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We continue the study of the Standard Model of Quantum Physics in the Clifford algebra of space. We get simplified mass terms for the fermion part of the wave. We insert the simplified equations in the frame of General Relativity. We construct the electromagnetic field of the photon, alone boson without proper mass. We explain how the Pauli principle comes from the equivalence principle of General Relativity. We transpose in the frame of the algebra of space the second quantification of the electromagnetic field. We discuss the changes introduced here.

We use the introduction, the notations and results of the first part [

The

are named contravariant. The vectors transforming like the gradient

Each Dirac wave is made of a right and a left wave. For instance the

We use similar notations for the wave

Each chiral wave allows us to construct a unique contravariant vector:

The wave equations of the electron and of the Lochak’s magnetic monopole [

We do not change anything in (1.6), (1.8) and (1.9) from the wave equations of the SM, where

We also use the value of the Weinberg-Salam angle obtained in [

where

We explain below why the electromagnetic potential

We have also:

We then get:

A similar calculation gives:

In each of the four wave Equations (1.6)-(1.9) the mass term has three parts while the

We have obtained in [

With the value obtained in [

The system (1.6)-(1.9) is then reduced to:

We have:

These chiral currents satisfy:

From the previous equalities we can get the result: the chiral currents are isotropic, and using the numeric equations equivalent to (1.19) we get the law of conservation of these four chiral currents. For instance the

The conservative currents are

The same calculation gives for the other terms in

This implies:

The space-time vector

The unitary space-time vector

We start from the wave equations obtained in [

The mass term

Like in the previous section the calculation of

The weak potentials satisfy:

A calculation similar to (1.14) gives:

The gauge bosons acting in (1.34) on the right part of the quark d are made of only the right colored d waves:

This gives:

then the wave Equation (1.34) of the right red d quark is reduced to :

Similarly the wave equations of the green and blue colors of the right d quark are:

The gauge bosons acting in (1.35) on the right part of the quark u are made of only the right colored u waves:

This gives:

then the wave Equation (1.35) of the right red u quark is reduced to :

And the wave equations of the green and blue colors of the same right u quark are:

Next for the left waves we get a double dependence for the gauge bosons, because the weak potentials also change with the color and the strong potentials change with the savor. Similarly to (1.18) we have:

The gauge bosons acting in (1.36) on the left part of the quark d are made of only the left colored d waves:

This gives:

then the wave Equation (1.36) of the left red d quark is reduced to :

And the wave equations of the green and blue colors of the left d quark are:

The gauge bosons acting in (1.37) on the left part of the quark u are made of only the left colored u waves:

This gives:

then the wave Equation (1.37) of the left red d quark is reduced to:

And the wave equations of the green and blue colors of the same d quark are:

So we describe all particles of the first generation in the SM with 8 left spinors and 8 right spinors, then 16 spinors together, we have obtained 16 simplified wave equations: four in (1.19), three in (1.40)-(1.41), three in (1.44)-(1.45), three in (1.49)-(1.50), three in (1.53)-(1.54). Each wave equation is equivalent to four numeric partial differential equations, we have the true number of numeric equations for our

In the frame of General Relativity (GR) the space-time manifold has in each point of the manifold a tangent space-time with a variable space-time basis

This point of view comes not directly from the ideas of A. Einstein, based on the concept of invariance, but from the ideas of his professor of mathematics, Minkowski. Naturally D. Hestenes and most of his followers [

V. Fock [

Fock emphasized also this: “The equations of the gravitational, or any other field, are partial differential equations, the solutions of which are unique only when initial, boundary or other equivalent conditions are given. The field equations and the boundary conditions are connected and the latter can in no way be considered less important than the former. But, in problems relating to the whole of space, the boundary conditions refer to distant regions and their formulation requires knowledge of space as a whole”. Consequently Fock used a space uniform at infinity. This also is implicit in the point of view of the

This is exactly the result of Fock: The wave equation of the electromagnetic front wave:

has the same form in two inertial frames only if:

We explained in the first part of this work [

We then have four different forms of gradient applying into the wave equations:

This appeared in the wave equations (1.6)-(1.9) and (1.27)-(1.30). Even in the case of a flat space with null Christoffel symbols we get again these four kinds of semi-variance, because the

Then the wave equations equivalent to (1.19) read:

Similarly the wave equations of the colored quarks read, with

The

We then get:

This gives:

Since

And it is the same for (2.8)-(2.9). It is this form invariance that is named “relativistic invariance” in the Dirac theory.

