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The main aim of this paper is to explain why the Weinberg-Salam angle in the electro-weak gauge group satisfies . We study the gauge potentials of the electro-weak gauge group from our wave equation for electron + neutrino. These potentials are space-time vectors whose components are amongst the tensor densities without derivative built from the three chiral spinors of the wave. The gauge invariance allows us to identify the four potential space-time vectors of the electro-weak gauge to four of the nine possible vectors. One and only one of the nine derived bivector fields is the massless electromagnetic field. Putting back the four potentials linked to the spinor wave into the wave equation we get simplified equations. From the properties of the second-order wave equation we obtain the Weinberg-Salam angle. We discuss the implications of the simplified equations, obtained without second quantification, on mass, charge and gauge invariance. Chiral gauge, electric gauge and weak gauge are simply linked.

L. de Broglie found [

We previously have obtained a wave equation for a pair electron + neutrino [

where

This is the awaited transformation, because the electromagnetic potential moves with the charges and must then be a contravariant space-time vector, while the electromagnetic field of two photons

and

we get a potential which changes sign when we exchange the indexes:

(

where e is the wave of the electron; n is the wave of the electronic neutrino; p is the wave of the positron and a, the wave of the anti-neutrino, satisfies:

Therefore searching for anti-symmetrical products with

The quantum wave of the pair electron + neutrino is a function

where

Therefore, with the 12 real parameters of the

Under the dilation D the wave is transformed with only one M not two like x or

We shall use also

Each chiral spinor L, R and n allows us to get

The

We get:

Next we shall use for the odd terms in space-time:

A double link exists between the wave equation and its Lagrangian density because the wave equation is obtained from the Lagrangian density by variational calculus and, this is new, the Lagrangian density is exactly the scalar part of the invariant wave equation ( [

This means that when we write the wave equation with R, L and n or when we use the variational calculus to get the wave equation, we encounter the same equations which read in the R case:

Next the L spinor satisfies:

And the n spinor satisfies:

In de Broglie’s work on light [

The tensor densities of the Dirac theory have been intensively studied by L. de Broglie [

A solution must exist to this problem. First, since Feynman graphs act equally for fermions and bosons, the dynamics of both fermions and bosons must come from same laws. Now we know that the Lagrangian mechanism, for all fermions, comes from the real part of the wave equation. None meta-physical principle rules that physical laws necessary come from a Lagrangian density. Then the dynamics of bosons must also come directly from the wave equation of fermions, and only from this wave equation. Now every user of Feynman graphs thinks that all this theory results from a unique Lagrangian density, giving not only the dynamics of the fermion wave, but also the dynamics of the gauge bosons. But the link between boson fields and potential terms is not deduced from the Lagrangian density, it is postulated as definition of the potentials, or to get a simplified second-order equation, or to follow the rules of the gauge group. In the lone electromagnetic domain two links

are used between potential and field:

classical electromagnetism. How must we choose? None of these two different derivatives comes from a Lagrangian density. Then we must consider only a derivative using the spinor wave equation. Naming

For instance with

we get

Therefore if we set

which means that

We name

From

and using (3.2) we have

With (2.6) and since

is the form of the exterior product of the space-time algebra in space algebra, we get

Next we get

Similarly we get

Now we let

This vector satisfies

and we get

This, with (1.7), gives

This vector is then similar to the space-time potential

A detailed study of the nine space-time vectors available from the 78 tensor densities without derivative shows that this field is the only field without mass term. Since the photon is moving at the limit velocity, we know that its mass is null, then we may identify A to the electromagnetic potential and F to the electromagnetic field. This means that these tensors are both quantum quantities and classical quantities. The electromagnetic field F is exactly the bivector

The previous calculation of

We have explained [

which gives

Then B may be any linear combination of

For the group generated by

which gives

while the gauge transformation of the wave gives

This implies that the identification

which gives

while the gauge transformation of the wave gives

The gauge derivative of the wave equation for electron + neutrino is then compatible with the identification:

The Weinberg-Salam angle and the

From this definition results:

We have seen that

Now the knowledge of the gauge potentials B,

We get also

Next for the wave equation of R we use

and we get

At the second order we get:

We get a similar relation with

This means: first that the standard model of quantum mechanics, where too many parameters are available, has now one of these parameters fixed. Secondly this comfort the identification that we have made between gauge potentials and space-time vectors linked to the spinor wave. It is also possible to get the value of

We get also

And we also have:

We then get for the neutrino part of the wave:

which gives with (3.4) and (7.11)

There are many simplifications in our previous calculations, for instance (5.12) becomes

The electromagnetic field appears as second behind the space-time vectors. From the beginning of quantum mechanics the potential terms are in the wave equations, the electromagnetic field is only used as coming from the potential space-time vector.

Now if we follow T. Socroun [

Then Equations (5.8), (6.10) and (7.11) imply the simple equality:

The electromagnetic boson

the key to understand the

and we have

The

This is true only with

The parameter

The theory of light built by L. de Broglie was able to account for the photon of A. Einstein, but his construction started from the linear Dirac equation, so his photon had a mass which was never observed. And this mass term was breaking the gauge symmetry. The problem of the mass was also present in the Weinberg-Salam model of electro-weak interactions. The electron and the neutrino of this model have lost mass. Since it was necessary somewhere to account for the mass of the electron, a complicated mechanism of spontaneous symmetry breaking was invented. Even if the Higgs boson is now observed, this symmetry breaking is not able to reduce the too great number of parameters of the standard model.

Actually the symmetry breaking is useless in the frame developed here, based on a wave equation with mass term which is both form invariant (then relativistic) and fully gauge invariant under the gauge group

The simplified wave Equations (7.10) and (7.13) give also a new understanding of mass and charge. The mass term of the Dirac equation links the derivative of the left wave to the right wave and vice-versa. Why? The reason comes from the structure of the wave, with left and right spinors in a different column, and from the transformation under a dilation of the derivative:

Then

of the

a degree of freedom which shall give the 3-dimensional gauge group

We have a

ClaudeDaviau,JacquesBertrand, (2015) Electro-Weak Gauge, Weinberg-Salam Angle. Journal of Modern Physics,06,2080-2092. doi: 10.4236/jmp.2015.614215