Also we would like to have a consistent description of the spin of the electron that in the nonrelativistic theory has to be added by hand. Tomotivatethediracequation,wewillstart by studying the appropriate representation of the lorentz group. Solutions of dirac equation the wavefunctions can be written as. The dirac equation we will try to find a relativistic quantum mechanical description of the electron. Majorana particles in physics and mathematics student theses. On charge conjugation, chirality and helicity of the dirac. In this paper we demonstrate that both of them can be considered as a special cases of the more general equation. How dirac and majorana equations are related murod abdukhakimov murod. Qm of majorana particles weyl, dirac and majorana relativistic equations.
This work clarifies the relation between maxwell, dirac and majorana neutrino equations presenting an original way to derive the dirac and neutrino equation from the chiral electrodynamics leading, perhaps, to novel conception in the mass generation by electromagnetic fields. In mathematical physics, the gamma matrices, also known as the dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the clifford algebra c. To sketch the derivation, note that the usual graph for. Within this paper the generic form of dirac equation will be written as. We discuss the dirac, majorana, and weyl fermion fields. A new decomposition of the dirac spinor field is suggested and achieved by means of projection operators based on charge conjugation, which is discussed here in a nonstandard way. In the majorana representation of the simple twosite hamiltonian right there are two crossing interaction lines. Dirac firstly introduced is the socalled chiral or weyl representation. Crucial to this will be using the hoa methods to solve the related minimalcoupled dirac equation. In mathematical physics, the gamma matrices,,, also known as the dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the clifford algebra c. Dirac and majorana edge states in graphene and topological. What is the usefulness of the majorana representation.
The paper of majorana about relativistic particles of arbitrary spin in the paper of majorana the following linear wave equation of the dirac type was introduced. We argue that our formalism can be useful to have a better understanding of possible majorana fermions. The so called twocomponent majorana equation 1,2 permitting a finite mass term has been used in modern quantum field theory for the description. Dirac attempted to overcome some of the problems of relativistic quantum mechanics by introducing a rstorder wave equation. So, as with all discussions surrounding quantum eld theory, it is probably best to start with looking at the dirac equation3. The twocomponent majorana equationnovel derivations and. Majorana representation for dissipative spin systems. When interpreted as the matrices of the action of a set of orthogonal basis vectors. Those of you who have studied diracs relativistic electron equation may know that the 4component dirac spinor is actually composed of two 2component spinors that weyl introduced to physics back in 1929. In modern gauge theories, chiral spinors answer to twocomponent majorana equations 9, 10. This equation was diracs way of formulating a description of elementary spin1 2. The physics of ettore majorana through just a handful of papers, ettore majorana left an indelible mark on the. In the wesszumino model the simplest susy model a supermultiplet is constructed from a complex scalar, auxiliary pseudoscalar field, and majorana spinor precisely because it has two degrees of freedom unlike a dirac spinor.
Pdf a new route to the majorana equation researchgate. The majorana spinor field is a spacetime dependent majorana spinor, solution of the free dirac equation. Formal analogy between the dirac equation in its majorana. I think this was because of the slightly unusual usage of these terms in physics parlance.
Both operators are associated with a single wave function. A real version of the dirac equation and its coupling to. Question about majorana fermion and majorana representation. In this introductory chapter we describe both and explain how they apply to graphene and topological superconductors. A selfcontained derivation of the formalism describing weyl, majorana and dirac fields from a unified perspective is given based on a concise description of the representation theory of the proper orthochronous lorentz group. Shortly afterwards, in 1938, majorana mysteriously disappeared, and for 70 years his modified equation remained a rather obscure footnote in theoretical physics box 1.
The standard representation of the dirac matrices 3is obtained from 5 by means of a similarity transformation. It is also possible to define higherdimensional gamma matrices. We revisit the chargeconjugation operation for the dirac equation in its chiral representation. Proposal for realization of the majorana equation in a. The majorana spinor eld is a spacetime dependent majorana spinor, solution of the free dirac equation. A majorana spin representation is essentially a real spin representation see at spin representation real representations but regarded as a complex spin representation equipped with real structure recalled as def.
