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A real version of the Dirac equation is derived and its coupling to the electromagnetic field considered. First the four-component real Majorana equation is briefly discussed. Then the complex Dirac equation including an electromagnetic field will be written as an eight-component real spinor equation by separating it into its real and imaginary parts. Through this decomposition, what becomes obvious is the way in which the electromagnetic field couples to charged fermions (electron and positron) when being described by real spinor fields. Thus, contrary to common expectation, charged fermions can also be described by a real Dirac equation if they are considered as a doublet related to the SO(2) symmetry group, which enables a matrix coupling to the electromagnetic field and corresponds to the usual U(1) gauge symmetry of the standard Dirac equation.

In modern elementary particle physics the complex Dirac equation [

It is well known that fermions described by the real four-component Majorana [

The outline of the paper is as follows: First, the four-component real Majorana equation is briefly discussed as the classical paradigm for a real spinor field equation. Second, the covariant derivative in electromagnetic gauge theory is reconsidered, and a way opened to introduce electromagnetism into a real spinor field equation. Third, the complex Dirac equation including an electromagnetic field will be transformed into a real Dirac equation for an eight-component real spinor field. Through this procedure it becomes obvious how the electromagnetic field couples to charged fermions (e.g., electrons and positrons) when being described by real spinor fields. The symmetries of the resulting real Dirac equation are also discussed. Fourth, the eigenfunctions of the real Dirac equation are derived. Fifth, the properties of bilinear forms are shortly addressed, and then we consider similarity transformations. Finally, the conclusions are presented.

At the start we rederive the real four-component spinor Majorana equation, without recourse to the Dirac equation (thereby following the recent work of Aste and Marsch [

In the subsequent algebra we make use of conventional symbols, notations and definitions, and use units of

for the scalar field

with appropriately defined real spinor field

Here

Apparently, the condition (3) can be fulfilled by the real

Obviously, all three sigma matrices mutually anti-commute with each other and, when being squared, are equal to the

In terms of the above lambda matrices the Majorana gamma matrices (see, e.g. Kaku [

whereby

transposed matrix. The standard representation of the Dirac matrices [

We emphasize that the Majorana equation, while being real, has two degrees of freedom (

As is well known from the classical work of Klein, Pauli and Weisskopf [

This equation obeys U(1) symmetry, meaning that the spinor field

However, apparently this procedure does not work for the real Majorana Equation (2) or (6), without destroy- ing their real nature. So, the real four-component Majorana equation can not be coupled to an electromagnetic field (and also not to other gauge fields when being associated with complex phase factors) by means of the minimal coupling in the above form.

Yet, there is another way to introduce electromagnetism in a real field equation, if we just assemble the real and imaginary parts of a complex spinor

which is a generator of the SO(2) group. Here use has been made of

As shown below, when the Dirac equation is brought into its real form the SO(2) symmetry is obeyed instead of U(1), and thus a Dirac fermion can be charged if it is described in terms of a real eight-component spinor field.

The transformation of the spinor field for a spatio-temporal varying phase

if the gauge field is transformed as

Following the reasoning in the previous sections, we now want to derive a real version of the Dirac equation. The complex standard Dirac equation for

The Dirac matrices in their standard form can be written concisely as

Using the above lambda matrices, we obtain

Note that the three components of the vector

We can now decompose the complex Dirac Equation (12) into its real and imaginary parts. Thereby we assemble again the real and imaginary parts of a complex four-component Dirac spinor,

The gamma matrices have a

These four matrices commute with

Let us discuss the general symmetries of the real Dirac equation. Concerning its space-time symmetries, we define conventionally the time and space coordinate inversion operations as

whereby we find for the product of Dirac matrices the result

The charge conjugation operation

which is found to obey

Finally, let us briefly discuss chiral symmetry. The real chiral matrix is defined appropriately as

which yields

jection operators by

left-chiral components, i.e.

and are coupled by the mass term which obviously breaks chiral symmetry.

