup of the versal deformation of the singular semisheaf pS

L (resp. pS

R )

of which sections are affected by degenerate singularities of type swallowtail generates

monomial base semisheaves (1)p

L

S

S T (resp. (1)p

R

S

S T ) of which 3( )L x (resp. 3( )R x ), case 3i , is

again a singular semisheaf noted 3( )pS

L x (resp. 3( )pS

R x ).

This singular semisheaf 3( )pS

L x (resp. 3( )pS

R x ) can then be submitted to a versal deformation

and a blowup of it generating the monomial base semisheaf (2) ( )L x (resp. (2) ( )R x ) covering

partially by patches the semisheaf (1)p

L

S

S T (resp. (1)p

R

S

S T ).

Thus, in the case of a singular semisheaf ( )pS

L LM (resp. ( )pS

R RM ) of swallowtail type, the

two semisheaves

15

p

p

S

(1) (1)

S

(2) (2) (2) (2)

(resp. )

and ( ) (resp. (x))

,

p

L R

p

L R

S

S T S T

S

S T L S T Rx

generated by versal deformation and blowup of this one, cover partially the residual singular

semisheaf ( )pS

L LM (resp. ( )pS

R RM ) leading to the embedding:

(1) (2)

(1) (2)

( )

(resp. ( ) ) .

p p p

L L

p p p

R R

S S S

L L S T S T

S S S

R R S T S T

M

M

Referring to section 3.2, the semisheaf ( )pS

L LM (resp. ( )pS

R RM ) is a space vacuum semifield of

the dark energy vacuum ''ST'' substructure of an elementary particle. The covering semifields

(1)p

L

S

S T (resp. (1)p

R

S

S T ) and (2)p

L

S

S T (resp. (2)p

L

S

S T ) are also of space nature.

The corresponding semifields of ''time'' type can be generated by a composition of morphisms

r t LE (resp. r t RE ) as described in section 2.5:

*

*

: ( ) ( ) ( )

(resp. : ( ) ( ) ( )) .

p p p

p p p

S T S

r t L L L L L L L

S T S

r t R R R R R R R

E M M M

E M M M

Let us use as in chapter 2 the more condensed notation p p

L

T S

STM

(resp. p p

R

T S

STM

) for *( ) ( )p pT S

L L L LM M (resp. *( ) ( )p pT S

R R R RM M ).

Similarly, time semifields can be generated by ( )r t LE (resp. ( )r t RE ) morphisms from

space semifields (1)p

L

S

S T (resp. (1)p

R

S

S T ) and (2)p

L

S

S T (resp. (2)p

R

S

S T ) leading to covering space-time

semifields (1)p p

L

T S

S T (resp. (1)

p p

R

T S

S T ) and (2)

p p

L

T S

S T (resp. (2)

p p

R

T S

S T ).

These two covering space-time semifields will be respectively labeled middle ground ''MG'' (or darkmass ''DM'') and mass ''M'', as introduced in [Pie1], and will be noted according to:

(1) (1)

- -- -

(1) (1)

-

DM

-

(2) (2)

MG or

(resp. or )

and (res . ) . p

p pp p p p p p

LL L L

p p p pp p p p

R RR R

p p p pp p p p

L RL R

MG D

T ST S T S T S

S T S T

T S T ST S T S

S T S T

T S T ST S T S

S T S T

M

M M

M M

M M

M M

Thus, degenerate singularities of corank 1 and codimension 3 on the semisheaf ( )

p p

L

S S

L LSTM M (resp. ( )p p

R

S S

R RSTM M ) are able to generate the three embedded

semisheaves:

R R R

-

ST DM M

ST D

-

M M

-(resp. ) .

p p p p p p

L L L

p p p p p p

T S T S T S

T S T S T S

M M M

M M M

Remark that instead of considering degenerate singularities on a vacuuum semifield of ''space type''

ST ( )p p

L

S S

L LM M , we could have envisaged them on the corresponding vacuum semifield of ''time

type'' ST ( )p p

L

T TL LM M leading similarly to the same three embedded semisheaves of type

ST DM M as described explicitly in [Pie1].

