^{1}

^{*}

^{2}

^{*}

With the right and the left waves of an electron, plus the left wave of its neutrino, we write the tensorial densities coming from all associations of these three spinors. We recover the wave equation of the electro-weak theory. A new non linear mass term comes out. The wave equation is form invariant, then relativistic invariant, and it is gauge invariant under the U(1)×SU(2), Lie group of electro-weak interactions. The invariant form of the wave equation has the Lagrangian density as real scalar part. One of the real equations equivalent to the invariant form is the law of conservation of the total current.

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 algebra^{1} of space-time [

We got with the 8 real parameters of the wave of the electron [^{2}. Now with 12 real parameters

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

With the only right spinor

we get the space-time vector ^{3}

Similarly with the left spinor

we get the space-time vector

It is well known [

With the left spinor

we get the space-time vector

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

where^{4}. To see this, we can consider the form invariance of the Dirac theory.

is [

We get

So

Then

Now with

Vectors

Finally with

Vectors

The main invariant term of the electron wave is [^{5}

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 [

where

Projectors

Noting

Operators

With

which results from our choice [

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

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

The Lagrange equation

Together these equations read

Multiplying by

Since

then using the conjugation

The Lagrange equation

The Lagrange equation

Together these equations read

Multiplying by

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 [

Lagrange equation

The Lagrange equation

Together these equations read

Multiplying by

Without the mass term, this equation is the wave equation of the electronic neutrino in the electro-weak theory [

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

With (2.2) and (2.42) we get

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

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

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

Under the

And we shall get the form invariance of the wave equation if and only if

From (1.13), (1.16), (1.18) we get with (1.11)

This gives (3.5) since

And the wave equation is form invariant under

We shall use a convenient form of the projector

We have proved ([

Equation (3.8) reads

This gives

And we get

We then get for

We must study

We get

and similarly

This gives

which reads

and we finally get

We may then say that the wave equation is gauge invariant under the gauge transformation generated by

This generator acts only upon left waves: we get

And with left waves we get

That reads

We then get for

The covariant derivative is here reduced to

We let

We get

Only the left column of

For

We then get

Since the same matrix multiplies the differential part and the mass part of the wave equation, we may say that this equation is invariant under the gauge generated by

This generator also acts only upon left waves: we get

which reads with

We get for

The covariant derivative is now reduced to

We let

We get

As previously, for

We then get

Since the mass term is changed exactly in the same way that the differential term we can say that this wave equation is gauge invariant under the gauge generated by

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

(2.36) on the left side by

We shall use

We got in [

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:

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

The conservative law

The mass term of (2.43) replaces

We previously studied [

The formalism with Dirac matrices ([

When we cancel