In the SM the electromagnetic field is a field of bivectors (antisymmetric second rank tensor) on the relativistic point of view, and a field of operators on the quantum point of view. The motion of this field comes from a special part of the Lagrangian density. We cannot use this way, because we do not believe into a meta-physical necessity of the Lagrangian mechanism for any physical field. The Lagrangian density comes from the real part of the invariant form of the wave equations. Consequently only the wave equations of the fermion part of the SM are compulsory and all must come from them. We know that the particles of this electromagnetic field, the photons, are bosons with no mass. And we shall prove that this is the case for:

We have explained in [

Now the covariant derivative is more general than the partial derivative, so we replace this by:

This derivative will be generalized in a further article. This kind of derivative is sufficient to avoid the infinity of tensorial densities coming from the use of simple partial derivatives in the Dirac theory [

We get with

We then get for the electromagnetic field created by the lepton wave:

These last terms cancel with (1.12) and (1.17) and we get:

Similarly the first wave Equation (2.9) gives:

Next the second Equation (2.9) gives:

We then get:

Next with the left waves and the third Equation (2.9) we get:

Next with the last Equation (2.9) we get:

We then get:

By subtracting this from (3.10) we get:

And

The electromagnetic field

Studying the anomalous Zeeman effect, Pauli found the necessity of the “spin number” of the electron in the frame of the Bohr’s model of atom. And he came to his principle [

Since its formulation is made in the frame of the non-relativistic wave of a system of electrons we cannot follow the usual explanation of this principle by the anti-symme- trization of the wave. The only compulsory wave equation is the wave of one electron and more generally of one fermion. Moreover the explanation by the four quantum numbers

quantum number j of the total kinetic momentum is only indirectly present:

The other quantum numbers are the degrees of the Gegenbauer’s polynomial giving the angular functions (the Legendre polynomials with degree

This last equality allows us to define a norm for the wave

Next the Dirac equation which is the linear approximation of our non-linear wave equation in the electron case is satisfied by the sum and the difference of two solutions. The previous norm defines a scalar product:

With the kind of functions used by the separation of variables in spherical coordinates, the condition of orthogonality for this Euclidean scalar product is automatically satisfied by the different solutions. Moreover the condition of orthogonality for the Euclidean scalar product is exactly equivalent to the cancellation of the Hermitian scalar product of quantum mechanics [

The quantized electromagnetic field is described with operators of creation and annihilation: the operator of creation adds one photon to the electromagnetic field and the operator of annihilation suppresses one photon. We previously studied the wave of a unique fermion and the potential present in the wave equations is linked to this unique fermion. The dilation generated by any

With

Then (5.1) gives:

Now if the field of a system of two photons

this field also transforms like the field of a unique photon:

This is generalized by using the group

The quantized electromagnetic field is the general element

Actually this field is independent on the chiral gauge and the scale parameter: we use again

We then get for any

Like all gauge fields the quantized electromagnetic field

Then this field is a scalar + pseudo-scalar field, similar to the Higgs field. More generally the field of an odd number of photons is a bivector field and the field of an even number of photons is scalar + pseudo-scalar field. This also implies that the quantized electromagnetic field is not reduced to the field

where

The unification of all interactions needs only two Clifford algebras, the

We have resolved the wave equations of the electron + neutrino in the case of the hydrogen atom [

A priori we do not expect a revolution with new particles or a new kind of interaction. The only new wave awaiting study is the magnetic monopole which uses the sixteenth

Since only the fermion wave equations are doubly linked to their Lagrangian density, since the gauge fields do not have the same double link to the Lagrangian density and since the momentum-energy tensor is the conservative tensor linked to the invariance of this Lagrangian density under the space-time translations we can predict this: All momentum-energy belongs to the fermion part of the quantum wave, there is none momentum-energy belonging to the boson part of the quantum world. We do not see the momentum-energy of an isolated photon, we see only the momentum-energy emitted or absorbed by fermions. The double equality

About anti-matter, we predict that any object made of anti-matter has the same gravitation as the corresponding object of matter. This comes from the wave equations for anti-particles which change only the sign of the differential term (see [

This work is not an end, it is a beginning. Now that we know the wave equations for all the first generation, the study of the two other generations must be done. It will be also necessary to resolve the equations and to confront the results with what has been learned in the laboratories. A great progress should be the description of the transition of an electron from one state to another one by emitting or absorbing a photon, allowing us to understand from where comes the probability of spontaneous and stimulated emission, the probability of absorption and the link with the number of photons in the vicinity of the electron.

What can we expect as experimental confirmation of this work? First the existence of leptonic magnetic monopoles is necessary in the

Daviau, C., Ber- trand, J. and Girardot, D. (2016) Towards the Unification, Part 2: Simplified Equations, Covariant Derivative, Photons. Journal of Mo- dern Physics, 7, 2398-2417. http://dx.doi.org/10.4236/jmp.2016.716207