16

3.7 Proposition (Dark matter structure)

The dark matter structure of an elementary fermion is composed of the tensor product of the right and left direct sums of its vacuum space-time semifields of type ''ST'' and ''DM'':

ST DM ST DM

ST ST DM DM

ST DM DM Sint

T

( ) (D D S )MS Mp p p p p p p p

R R L L

p p p p p p p p

R L R L

p p p p p p p p

R L R L

T S T S T S T S

R R L L

T S T S T S T S

T S T S T S T S

F F M M M

M M M M

M M M

M

M

where:

DMS ( )L LF (resp. DMS ( )R RF ) denotes the left (resp. right) dark matter structure of an

elementary semifermion LF (resp. RF );

ST ST

p p p p

R L

T S T SM M

and DM DM

p p p p

R L

T S T SM M

denote respectively the vacuum space-time ''ST''

and dark mass ''DM'' fields of this bisemifermion ( )R LF F ;

i

ST DM DM STnt

p p p p p p p p

R L R L

T S T S T S T SM M M M

are interaction fields between the ''ST'' and

''DM'' substructures of ( )R LF F .

This dark matter structure is generated from degenerate singularities of codimension inferior to 3

on the space (time) semifields )

ST

( p p

R

T SM

and

)

ST

( p p

L

T SM

of type ''ST''.

Proof. It was seen in sections 3.4 and 3.5 that degenerate singularities of corank 1 and codimensions 1 and 2 as well as singularities of corank 2 and codimension 3 on the space (time)

semifields )

ST

( p p

L

T SM

(resp.

)

ST

( p p

R

T SM

) of the dark energy structure ''ST'' of an elementary left (resp.

right) semifermion LF (resp. RF ) can generate by versal deformations and blowups of these the

space (or time) semifields )

DM

( p p

L

T SM

(resp.

)

DM

( p p

R

T SM

) of its dark mass covering substructure:

( ) ( ) ( )

( ) ( -) ( )

ST ST DM

ST ST DM

( ) :

(resp. ( ) : )

p p p p p p

L L LL

p p p p p p

R R RR

T S T S T S

L S

T S T S T S

R S

S T D M M M

S T D M M M

where LSD (resp.

RSD ) denotes the versal deformation and LS T (resp. RS T ) its blowup.

So, the tensor product of the right and left direct sums of the dark energy semifields ''ST'' and of the

dark mass ''DM'' semifields generates the dark energy substructure ST ST( )p p p p

R L

T S T SM M and the dark

mass covering substructure DM DM( )

p p p p

R L

T S T SM M

of the considered elementary bisemifermion

( )R LF F as well as the interaction fields ST DM( )

p p p p

R L

T S T SM M

and

DM ST( )p p p p

R L

T S T SM M

between these

two substructures. As developed in section 2.5, each of these four fields decompose into bilinear diagonal, magnetic and electric subfields.

17

As mentioned in section 2.8, it is likely that the interactions between the dark energy subfields and the dark mass subfields are not very important since their frequencies are supposed not to be of the same order.

The dark mass substructure DM DM( )

p p p p

R L

T S T SM M

of an elementary bisemifermion is a ''mass''

structure field because it is of contracting nature limiting the expanding nature of the internal

vacuum dark energy substructure ST ST( )

p p p p

R L

T S T SM M

. Furthermore, it is generated from its

vacuum dark energy field by versal deformation and blowup of this one as the visible mass covering field of an elementary bisemifermion will be seen to be generated from its vacuum dark mass field by versal deformation and blowup of it.

Remark that the ''dark mass'' would refer to the time field DM DM( )

p p

R L

T TM M of the dark mass

substructure, as it is commonly envisaged for the visible mass, i.e. the proper mass, of a fermion. ■

3.8 Proposition (Visible matter structure)

The visible matter structure of an elementary fermion is composed of the tensor product of the right and left direct sums of its vacuum semifields of dark energy and dark mass and of its visible mass semifields:

ST DM M ST DM M

ST ST DM DM M M

ST DM ST M

DM ST DM

VMS V )M( S) (

p p p p p p p p p p p p

R R R L L R

p p p p p p p p p p p p

R L R L R L

p p p p p p p p

R L R L

p p p p

R L R

R R L L

T S T S T S T S T S T S

T S T S T S T S T S T S

T S T S T S T S

T S T S

F F

M M M M M M

M M M M M M

M M M M

M M M

M

M Sint

T M DM

p p p p

L

p p p p p p p p

R L R L

T S T S

T S T S T S T S

M

M M M M

where: VMS ( )L LF (resp. VMS ( )R RF ) denotes the left (resp. right) visible matter structure of an

elementary semifermion LF (resp. RF );

M M( )

p p p p

R L

T S T SM M

denotes the visible mass covering substructure ''M'' of this bisemi-

fermion ( )R LF F ;

the fields inside int[·] are interaction fields between the three different right and left

semifields ''ST'', ''MG'' and ''M''. The visible matter structure is generated from degenerate singularities of corank 1 and

codimension 3 on the space (time) semifields ST

p p

R

T SM

(resp. ST

p p

L

T SM

) of the dark energy ''ST''

substructure.

