The standard model of quantum physics uses the Dirac equation for the wave of the electron and the wave of the electronic neutrino. Weak interactions mix the left wave of the electron and the left wave of its neutrino. We start from the rewrite of this part of the standard model in the Clifford algebra¹ of space-time [1] . We use a nonlinear homogeneous wave equation which has the Dirac equation as linear approximation. This wave equation has been extended to account for the electro-weak gauge group. It is a wave equation for both the right and left spinors of the electron and for the left spinor of the electronic neutrino. The wave is then a function of space and time with 12 real parameters.

We got with the 8 real parameters of the wave of the electron [4] tensorialdensities². Now with 12 real parameters tensorial densities are awaited. We shall need most of them, so the first part of this study reviews these 78 densities.

In the case of the electron alone the wave equation comes from a Lagrangian density and this is true both with the nonlinear homogeneous equation and with its linear approximation. The difference between these two densities is in the mass term. We got an invariant form for both equations. Since there are 8 real parameters, the wave equation is equivalent to 8 equations. The real one is the cancellation of the Lagrangian density, and another one is the conservation of the current of probability.

It is well known that the electro-weak theory has a problem with the mass term of the Dirac equation, because this term links the left spinor and the right spinor that behave differently in the electro-weak gauge. Then this part of the standard model begins with an electron without mass term and a complicated model of symmetry spontaneously broken is necessary to get a mass term. The aim of this article is to study a wave equation with a mass term that is both relativistic invariant and gauge invariant under the electro-weak gauge group. This was previously thought as impossible.

An invariant form of this wave equation also exists; the Lagrangian density is the real part of this invariant form. We get the wave equation by the Lagrangian mechanism. The conservation of the total current is one of the numeric equations equivalent to the invariant form. The Dirac equation with mass term is the linear approximation of our wave equation for electron + neutrino when we cancel the wave of the neutrino.

Each of the three spinors has 4 parameters, and then each gives components of tensors, a space- time vector (4 components) and a space-time bivector (6 components).

With the only right spinor of the electron

we get the space-time vector and the space-time bivector satisfying³

is a space-time vector, because it satisfies.

Similarly with the left spinor of the electron

we get the space-time vector and the space-time bivector satisfying

It is well known [6] that these currents, are the fundamental ones in the Dirac theory, and the usual currents and are sum and difference of these chiral currents

With the left spinor of the electronic neutrino

we get the space-time vector and the space-time bivector satisfying

Next with two of these three spinors we can get 16 densities. We begin with and. We let

and are known in the Dirac frame:

where, are the relativistic invariants of the Dirac theory, and is the Yvon-Takabayasi angle [1] [7] . [5] is the bivector which, with, , and, gives the 16 components of tensors known in the complex Dirac theory. They are those invariant under the electric gauge [3] . and are space-time vectors⁴. To see this, we can consider the form invariance of the Dirac theory. being any complex

matrix, the transformation from the space-time in to itself, which to any associates satisfying

is [4] [5] [8] a Lorentz dilation, product of a Lorentz rotation and of a homothety with ratio satisfying

We get

So is a quantity varying like, a contravariant vector. It is the same for or. We get also

Then

Now with and we let

Vectors and are contravariant vectors, is a bivector. We shall need

Finally with and we let

Vectors and are contravariant vectors, is a bivector. We shall need

The main invariant term of the electron wave is [1] . Since we get now not only one, but three similar terms, the natural generalization to the wave of the electron and its neutrino is⁵

Since this invariant term is the generalization of the invariant mass term of the electron wave, since this term is the mass term of the Lagrangian density [1] , the Lagrangian density of the wave of the electron and its neutrino is the real scalar part of

where, is the reverse and is the covariant derivative satisfying

Projectors satisfy

Noting they satisfy

Operators are then generators of the Lie group of electro-weak interactions. We shall use

With and. The last equality (2.5) comes from:

which results from our choice [1] of Dirac matrices:

and we get

With (2.2) we get

which gives

Next we get, with

From (2.8) and (2.9) we get with

Next we get, with

We then get

We next get

With our matrix representation (2.7) of the space-time algebra, the real part of a multivector is the real part of the scalar part of the matrix. Therefore we get

Next we get

From (2.10) and (2.14) we get

which gives

(2.19)

