Abstract
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.
Highlights
In modern elementary particle physics the complex Dirac equation [1] plays a fundamental role and is used in the standard model (SM) to describe the charged fermions, which are represented in terms of four-component complex spinor fields or two-component complex Weyl spinor fields [4] in the chiral representation for the massless case
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
Summary
In modern elementary particle physics the complex Dirac equation [1] plays a fundamental role and is used in the standard model (SM) to describe the charged fermions (see, e.g., the textbooks by Lee [2] and Kaku [3]), which are represented in terms of four-component complex spinor fields or two-component complex Weyl spinor fields [4] in the chiral representation for the massless case. The aim of the present work is to show that charged fermions can be described by a purely real Dirac equation that does not involve any complex numbers at all but only real matrices and spinors. The gauge symmetry considered thereafter for the coupling of fermions to an electromagnetic field is based on the SO(2) group It is, while being its adjoint representation, closely related to the U(1) group which is commonly used in electrodynamic gauge theory of the standard complex Dirac equation. That for the real version of the Dirac equation the coupling of charged fermions to an electromagnetic field is possible, and can be retained if in that equation the real and imaginary part of a complex Dirac spinor field are assembled in a doublet associated with the SO(2) symmetry group.
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