18

Proof. It was seen in section 3.6 that degenerate singularities of corank 1 and codimension 3 on the

space (or time) semifields )

ST

( p p

L

T SM

(resp.

)

ST

( p p

R

T SM

) of the dark energy structure of an elementary

left (resp. right) semifermion LF (resp. RF ) can generate by two successive versal deformations

and blowups of these the space (or time) semifields )

DM

( p p

L

T SM

(resp.

)

DM

( p p

R

T SM

) and (

M

)p p

L

T SM (resp.

)

M

()

p p

R

T SM

of its dark mass and visible mass covering substructures.

So, the tensor product of the right and left direct sums of the dark energy semifields ''ST'', dark mass semifields ''DM'' and visible mass semifields ''M'' generate the visible matter structure ( ( ) ( ))VMS VMSR R L LF F of the considered elementary bisemifermion ( )R LF F in a way similar

to that envisaged in proposition 3.7 with the difference that the visible mass covering substructure ''M'' is generated from the vacuum substructure ''ST DM '' of this bisemifermion ( )R LF F

giving it a mass.

This visible mass substructure M M( )

p p p p

R L

T S T SM M

is a ''mass'' structure field because it is of

contracting nature stabilizing its vacuum substructure, i.e. its dark matter structure D ( ) ( )MS DMSR R L LF F .

This mass substructure ''M'' is visible because the frequency (or rotation) of its (bi)sections is inferior to that of the dark energy and dark mass substructures taking into account that the number of quanta on the sections of the mass substructure is inferior to the number of quanta on the respective sections of the dark mass and dark energy substructures: this results clearly from the generation of these substructures (see sections 3.4 and 3.5). ■

3.9 How can we explain the preponderance of dark matter over ordinary visible matter in our universe?

The hypothesis developed in this paper concerning the generation of dark matter and visible matter on the basis of embedded substructures constituting the internal structure of elementary particles allows to explain the preponderance of dark matter over visible matter. Indeed, the visible matter structure of elementary particles is composed of the three embedded shells ST DM M in such a way that the external shell ''M'' is generated from the middle ground shell ''DM'' on the basis of degenerate singularities of corank 1 and codimension 3 on the vacuum shell ''ST'', when the dark matter structure is composed of two embedded shells ST DM of which shell ''DM'' is generated from degenerate singularities of corank 1 or 2 and codimension 3 on the vacuum shell ''ST''. The generation of the visible mass shell ''M'' needs thus much more energy than the generation of the dark mass shell ''DM''. This has been verified in galaxies where dark matter is observed outside a certain radius R while visible matter seems localized inside this radius where the acceleration rate is much more important.

19

3.10 Dark matter and visible matter structure for all kinds of particles

This chapter was devoted to the study of the generation of dark matter and visible matter of elementary (bisemi)fermions, i.e. the (bisemi)leptons and their neutrinos as well as the (bisemi)quarks, from their vacuum dark energy substructures. It is evident that the dark mass ''DM'' and the visible mass ''M'' substructures of the (bisemi)-baryons and of the (bisemi)bosons (i.e. the mesons and the (bisemi)photons which are exchange (bisemi)particles), are generated similarly from their vacuum dark energy substructures ''ST'' according to [Pie1], noticing for example that the bisections (which are open bistrings) of the fields ''DM'' and ''M'' are bound bisemiphotons at these corresponding levels.