We next get

For (2.17) we have

And for (2.19) we have

Therefore the Lagrangian density is

Therefore the Lagrangian density is

The Lagrange equation gives

The Lagrange equation gives

Together these equations read

Multiplying by we get

Since and this also reads

then using the conjugation we get

The Lagrange equation gives

The Lagrange equation gives

Together these equations read

Multiplying by this reads

Adding (2.31) and (2.35) we get the wave equation

Without its mass term, this equation is the wave equation of the electron in the electro-weak theory [1] (6.57). The great difference is that there is now a mass term, and since the Lagrangian density is both relativistic and gauge invariant, we shall see that the wave equation with mass term conserves these invariant properties. The

Lagrange equation gives

The Lagrange equation gives

Together these equations read

Multiplying by this reads

Without the mass term, this equation is the wave equation of the electronic neutrino in the electro-weak theory [1] (6.58). Following backwards the calculation from (2.10) to (2.15) we can see that the system (2.36) (2.40) is equivalent to the wave equation

where

or to the invariant equation

Since this wave equation is not exactly our starting one, we must explain how this equation has exactly the Lagrangian equation as real part. We shall next prove the relativistic invariance and the gauge invariance of this wave equation under the electro-weak gauge group.

With (2.2) and (2.42) we get

We get

(3.2)

Therefore the Lagrangian density (2.25) is also the real part of the invariant form (2.43) of the wave equation. The value of is more simple because we get

Therefore the Lagrangian density (2.25) is also the real part of the invariant form (2.43) of the wave equation. The value of is more simple because we get

We now review the form invariance of this wave equation; next we shall prove its gauge invariance.

We start from (2.36) and its conjugated equation:

and from (2.40) and its conjugated equation

The differential term of (2.44) is

Then (2.43) reads, which is equivalent to. To get the first equation we multiply

(2.36) on the left side by and (4.2) by, this gives

(4.4)

(4.5)

We shall use

We got in [1] (A.18), for any space-time vector, the following equality:

Then the first part of (4.5) reads

Since the second part of (4.5) satisfies

It is a pseudo-vector in space-time and we let

This gives

Similarly we let

which gives

Next we get, with Appendix A:

This gives

We also need

We get with Appendix A:

(4.15)

We get with Appendix A:

which gives

which gives

We get also with Appendix A:

We get also with Appendix A:

Since the mass term satisfies (3.2) the wave equation (4.4) is equivalent to the system:

Since the mass term satisfies (3.2) the wave equation (4.4) is equivalent to the system:

A conservative current exists: it is the total current where is the current of probability in the case of the alone electron, and is the isotropic current of the neutrino. Now these currents are not separately conservative, only the total current is conservative. The calculation is similar for. We get

The conservative law is obviously the simplest equation in (4.19). Contrarily to the case of the electron alone, where is also a conservative current [4] , we get here only one conservative current. When the electron is alone, the conservative current is interpreted as the probability density of presence of the electron. It is really the relativistic generalization of the probability density of the Schrödinger wave. But it is nonsense to think the same for the current, since we have here both an electron and its neutrino.

The mass term of (2.43) replaces by. This is enough to get a wave equation both form invariant and gauge invariant. To get this important novelty in the electro-weak gauge theory, the use of the Clifford algebra of the physical space is essential. It should be impossible to get this using only the algebra of Dirac matrices or the Clifford algebra of space-time. The physical reason is the difference between left and right waves.

We previously studied [9] a wave equation for the electron, with a mass term similar to the mass term obtained here to account for the electron and its neutrino. Another difference between this study and the present one is the role of the Lagrangian density. In the case of the electron alone, or here, the link between the Lagrangian density and the wave equation is double: the Lagrangian density is obtained from the real part of the invariant wave equation, and the wave equation is obtained from the Lagrange equations. Developing the first equation (4.19) and using (2.25) we can see that the first equation (4.19) is. In [9] , we studied a case, which could be extended to the present study, where this double link is cut. The Lagrangian density is again obtained from the real part of the invariant wave equation, but the wave equation cannot be obtained from the Lagrange equations.

The formalism with Dirac matrices ([1] (6.74)) uses and, but we prefer and: because they are true space-time vectors, because the term is here not the usual of quantum fields, generator of the electric gauge, but the of the chiral gauge. Moreover calculations are more complicated with.

When we cancel, we get,; the wave equation is reduced to the homogeneous nonlinear equation previously studied [1] [3] . This equation has the Dirac equation as linear approximation and in the case of the H atom, a set of orthonormal solutions exists with a angle everywhere defined and small. It is exactly the sufficient condition allowing the Dirac equation to approximate our nonlinear homogeneous wave equation. The mass term that we obtained here is also available for the magnetic monopole studied in [1] and [6] [7] [10] .