The spinor elds, spacetime dependent spinors, are solutions of the free dirac equation 16. Pdf weyl, majorana and dirac fields from a unified. The dirac equation for the wavefunction of a relativistic moving spin1. When dirac first wrote down his relativistic equation for a fermion field. This has rekindled the interest in majorana dynamics and, recently, it has been proposed both a procedure to implement nonphysical operations related to majorana physics 11 and a scheme to simulate the majorana. The majoranafourier and majoranahankel transforms of majorana spinor fields are defined.
Multiply the nonconjugated dirac equation by the conjugated wave function from the left and multiply the conjugated equation by the wave function from right and subtract the equations. Thus, two separate twocomponent majoranatype field equations for the eigenfields of the chargeconjugation operator. This is a useful representation called the weyl representation or chiral. Majorana and dirac equations are usually considered as two di erent and mutually ex clusive equations. And one fermionic state is described by two majorana states. The matrices in equation 15 provide one example, known as the weyl or chiral representation for reasons that will soon become clear. Fermions and the dirac equation in 1928 dirac proposed the following form for the electron wave equation. The tensor dirac equation extends immediately to general coordinate systems, thus to noninertial e. The eigenfunctions of the majorana equation are calculated in a concise way. The majorana representation of spin operators allows for efficient fieldtheoretical description of spinspin correlation functions. A full derivation of the results in equations 89,90 can be found, e. The bestknown ones are the dirac, weyl and majorana representations. Once the dirac solutions are known, the majorana solutions follow directly. The hilbert space of dirac spinor elds is complex, while the hilbert space of majorana spinor elds is real.
Majorana spinors are used frequently supersymmetric theories. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory, which are commonly not included in. For a certain class of npoint spin correlation functions including auto and pairwise correlations a further simplification is possible, as they. In this situation, these solutions are realvalued and describe a onedimensional majorana single particle. The fourcomponent real dirac equation in its majorana representation is shown to be the natural outcome of the twocomponent complex majorana equation. Derivation of the real fourcomponent majorana equation.
It is pointed out that these definitions have to do with the proper lorentz group and not with any discrete symmetry. There are several choices of signature and representation that are in common use in the physics literature. The majorana equation at the start we rederive the real fourcomponent spinor majorana, without recourse to the dirac equation equation thereby following the recent work of aste and marsch 1210. In the present article, it is shown that maxwell equations can be written in the same form as the two components dirac. It is well known that at times we do need explicit representations for the dirac gamma matrices while doing calculations with fermions. The majorana equation with a mass term provides a useful description of massive neutrinos. Any npoint spin correlation function is equivalent to a 2 npoint correlator of majorana fermions. The definitions and motivations for introducing each kind of field is discussed, along with the connections between them. The majorana spinor representation of both sl2,c and pin3,1 is irreducible 12. The majorana spinor representation of the poincare group. Recently i found two different expressions for majorana representation for the gamma matrices.
The schrodinger equation is not relativistically invariant. Question about majorana fermions physics stack exchange. However, both of them can be considered as a special cases of the more general equation. The dirac equation is one of the two factors, and is conventionally taken to be p m 0 31 making the standard substitution, p. Notice that there are other spinor fields which would not be called dirac fields, such as those transforming in a majorana representation or a weyl representation, or for example the gravitino field which would be called a raritaschwinger field. The action of discrete symmetries, such as charge conjugation and cp on various types of fermion. The lor entzinvariant complex conjugation operation involves the spinflip operator, and its connection to chiral symmetry is discussed.
Dirac equation, fourvector wave function, bargmannpauli hermitizing matrix, dirac gamma matrices. Formal analogy between the dirac equation in its majorana form and the discretevelocity version of the boltzmann kinetic equation f. For subsequent use we also introduce another name for the real gamma matrix, writing it as. The twocomponent majorana equationnovel derivations. The dirac equation can be thought of in terms of a square root of the kleingordon equation. Thus we can readily build up the majorana version of the dirac equation in its chiral representation. In this section we will describe the dirac equation, whose quantization gives rise to fermionic spin 12particles.
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