_{1}

^{*}

This contribution is the third of a series of articles devoted to the physics of discrete spaces. After the building of space-time [1] and the foundation of quantum theory [2] one studies here how the three fundamental interactions could emerge from the model of discrete space-time that we have put forward in previous contributions. The gauge interactions are recovered. We also propose an original interpretation of gravitational interactions.

the particle to particle interactions are carried by three sorts of fields, the electroweak field, the strong field, and the gravitation field. The most striking feature is the enormous difference between their intensities. The electric force is stronger than the gravitation force by more than forty orders of magnitude. This is the hierarchy problem.

We have put forward in [

The article is, accordingly, divided in two main sections. In the first section we show how the gauge symmetry interactions naturally emerge from the model of discrete spaces that we propose. In the second section we introduce a new interpretation of gravitation based on a mechanism, somehow similar to the Van der Waals interaction, where quantum wave fluctuations play the role of electric dipole fluctuations.

According to Yang-Mills theory [

Whereas a position increment dx is enough to define the derivative operator in a 4-dimensional continuous space that is into a fibre, in a fibre bundle one must also take into account an increment

In this section we show that the Yang-Mills theory can be transposed in the framework of the model of discrete spaces that we propose. The ill-defined concepts used by the Yang-Mills theory, such as the notion of internal spaces, are now given a physical meaning. Moreover unanswered questions posed by this theory, for example the choice of the relevant Lie groups, are also given a response.

In the model of discrete spaces that we put forward, the universe is made of basic cells called world points, and the physical points of the Yang-Mills theory are similar to world points. The internal space of a world point is the space spanned by its possible states. This space is d-dimensional with _{4} (since_{4} and U(4) are gauge symmetry groups of the model of discrete spaces. The relevant symmetry groups, however, must comply with both gauge groups. The group S_{4} has five irreducible representations, namely two 1-dimensional representations (

a) U(1) which is associated with irreducible representations

b) SU(2) which is associated with irreducible representation

c) and, finally, SU(3) which is associated with irreducible representations

There are no other gauge groups.

All properties of discrete spaces are derived from a very general Lagrangian

The Lagrangian writes

The notion of partial derivatives is introduced in discrete spaces through the matrix D obtained by factorizing the operator

According to the LDU (Lower triangular, Diagonal, Upper triangular) Banaciewicz theorem

and the partial derivative by

an operation that can be symbolized by

Physics must be left unchanged under the operations

_{a} are the generators of

the internal space of world point i. We consider infinitesimal transformations

first order expansion approximation of

Then the derivation operation becomes

Finally, one finds

parameters, materializes a gauge field.

In the Lagrangian (1), rewritten as

one replaces the operators D by their covariant expressions (3). Then (4) is made of three terms.

a) We first recover the Lagrangian term of the free quantum particle field:

b) Then we obtain the Lagrangian term of gauge fields

If we only consider local contributions we must look at terms with

c) and finally, a local interaction term between the particle and the gauge fields:

In this section we recover the results of the GSW (Glashow, Salam, and Weinberg) theory of electroweak interactions [

According to the GSW theory the gauge group for leptons would not be U(1) or SU(2) or SU(3) but a combined group, namely

with

under the constraint

The contribution writes

Here the matrices

From a mathematical point of view this expression is meaningless because it mixes a scalar with two dimensional matrices. The problem is resolved by introducing the simple following transformation between the scalar 1 and the two dimensional matrix

The value of the electroweak Lagrangian in vacuum is then given by (with c = 1)

Instead of a 4-dimensional real representation of

In this representation the vacuum state is written

By introducing this state in Equation (7) and by defining

and

one has

Finally with

the results of the Glashow, Salam and Weinberg (GWS) theory are recovered. The eigenvalues of the last 2-dimensional matrix are

a) What is the mechanism that binds the groups U(1) and SU(2)?

We shall not treat that subject here in detail but it can be shown that the entire organization of particles of the Standard Model can be recovered by assuming that the seed of a particle does not involve a single world point but a pair of world points, instead, one with a bosonic character, the other with a fermionic character, an idea close to super-symmetric (Suzy) approaches. According to this interpretation the leptons would transform as

b) What is the vacuum state?

The vacuum state

with

In our interpretation the vacuum state of an isolated world point is necessarily asymmetric. It is given by Equation (6) which is precisely the vacuum state used in the GSW theory.

c) How to determine the Weinberg parameter

The vacuum state

The experimentally accessible parameter is

Since

The experimental value is

which is consistent with the prediction. With this value as a datum we find

A state of vacuum with

Gaussian function. Then vacuum is defined as a state

with

The central hypothesis of general relativity is that the metric matrices are site dependent in space-time and that these modifications are caused by masses and, more generally, by non vanishing energy densities. In our model this hypothesis transforms the vacuum metric matrix into

The length increments

where l^{*} is the size of a world point. This gives

In the continuous limit the expression reads

This action is the starting point of General Relativity. The main conclusions of General Relativity, in particular the Einstein equation, may then be derived by using the usual (covariance) arguments. More precisely General Relativity may also be seen as a gauge field theory where space-time is analogous to a fiber bundle. The fibers of the bundle are (approximate) copies of the Minkowskian metrics and the basis of the bundle is space- time itself. This is exactly the scheme that appears in the present approach. The fibers are (approximate) copies ^{*}, quantum theory because a quantum state

Let us now consider weak gravitation fields. In weak gravitation fields the local metric matrix

where

The second term on the right hand side may be seen as a three-body interaction

that can be satisfied only if

This expression is a propagation equation that describes the dynamics of a massless tensor field