3.11 Proposition

The dark matter structure and the visible matter structure of an elementary particle are universal singular mathematical structures. Proof. According to proposition 2.7, the dark energy structure ''ST'' of an elementary particle is a universal mathematical structure. Due to strong external perturbations, this dark energy structure ''ST'' will join up the dark mass ''DM'' and/or the visible mass ''M'' covering substructures according to the nature of the degenerate singularities on the internal vacuum substructure ''ST''. Each space or time substructure ''ST'' or ''DM'' is then a singular bisemisheaf of differentiable bifunctions, i.e. a physical field, according to section 3.5. This singular bisemisheaf, being the analytic representation of the product, right by left, of global Weil groups in the frame of the global program of Langlands, gives rise to the product, right by left, of (truncated) holomorphic functions (and/or cuspidal forms) if the suitable desingularizations are envisaged on these as developed in [Pie4]. This explains why the dark matter structure

ST ST DMDM( ) ( )DMS DMSp p p p p pp p

R L LR

T S T S T ST S

R R L LF F M M M M

and the visible matter structure

ST DM M ST DM M( ) ( )VMS VMSp p p p p p p p p p p p

R R R L L L

T S T S T S T S T S T S

R R L LF F M M M M M M

of an elementary (bisemi)fermion (but also of every elementary (bisemi)particle) are respectively universal twofold and threefold singular mathematical structures. ■

4 The dark energy and the dark matter structures of elementary particles interacting with the visible matter structure of other elementary particles

4.1 Bilinear interactions (dark energy case of bisemifermions)

Let ST ST( )

p p p p

R L

T S T SM M

,

ST MG ST( )p p p p

R R L L

T S T S

MGM M

and MST MG M MGST( )

p p p p

R R R L L R

T S T SM M denote

respectively the dark energy, dark matter and visible matter structures of an elementary bisemifermion (see section 3.1), the notation ''MG'' replacing here the notation ''DM''.

20

Let ST ST( ( ) ( ))

p p p p

R L

T S T SM I M I

denote the dark energy structure of '' I '' interacting elementary

bisemifermions on this unique level. It is the completely reducible nonorthogonal functional

representation space of the bilinear complete (algebraic) semigroup ST2 (G )L p pT S

I v vL L [Pie6]:

ST S ST2T FREPs( ) ( ) (p GL ( )) . p p p p p p

R L

T S T S T S

I v vM I M I L L

Its time part 2 ST( (F ))R P L E sp G pT

I v vL L can be expressed as the cross binary product '' '' of

its ''I '' bielements according to [Pie4], [Pie7]:

1

2

2

2

2

2

ST

ST

ST

ST

ST ST

ST

21 1

2

1

FREPsp GL

FREPsp GL

FREPsp GL

FREPsp GL

FREPsp FREPsp

FREPsp

( ( ))

( ( ))

( ( ))

( ( ))

( ( ) ) ( ( )

( (GL

)

))

p

p

p

i

p p

I

p p

i i

p

i

T

I v v

T

v v

T

v v

T Sv v

I IT Tt t

v vi i

IT

v vii

L L

L L

L L

L L

T L T L

L L

ST

ST ST ST S1

T

2 21

1

( ( ) ( ))FREP

( ( ) ( ))

p

( ( ( ))

s

)

p

i i

p p p p

R L R L

ITt

v vi

I IT T T T

i i i

T L T L

M i M i M i M i

splitting ST ST( ( ) ( ))

p p

R L

T TM I M I into '' I '' free time fields ST ST 1{( ( ) ( ))}

p p

R L

T T IiM i M i of '' I ''

bisemifermions and into 2( )I I interacting time semifields ST 1ST{( ( ) ( ))}

p p

R L

T T Ii iM i M i of the

right and left semifermions belonging to different bisemifermions.

4.2 Proposition (Dark energy structure of I bisemifermions)

The dark energy structure of '' I '' interacting elementary bisemifermions is given by:

ST ST ST ST ST ST1 1

( ( ) ( )) ( ) ( ) ( ) ( )p p p p p p p p p p p p

R L R L R L

I IT S T S T S T S T S T S

i i i

M I M I M i M i M i M i

where:

a) each free space-time field ST ST( ( ) ( ))

p p p p

R L

T S T SM i M i

of the dark energy structure of an

elementary bisemifermion decomposes into a four-dimensional bilinear diagonal subfield, a three-dimensional off-diagonal magnetic subfield and into a three-dimensional off-diagonal electric subfield;

b) each product, right by left, of space-time interacting semifields ST ST( ( ) ( ))

p p p p

R L

T S T SM i M i

of

the right and left dark energy semistructures of the interacting bisemifermions i and i decomposes into a four-dimensional bilinear diagonal gravitational field, a three-dimensional interacting magnetic subfield and into a three-dimensional interacting electric subfield.

