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A wave equation with mass term is studied for all fermionic particles and antiparticles of the first generation: electron and its neutrino, positron and antineutrino, quarks u and d with three states of color and antiquarks and . This wave equation is form invariant under the group generalizing the relativistic invariance. It is gauge invariant under the U(1)×SU(2)×SU(3) group of the standard model of quantum physics. The wave is a function of space and time with value in the Clifford algebra Cl_{1,5}. Then many features of the standard model, charge conjugation, color, left waves, and Lagrangian formalism, are obtained in the frame of the first quantization.

We use here all notations of “new insights in the standard model of quantum physics in Clifford algebra” [

with

Here

The link with the usual presentation of the standard model is made by the left and right Weyl spinors used for waves of each particle. These right and left waves are parts of the wave with value in

We used previously the same algebra

We have noticed, for the electron alone firstly (see [

Moreover we generalized the non-linear homogeneous wave equation of the electron, and we got a wave equation with mass term [

The standard model adds to the leptons (electron

The electro-weak theory [

Waves

The form (2.3) of the wave is compatible both with the form invariance of the Dirac theory and with the charge conjugation used in the standard model: the wave

We can then think the

And the antineutrino has only a right wave. The multivector

we got

Most of the preceding presentation is easily extended to quarks. For each color

The

The link between the reverse in

The mass term reads

where we use the scalar densities

The covariant derivative

We use two projectors satisfying

Three operators act on quarks like on leptons:

The fourth operator acts differently on the leptonic and on the quark sector. Using projectors:

we can separate the lepton part

and we get (see [

This last relation comes from the non-existence of the right part of the

We start from generators

To simplify here notations we use now l, r,

Then (2.1) gives

We name

Everywhere the left up term is 0, so all

We can extend the covariant derivative of electro-weak interactions in the electron-neutrino case:

to get the covariant derivative of the standard model

where

Now we use 12 real numbers

and we get, using exponentiation

in any order. The set of these operators

the gauge transformation reads

The

The physical translation is: Leptons do not act by strong interactions. This comes from the structure of the wave itself. It is fully satisfied in experiments. We get then a

The wave equation (3.19) is equivalent to the wave equation

studied in [

This wave equation is equivalent to the invariant equation:

This wave equation is form invariant under the Lorentz dilation

We begin by the double link between wave equation and Lagrangian density that we have remarked firstly in the Dirac equation [

The existence of a Lagrangian mechanism in optics and mechanics is known since Fermat and Maupertuis. This principle of minimum is everywhere in quantum mechanics from its beginning, it is the main reason of the hypothesis of a wave linked to the move of any material particle made by L. de Broglie [

We shall establish the double link now for the wave equation (2.16). It is sufficient to add the property for (3.20). This equation is equivalent to the invariant equation:

We get from the covariant derivative (2.19) with the operators

Next we get

The calculation of the Lagrangian density in the general case is similar to the lepton case. We get

The calculation of

Since three

This new link between the wave equation and the Lagrangian density is much stronger than the old one, because it comes from a simple separation of the different parts of a multivector in Clifford algebra. The old link, going from the Lagrangian density to the wave equation, supposes a condition of cancellation at infinity which is dubious in the case of a propagating wave. On the physical point of view, there are no difficulties in the case of a stationary wave. Difficulties begin when propagating waves are studied. Our wave equations, since they are compatible with an oriented time and an oriented space, appear as more general, more physical, than Lagran- gians. These are only particular consequences of the wave equations.

On the mathematical point of view the old link is always available. It is from the Lagrangian density (4.12) and using Lagrange equations that we have obtained the wave equation (2.16).

Under the Lorentz dilation

We then let

which implies

Then we get

and we shall now study the form invariance of the mass term. All

This gives

Then the form invariance of the wave equation is equivalent to the condition on the mass term

linked to the existence of the Planck factor [

Since we have previously proved the gauge invariance of the lepton part of the wave equation, it is reason enough to prove the gauge invariance of the quark part of the wave equation.