Its dynamics is then given by

This is the propagation equation of a massless gravitation wave travelling at the speed of light c. Its propagator writes

In this section we look for the quantum dynamics of a particle evolving in such curved spaces. The Lagrangian

is minimized under the set of N constraints

where

Making explicit every component of world point polarizations, this equation gives

Following the argument already given in [

This equation describes the quantum dynamics of a particle in the framework of the theory of general relativity. In flat spaces, the Klein-Gordon equation is recovered. This description of quantum states in gravitation fields is similar to the approach called “quantum mechanics in curved spaces” [

Particles interact through gauge interactions by an exchange of bosonic particles (photons, vector bosons or gluons) that result from the quantization of gauge fields. These are direct interactions, but besides those interactions there are also indirect interactions where two physical systems interact through fluctuations of their internal structures.

The best known example of indirect interactions is the Van der Waals interaction. In physical systems housing positive and negative electric charges, the centres of gravity of positive and negative charges may not match, giving rise to fluctuating electric dipoles. The dipoles interact and some dipole orientations lower the energy, the closer the systems the larger the energy lowering. The result is an attractive force that decreases as

Another example is the nuclear indirect interaction between nucleons. The mechanism at work arises from the fluctuations of quark colour charges. The fluctuations are transmitted between the various quarks that compound a nucleus through gluons (which are colour charged). The result is a strong, attractive, short range interaction.

Since the fluctuation interactions are always attractive and since the gravitation interaction is also attractive, we suggest that the gravitational interaction is an indirect interaction. The polarization of a world point, in fact the quantum wave

The polarization amplitude

with

where

The polarization

The fluctuations vanish when n, or b, or J, goes to infinity. The vanishing of fluctuations characterizes the so- called mean field approximation. Then the realized state is the state

The eigenvalue of a system where a world point i houses a particle P is

where

Similarly, the polarization perturbation of a world point j housing another particle Q is

The propagation of gravitation waves creates an interaction between the two particles P and Q that, owing to the weakness of vertices, may be calculated by using a low order perturbation expansion. The lowest order writes

By using Equation (8) the static

The Fourier transform of

Finally, the Newton expression of attractive gravitational forces is recovered:

The gravitation constant ^{*} the size of a world point.

The Planck length is given by

In other respects, the electric potential between two electrons at a distance

whereas the gravitational potential between these two electrons is

With

(

a very large number indeed. n is determined by the ratio of second order

fourth order

One has

This large difference between

The number

that is

the size scale of world points that has been used so far. The energy corresponding to this size is

very far (by four orders of magnitude) from the possibilities of available machines even those of the LHC.

Finally the size

about two hundred times the size of an hydrogen atom.

The anomalous motion of outer stars in a galaxy has led the astrophysicists to introduce an invisible matter that they called dark or hidden matter [

Milgrom suggests that one must write

with

In Milgrom dynamics

a constant as observed.

Since the motion of stars in the disks of galaxies is determined by the Mond dynamics the Milgrom parameter

The model of discrete universe that we propose can provide an explanation of Mond theory based upon the possible modifications of world point dimensionality d under the influence of polarization fluctuations. Let us recall that the internal space of a world point may be considered as a d-dimensional space where the polarization

and

has been computed in [

Every component

If the fluctuations of one component

where

Then by letting

Since

The form of the gravitation interaction associated with (2 + 1)-dimensional world points is modified because the Fourier transform (7) is now 2-dimensional. The potential in a 2-dimensional space becomes

Finally, the Milgrom attractive gravitational forces is

As a whole the gravitation interaction becomes

with

grom.

The disappearance of a dimension may be interpreted as the shrinking of the lengths associated to that dimension by a factor

A world point i eventually loses another dimension if the fluctuations perturb simultaneously two field components

The associated gravitation force is

a repulsive constant force that acts as a negative pressure exactly as does the cosmological constant.

The formula gathering the various contributions to the gravitation forces is

In Equation (13)

Since the distance travelled by light in one year is

The distance r_{M} where the cosmological expansion takes the lead over the Milgrom dynamics is

Up to now the possible effect of dark matter has not been taken into account. Dark matter has been introduced to account for the rotation curves of stars gravitating at the peripheries of galaxies. The Mond theory proposes another explanation and dark matter seems to be no longer necessary. The study of galaxies clusters shows that this is not the case. The gravitation forces are, in the Mond theory, exactly known. They are central and enable an exact calculation of the motions of galaxies in a cluster of galaxies to be carried out. The observation of the galaxy cluster 1E0657-56 (the Bullet) does not support the calculations. Dark matter is still necessary. Moreover dark matter is also necessary to account for the formation of galaxies. There is, however, no direct experimental evidence for their material existence except for their gravitational lens effects.

The model of discrete space-time that we put forward may provide another interpretation. Let us consider the metric matrix of the model [

This metric matrix is sensitive to variations of cosmic noise b and since the speed of light is given by

The parameter

In the classical limit this mass is the mass parameter that appears in the Schrödinger equation and, finally, through the Erhenfest equations, the mass of the Newton equation

Therefore

In contribution [

Obviously, the introduction of a cosmic noise must have large consequences in cosmology. Although this issue is out of the scope of this article we would like to mention briefly a few effects of b.

Below

If

Finally the speed of light diverges for

I would like to heartily thank Pr. Bart Van Tiggelen. He does not believe in my ideas but he strongly incited me to continue.