Proof.

1. The space dark energy structure of I interacting bisemifermions can be handled similarly as it has been done for their time part in section 4.1 taking into account the morphism

21

( ) ( )t r R t r LE E transforming the time dark energy structure into its complementary

space dark energy structure as developed in section 2.5.

2. The decomposition of the space-time internal fields ST ST( ( ) ( ))

p p p p

R L

T S T SM i M i

of an

elementary bisemifermion '' i '' into diagonal, magnetic and electric subfields has been recalled in section 2.5.

3. The product, right by left, of the space-time interacting semifields i and i decomposes similarly according to:

ST ST ST ST

S

( ) ( )

magn elT ST T ecS ST

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

p p p p p p p p

R L R L

p p p p p p

R L R L

T S T S T S T S

D

S S S T T S

M i M i M i M i

M i M i M i M i

where:

ST ST( ( ) ( ))

p p p p

R L

T S T S

DM i M i

is a space-time gravitational field between the dark energy

structures of the i -th right semifermion and the i -th left semifermion as justified in [Pie1]. This gravitational field is generated by the exchange of (mixed) orthogonal gravitational biquanta;

magnST ST( ( ) ( ))

p p

R L

S SM i M i is a spatial magnetic field between the dark energy structures

of the i -th right semifermion and the i -th left semifermion. This magnetic field is generated by the exchange of off-diagonal spatial magnetic biquanta.

( ) ( )

eleST Sc T( ( ) ( ))p p p p

R L

S T T SM i M i is a space-time (or time-space) electric field generated by

the exchange of space-time (or time-space) electric biquanta between the dark energy structures of the i -th right semifermion and the i -th left semifermion. ■

4.3 Proposition (Dark mass structure of J (interacting) bisemifermions)

The dark mass structure of ''J '' interacting elementary bisemifermions is given by:

ST MG ST MG

ST MG ST MG

ST ST MG MG

ST M

1

1G

1

( ( ) ( ))

( ( ) ( )) ( ( ) ( ))

( ( ) ( )) ( ( ) ( ))

( ( ) ( ))

p p p p

R R L L

p p p p p p p p

R R L L

p p p p p p p p

R L R L

p p p p

R L

T S T S

T S T S T S T S

J JT S T S T S T S

j j

J T S T S

j j

M J M J

M J M J M J M J

M j M j M j M j

M j M j

1

1 1

1

MG ST

ST ST MG MG

ST MG MG S1

T

( ( ) ( ))

( ( ) ( )) ( ( ) ( ))

( ( ) ( )) ( ( ) ( ))

p p p p

R L

p p p p p p p p

R L R L

p p p p p p p p

R L R L

J T S T S

J JT S T S T S T S

j j j j

J JT S T S T S T S

j j j j

M j M j

M j M j M j M j

M j M j M j M j

where: a) the direct sum of fields inside the first parenthesis {·} refers to the internal structure

fields ''ST'' and ''MG'' of the J free dark massive bisemifermions as well as to the

22

interacting internal electro-magnetic fields between the right ''ST'' (resp. right ''MG'') semistructures and the left ''MG'' (resp. left ''ST'') semistructures of these free dark massive bisemifermions;

b) the direct sum of interacting bilinear fields inside the second parenthesis {·} refers to the gravito-electro-magnetic fields between the ''ST'' and ''MG'' right and left shells belonging to different bisemifermions as well as the gravito-electro-magnetic fields between mixed right and left ''ST'' and ''MG'' shells belonging to different bisemifermions.

Proof.

a) The dark mass structure of J interacting elementary bisemifermions

ST MG ST MG( ( ) ( ))p p p p

R R L L

T S T SM J M J

is given by the tensor product of right and left direct sums of

their space-time ''ST'' and middle-ground ''MG'' semistructures on the consideration that it is a completely reducible non-orthogonal functional representation space of the mixed bilinear

complete (algebraic) semigroup S G2 T M( )GL p pT S

J v vL L according to section 4.1.

b) Let us analyze the different fields responsible for the dark mass structure of these interacting bisemifermions

1) T T

1S S( ( ) ( ))

p p p p

R L

J T S T S

j

M j M j

and G G

1M M( ( ) ( ))

p p p p

R L

J T S T S

j

M j M j

are respectively the ''ST''

and ''MG'' internal fields of these J bisemifermions decomposing into diagonal, magnetic and electric subfields as envisaged in proposition 4.2.