We have here

To get the gauge invariance of the wave equation we must get

This is satisfied because

All up terms in the matrix

And we finally get

The wave equation with mass term is gauge invariant under the group generated by

We have here

Since

We let

Then (5.31) is equivalent to the system

or to the system

We then get

This implies

Similarly, permuting colors, we get

This implies

and also

This implies

Moreover we get

We then get

Next we have

and we get

Since we get the same relation for g and b colors we finally get

The wave equation with mass term is then gauge invariant under the group generated by

We have here

Since

We let

Then (5.67) is equivalent to the system

or to the system

We then get

This implies

Similarly, permuting colors, we get

This implies

and also

This implies

Moreover we get

We then get

Next we get with (5.56)

Since we get the same relation for g and b colors we finally get

The wave equation with mass term is then gauge invariant under the group generated by

We have here

Since

Then (5.97) is equivalent to the system

or to the system

We then get

This implies

Next we get with (5.56)

Since we get the same relation for g and b colors we finally get

The wave equation with mass term is then gauge invariant under the group generated by

We use now the gauge transformation

We can then forget here

must be equivalent to the system

Using relations (5.117) and (5.118) the system (5.121) is equivalent to (5.120) if and only if

We name

which implies with

The equality (5.117) is equivalent to the system

The equality (5.118) is equivalent to the system

This gives for the invariant scalars

We then get

Next we let

and we get with (B.17) and (B.18)

This gives the awaited result

The change of sign of the phase between (5.117) and (5.152) comes from the anticommutation between

We use with

The gauge invariance signifies that the system

must be equivalent to the system

Using relations (5.154) and (5.155) the system (5.158) is equivalent to (5.157) if and only if

because we get

The case

From experimental results obtained in the accelerators physicists have built what is now known as the “standard model”. This model is generally thought to be a part of quantum field theory, itself a part of axiomatic quantum mechanics. One of these axioms is that each state describing a physical situation follows a Schrödinger wave equation. Since this wave equation is not relativistic and does not account for the spin 1/2 which is necessary to any fermion, the standard model has evidently not followed the axiom and has used instead a Dirac equation to describe fermions. Our work also starts with the Dirac equation. This wave equation is the linear approximation of our nonlinear homogeneous equation of the electron.

^{1}The reversion is an anti-isomorphism changing the order of any product (see [

The wave equation presented here is a wave equation for a classical wave, a function of space and time with value into a Clifford algebra. It is not a quantized wave with value into a Hilbertian space of operators. Never- theless and consequently we get most of the aspects of the standard model, for instance the fact that leptons are insensitive to strong interactions. The standard model is much stronger than generally thought. For instance we firstly did not use the link between the wave of the particle and the wave of the antiparticle, but then we needed a greater Clifford algebra and we could not get the necessary link between reversions^{1} that we used in our wave equation. We also needed the existence of the inverse to build the wave of a system of particles from the waves of its components. And we got two general identities which existed only if all parts of the general wave were left waves, only the electron having also a right wave.

The most important property of the general wave is its form invariance under a group including the covering group of the restricted Lorentz group. Our group does not explain why space and time are oriented, but it respects these orientations. The physical time is then compatible with thermodynamics, and the physical space is compatible with the violation of parity by weak interactions.

The wave accounts for all particles and anti-particles of the first generation. We have also given [

Since the wave equation with mass term is gauge invariant, there is no necessity to use the mechanism of spontaneous symmetry breaking. The scalar boson certainly exists, but it does not explain the masses.

A wave equation is only a beginning. It shall be necessary to study also the boson part of the standard model and the systems of fermions, from this wave equation. A construction of the wave of a system of identical particles is possible and compatible with the Pauli principle [

Here indexes ^{2} the following matrix representation of

where

We get also

^{2},

^{3} anti-commutes with any odd element in space-time algebra and commutes with any even element.

Similarly we get^{3}

Scalar and pseudo-scalar terms read

For the calculation of the 1-vector term

we let

This gives

For the calculation of the 2-vector term

we let

This gives

For the calculation of the 3-vector term

we let

This gives with (A.3) and (A.9)

For the calculation of the 4-vector term

we let

This gives with (A.4) and (A.10)

For the calculation of the pseudo-vector term

we let

This gives with (A.7) and (A.12)

We then get

(A.25)

This implies

In

is

we must change the sign of bivectors

The reverse, in

is

Only terms which change sign, with (A.13), (A.18) and (A.20), are scalars

(A.34)

This link between the reversion in

There are

We used in [

with

We name

which implies

The equality (C.3) is equivalent to

The equality (C.4) is equivalent to

We get

This gives

from which we get

These relations are the awaited ones because

We name

which implies

This gives

We then get

This implies

We then get the awaited results