We return to the real Dirac Equation (16) with the aim to derive its eigenfunctions for

It resembles very much the Majorana Equation (6), with the difference being in the dimension of the gamma matrix representation and the associated spinors. Since Equation (24) is real, and as each component of

with free real amplitude

By insertion of the first into the second equation, or vice versa, the relativistic dispersion relation is obtained from:

which yields the two eigenvalues:

The negative root in Equation (29) is related to antiparticles, the positive one to particles. By solving (26) for

The second solution is obtained from first one if the free phase is chosen as

where the polarization vector is defined by

Returning now to the standard Dirac Equation (7), we can chose for it the standard basis

corresponding to particles

How many physical degrees of freedom does the real Dirac equation have? Of course it should not be more than the complex one, which has four, as the complex spinor fields

Let us now determine the basis spinors of the real Dirac equation. Obviously, in the above general solutions (32) and (33) the polarization spinors can be chosen freely, and the question then arises how to chose them adequately for the real Dirac equation. For that purpose we make use of the standard basis vectors (34), and with their help construct the new extended fundamental basis as follows

corresponding to particle and antiparticle states arranged in doublets of four-component spinors. Note that these basis spinors are eight-dimensional, corresponding to the dimension of the phase space of the complex standard Dirac equation, and form an orthonormal basis of the vector space of the real Dirac equation. Its solution is the doublet spinor field

Let us also discuss briefly how the charge conjugation and the chiral operator act on the basis spinors. We recall the definition of the charge conjugation, which is given by

showing the tight connection between particle and antiparticle spinors. As the charge conjugation operator

which is in consistent with the fact that

Like for the complex Dirac equation, bilinear forms involving the spinor field and its adjoint play an important role in a relativistic field theory. Therefore, we must consider such Lorentz-invariant bilinear forms (see, e.g., [

where

means

is intimately connected with the negative sign in front of the metric (3), which requires

have

Using these mathematical properties one finds after some algebra by means of (32) and (33) that for any pair of solution field spinors we have

Let us consider a general polarization spinor of the real Dirac equation

which vanishes identically, in particular for any pair of the basis spinors in Equation (34), since beta is sym- metric:

since

The transition to quantization is readily achieved, if we rewrite (omitting the index P here) the real spinor field (32) by replacing the trigonometric by exponential functions. We then obtain

with a new abbreviation that is the operator

properties

is

To render (42) becoming a quantum field, we just have to promote the amplitudes to fermion annihilation and creation operators that obey the canonical anti-commutation rule

With these comments and conclusions we shall close the discussion of the bilinear forms which determine the mass term in the Lagrangian density.

Finally, we will return to the Majorana equation and present its eight-component version which can be obtained by a similarity transformation from the real Dirac Equation (16). The similarity transformation matrix associated with the standard Dirac equation can be written as

By definition it is unitary and has the property

In a similar way we can now subject the real Dirac Equation (16) to a corresponding similarity transformation defined as

where use is made of the y-component of the gamma matrix four vector as given in Equation (17). Note that again the matrix

Remember that

which includes the coupling to the electromagnetic field via the matrix

and the Majorana spinor field is connected to the Dirac spinor field by the transformation

Without an electromagnetic field for

In this paper we have shown that the standard complex Dirac equation can be transformed into a real Dirac equation, which still permits the electromagnetic field to be introduced by minimal coupling, and which enables the real nature of that equation to be preserved in an eight-component spinor representation. This coupling to the electromagnetic field is established via the SO(2) symmetry group, which is equivalent to the U(1) symmetry of the complex Dirac equation. Thus the real Dirac equation also describes massive charged fermions coupled to an electromagnetic field. Coupling to other non-abelian gauge symmetry groups may also be possible, while pre- serving the real nature of that equation, if for the involved group use is made of its adjoint representation, which is purely imaginary and thus yields a real covariant derivative.

The real Dirac equation has a beta matrix that is antisymmetric, a property which causes problems for the fundamental Lorentz-invariant bilinear forms, such as

The real Dirac equation as here derived has essentially the same physical content as its complex ancestor, but looks somewhat cumbersome due to the larger gamma matrices involved. This is the mathematical price to be paid if complex numbers are fully avoided. The related Majorana version appears to be more transparent and seems easier to handle. The electromagnetic gauge field coupling in this equation is of matrix nature, as the imaginary unit is effectively replaced by the matrix

The author gratefully acknowledges the financial and institutional support provided for his work by the Extrater- restrial Physics Division headed by Robert Wimmer-Schweingruber at the Institute for Experimental and Ap- plied Physics of Kiel University in Germany.