2) T G

1S M( ( ) ( ))

p p p p

R L

J T S T S

j

M j M j

and G T

1M S( ( ) ( ))

p p p p

R L

J T S T S

j

M j M j

are respectively mixed

interaction semifields between the right ''ST''-left ''MG'' and the right ''MG''-left ''ST'' semistructures of these free bisemifermions. These gravito-electro-magnetic interacting semifields between mixed subsemistructures are likely not very relevant because the frequencies of the ''ST'' and ''MG'' shells are not of the same order preventing a real exchange of biquanta.

3) 1

ST ST( ( ) ( ))p p p p

R L

J T S T S

j j

M j M j

and 1

MG MG( ( ) ( ))p p p p

R L

J T S T S

j j

M j M j

are respectively

interaction semifields between the ''ST'' and ''MG'' right and left semistructures belonging to different bisemifermions. They decompose usually respectively into ''ST'' and ''MG'' gravito-electro-magnetic fields.

4) 1

ST MG( ( ) ( ))p p p p

R L

J T S T S

j j

M j M j

and 1

MG ST( ( ) ( ))p p p p

R L

J T S T S

j j

M j M j

are respectively mixed

''ST MGR L '' and '' MG STR L '' interaction semifields between different right and left

semifermions. Their decomposition into GEM (gravito-electro-magnetic) fields does not probably give rise to an effective exchange of biquanta. ■

4.4 Proposition (Visible mass structure of K (interacting) bisemifermions)

The visible mass structure of '' K '' interacting elementary bisemifermions is given by:

23

ST MG M ST MG M

ST MG M ST MG M

ST ST MG MG M1

M1 1

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

p p p p

R R R L L L

p p p p p p p p p p p p

R R R L L L

p p p p p p p p p p p

R L R L R L

T S T S

T S T S T S T S T S T S

K K KT S T S T S T S T S T S

k k k

M K M K

M K M K M K M K M K M K

M k M k M k M k M k M

ST MG MG ST ST M

M ST

1 1 1

1 1 1MG M M MG

( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

p

p p p p p p p p p p p p

R L R L R L

p p p p p p p p p p p p

R L R L R L

K K KT S T S T S T S T S T S

k k k

K K KT S T S T S T S T S T S

k k k

k

k

M k M k M k M k M k M k

M k M k M k M k M k M k

ST ST MG MG

M M

ST MG ST M

1

1 1

M

1

1

1

( ) ( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

(

p p p p p p p p

R L R L

p p p p

R L

p p p p p p p p

R L R L

p p

R

K KT S T S T S T S

k k k

K T S T S

k k

K KT S T S T S T S

k k k k

K T S

k k

M k M k M k M k

M k M k

M k M k M k M k

M

MG) ( )p p

L

T Sk M k

where: a) the direct sum of fields inside the first parenthesis {·} refers to the internal structure fields

''ST'', ''MG'' and ''M'' of the ''K'' free (visible) massive bisemifermions as well as to the interacting internal mixed electro-magnetic fields between the right ''ST'', ''MG'' and ''M'' semistructures of these free massive bisemifermions and their corresponding left mixed equivalents;

b) the direct sum of interacting bilinear fields inside the second parenthesis {·} refers to the

gravito-electro-magnetic fields between the ''ST'', ''MG'' and ''M'' right and left shells belonging to different bisemifermions as well as to the GEM fields between mixed right and left ''ST'', ''MG'' and ''M'' shells of these.

Proof. The visible mass structure of interacting bisemifermions corresponds to their dark mass

structure to which the covering ''M'' fields M M( ( ) ( ))

p p p p

R L

T S T SM k M k

have been added.

Thus, the different fields responsible for this visible mass structure are those of the corresponding (i.e. at '' K '' interacting massive bisemifermions) dark mass structure plus:

1) M M

1

( ( ) ( ))p p p p

R L

K T S T S

k

M k M k

which are the ''M'' internal fields of the K bisemifermions

decomposing into diagonal, magnetic and electric subfields;

2) the mixed interaction semifields T M

1T

1S M S( ( ) ( )) ( ( ) ( ))

p p p p p p p p

R L R L

K KT S T S T S T S

k k

M k M k M k M k

G M

1G

1M M M( ( ) ( )) ( ( ) ( ))

p p p p p p p p

R L R L

K KT S T S T S T S

k k

M k M k M k M k

between the right ''ST'' and ''MG''-

left ''M'' and the right ''M''-left ''ST'' and ''MG'' semistructures of these free bisemifermions;

24

3) the interactions fields 1

M M( ( ) ( ))p p p p

R L

K T S T S

k k

M k M k

between the ''M'' right and left

semistructures belonging to different bisemifermions; 4) the mixed ''ST MR L '', '' M STR L '', '' MG MR L '' and '' M MGR L '' interaction semifields

ST M M G

1 1M( ) ( ) ( ) ( )

p p p p p p p p

R L R L

K KT S T S T S T S

k k k k

M k M k M k M k

between different left and

right semifermions. ■

4.5 Proposition (Interaction between '' I '' dark energetic bisemifermions and '' K '' visible massive bisemifermions)

The interactions between '' I '' dark energetic bisemifemions and '' K '' visible massive bisemifermions decompose into:

a) '' I '' interacting dark energetic (elementary) bisemifermions and '' K '' interacting visible massive (elementary) bisemifermions;

b) plus the GEM interaction fields between the right (resp. left) dark energy semistructures

of the '' I '' right (resp. left) semifermions and the left (resp. right) ''ST'', ''MG'' and ''M'' subsemistructures of the '' K '' visible semifermions.

Proof. The interaction between the dark energy structure ST ST( ( ) ( ))

p p p p

R L

T S T SM I M I

of I

elementary bisemifermions and the visible mass structure ST MG M ST MG M( ( ) ( ))

p p p p

R R R L L L

T S T SM K M K

of

K different elementary bisemifermions is described by:

ST ST MG M ST ST MG M

ST ST ST MG M ST MG M

ST ST MG M ST MG M ST

( ) ( ) ( ) (

)

( ) ( ) ( ) ( )

( ) ( ) ( )

p p p p p p p p

R R R R L L L L

p p p p p p p p

R L R R R L L L

p p p p p p p

R L L L R R R L

T S T S T S T S

T S T S T S T S

T S T S T S T

M I M K M I M K

M I M I M K M K

M I M K M K M

( )pS

I

where:

a) the set of fields ST ST( ( ) ( ))

p p p p

R L

T S T SM I M I

and

ST MG M ST MG M( ( ) ( ))p p p p

R R R L L L

T S T SM K M K

refers

respectively to the dark energy structure of '' I '' interacting elementary bisemifermions as described in proposition 4.2 and to the visible mass structure of K interacting elementary bisemifemions as described in proposition 4.4;

b) the set of fields ST ST MG M( ( ) ( ))

p p p p

R L L L

T S T SM I M K

(resp. ST MG M ST( ( ) ( ))

p p p p

R R R L

T S T SM K M I

)

refers to the interacting fields between the right dark energy semistructures (resp. right visible mass semistructures) of the '' I '' right semifermions (resp. the '' K '' right semifermions) and the left visible mass semistructures (resp. the left dark energy semistructures) of the '' K '' left semifermions (resp. the '' I '' left semifermions).

We shall only focus on the fields of interaction ST ST MG M( ( ) ( ))

p p p p

R L L L

T S T SM I M K

between right dark

energy semistructures and left visible mass semistructures, the fields

ST MG M ST( ( ) ( ))p p p p

R R R L

T S T SM K M I

being strictly symmetric to the firsts.

25

They develop according to:

ST ST MG M

ST1 1 1

ST MG M

ST ST S

1

T1 1 1

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

p p p p

R L L L

p p p p p p p p

R L L L

p p p p p p

R L R

T S T S

I K K KT S T S T S T S

i k k k

I K IT S T S T S

i k i

M I M K

M i M k M k M k

M i M k M i

G

ST

1

1M

1

M ( )

( ) ( )

p p

L

p p p p

R L

K T S

k

I KT S T S

i k

M k

M i M k

where:

a) T T

1S

1S ( )) ( ( )

p p p p

R L

I KT S T S

i k

M i M k

are fields of interaction between the right dark energy

semistructures of the '' I '' right semifermions and the left '' ST'' subsemistructures of the '' K '' left semifermions.

The ( )i k -th interaction fields ST ST( ( ) ( ))

p p p p

R L

T S T SM i M k

decompose into GEM (gravito-

electro-magnetic) fields at this ''ST'' level as developed in proposition 4.2.

b) Similarly, the ( )i k -th interaction fields ST MG( ( ) ( ))

p p p p

R L

T S T SM i M k

and

ST( ( )p p

R

T SM i

M ( ))

p p

L

T SM k

between the right dark energy semistructure of the i -th right semifermion and,

respectively, the left ''MG'' and ''M'' subsemistructures of the k -th left visible massive semifermion decompose into GEM fields at these ''MG'' and ''M'' levels as noticed in proposition 4.4. It is likely that these interaction fields are not very important as previously noted. ■

4.6 Proposition (Interaction between '' J '' dark massive bisemifermions and '' K '' visible massive bisemifermions)

The interaction between '' J '' dark massive bisemifermions and '' K '' visible massive bisemifermions decomposes into:

a) '' J '' interacting dark massive bisemifermions and '' K '' interacting visible massive bisemifermions;

b) plus the GEM interaction fields between the right (resp. left) dark massive semistructures of the '' J '' right (resp. left) semifermions and the left (resp. right) visible massive semistructures of the '' K '' left (resp. right) semifermions.

Proof. The interaction between the dark mass structure ST MG ST MG( ( ) ( ))

p p p p

R R L L

T S T SM J M J

of '' J ''

elementary bisemifermions and the visible mass structure ST MG M ST MG M( ( ) ( ))

p p p p

R R R L L L

T S T SM K M K

of

'' K '' different elementary bisemifermions is given by:

26

ST MG ST MG M ST MG ST MG M

ST MG ST MG ST MG M ST MG M

ST MG ST MG M ST MG

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

p p p p p p p p

R R R R R L L L L L

p p p p p p p p

R R L L R R R L L L

p p p p

R R L L L R R

T S T S T S T S

T S T S T S T S

T S T S

M J M K M J M K

M J M J M K M K

M J M K M

M ST MG( ) ( )p p p p

R L L

T S T SK M J

where: a) the set of fields in the first parenthesis [(·) (·)] refers respectively to the dark mass

structure of '' J '' interacting elementary bisemifermions as described in proposition 4.3 and to the visible mass structure of '' K '' interacting elementary bisemifermions;

b) the set of fields in the second parenthesis ST MG ST MG M( ( ) ( ))

p p p p

R R L L L

T S T SM J M K

and

ST MG M ST MG( ( ) ( ))

p p p p

R R R L L

T S T SM K M J

refers to the interaction fields between the right dark

matter semistructures (resp. right visible matter semistructures) of the '' J '' right semifermions (resp. '' K '' right semifermions) and the left visible matter semistructures (resp. the left dark matter semistructures) of the '' K '' left semifermions (resp. the '' J '' left semifermions).

As in proposition 4.5, we shall only analyze the fields of interaction

ST MG ST MG M( ( ) ( ))p p p p

R R L L L

T S T SM J M K

between the right dark matter semistructures and the left visible

matter semistructures. They develop according to:

ST MG ST MG M

ST ST MG1 1 1 1

1 1

MG

ST MG

( ) ( )

( ) ( ) ( ) ( )

( ) ( )

p p p p

R R L L L

p p p p p p p p

R L R L

p p p p

R L

T S T S

J K J KT S T S T S T S

j k j k

J KT S T S

j k

M J M K

M j M k M j M k

M j M k

1 1

1 1 1

MG ST

ST M MG M1

( ) ( )

( ) ( ) ( ) ( )

p p p p

R L

p p p p p p p p

R L R L

J KT S T S

j k

J K J KT S T S T S T S

j k j k

M j M k

M j M k M j M k

where:

a) ST

1 1ST( ) ( )

p p p p

R L

J KT S T S

j k

M j M k

and

MG1 1

MG( ) ( )p p p p

R L

J KT S T S

j k

M j M k

are mixed

GEM interaction fields between the ''ST'' right and left semistructures of the '' J '' right and '' K '' left semifermions and between the ''MG'' right and left semistructures of the same set of semifermions;

b) the four other interaction fields are mixed GEM interaction fields between mixed right and left semistructures of the J right and K left semifermions. ■

4.7 Remark

This chapter has treated the interaction between sets of elementary bisemifermions, i.e. the (bisemi)leptons and the (bisemi)quarks, being visible massive, dark massive or dark energetic.

27

The extension of this treatment to (bisemi)baryons will not be envisaged here but can be handled easily by taking into account my preprint [Pie8] and section 2.5.

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