Weyl, Majorana and Dirac fields from a unified perspective

A self-contained 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. Lagrangian methods play no role in the present exposition, which covers several fundamental aspects of relativistic field theory which are commonly not included in introductory courses when treating fermionic fields via the Dirac equation in the first place.


Introduction
There can be great advantages in choosing a suitable notation when formulating a theory. E.g., it is customary to denote the contravariant components of the Cartesian coordinates of an event in flat 3 + 1dimensional Minkowski spacetime M ∼ = (R 1+3 , η) by a column four vector where the speed of light c accounts for equal physical units of the components x µ defined by the event time t = x 0 /c and the corresponding space coordinates given by a column vector x = (x 1 , x 2 , x 3 ) T . The Lorentz-invariant Minkowski bilinear form η is then defined by with the metric tensor g = diag(1, −1, −1, −1). However, one could also hit upon the idea to arrange the spacetime coordinates in matrix form according to [1] x = Then, the indefinite Minkowski norm squared can be written in an elegant manner as a determinant. Furthermore, with one finds a compact expression for the Minkowski scalar product η(x, y) = 1 2 tr(x y) .
Based on notational tricks of this kind, it is possible to derive and discuss the field equations of the fundamental fermionic fields appearing in the Standard Model and its modern extensions in a very elegant manner, as will be demonstrated below. Furthermore, the following discussion includes a thorough analysis of the basic properties of Weyl [2], Dirac [3] and Majorana [4] fields emerging from first principles like Lorentz symmetry and causality.
In Section 2, the most relevant topological and group theoretical properties of the Lorentz group are revisited, resulting in the construction of the fundamental ray representations of the proper orthochronous Lorentz group in Section 3. Section 4 deals with Weyl

Structure of the Lorentz group
In the following, Lorentz transformations will be interpreted as passive transformations, i.e. when an observer in the inertial system (or inertial frame of reference) IS assigns the contravariant Cartesian Minkowski coordinates x µ to an event, an observer in another inertial system IS' with a common point of origin will assign coordinates x µ to the same event. Then, the coordinates are related by a Lorentz transformation expressed by a matrix Λ according to x = Λx or Since the Minkowski metric is preserved under such transformations, one has for all x, y ∈ (R 1+3 , η) η(x, y) = x T gy = η(Λx, Λy) = x T Λ T gΛy , and consequently Λ T gΛ = g .
called the proper Lorentz group L + which is (isomorphic to) the special indefinite orthogonal group SO (1,3).
Representing a matrix Λ ∈ O(1, 3) according to the decomposition where γ is a real number, a and b are column vectors, and M is a 3 × 3 matrix, a short calculation using (Λ −1 ) T = gΛg −1 gives Furthermore L ↑ + is the identity component of the Lorentz group, containing the identity element denoted in the following simply by '1', by 1 4 , or diag(1, 1, 1, 1). L ↑ − contains the space reflection P with det P = −1, L ↓ + contains the spacetime reflection P T = −1 4 with det(P T ) = 1. The four transformations {1 4 , P , T , P T } constitute the discrete Klein four group V .
The so-called complex Lorentz group O(4, C) = {Λ ∈ GL(4, C) | Λ T Λ = 1 4 } consists of two connected components SO(4, C) = O + (4, C) and O − (4, C) only, which can be characterized by the determinant of their elements. The identity component SO(4, C), which contains the unimodular matrices with positive determinant, is also sometimes called the proper complex Lorentz group. One may note that the signature of the metric tensor does not play any rôle for the definition of the abstract group structure of the complex Lorentz group. A matrixΛ ∈ O(1, 3; C), which fulfills the conditionΛ T gΛ = g involving the metric tensor g = diag(1, −1, −1, −1), becomes via a similarity transformation and therefore Λ ∈ O(4, C).
3 Fundamental ray representations of the proper orthochronous Lorentz group 3.1 Explicit construction of the two two-dimensional inequivalent irreducible fundamental ray representations In order to construct the lowest-dimensional non-trivial irreducible representations of the proper orthochronous Lorentz group, one may introduce the relativistically generalized Pauli matrices where the three components in σ are given by the Pauli matrices The symbol σ 0 used for the identity matrix 1 2 in two dimensions fits nicely into the notation used above.
Arbitrary four vectors x with contravariant components onto the set Herm(2, C) of Hermitian 2×2 matrices. The inverse mapping can be easily obtained from a generalization of the well-know trace identity i.e. 1 2 tr (σ µσν ) = g µν .
Then one has The Minkowski scalar product can be transferred from M onto Herm(2, C) via and the identity leading to Alike, with y =σ µ y µ one obtains the compact expression for the Minkowski scalar product The special linear group SL(2, C) in two complex dimensions is defined by Now, a handy trick relies on the possibility to act with a matrix A ∈ SL(2, C) on x ∈ Herm(2, C) according to where + denotes Hermitian conjugation. Obviously, x is Hermitian again, and the Minkowski scalar product is preserved in the following sense Thus again, x can be represented by a real linear combination of generalized Pauli matrices and A explicitly acts as a Lorentz transformation due to Since x and x are obviously related by a Lorentz transformation, one has Λ ∈ O (1, 3), and a closer inspection shows that Λ ∈ SO + (1, 3) holds indeed. Actually, SL(2, C) is simply connected as will be demonstrated later. Since the mapping λ : A → Λ(A) is obviously continuous, it is also a homomorphism of the group SL(2, C) into the proper orthochronous Lorentz group L ↑ + = SO + (1, 3). Furthermore, λ is surjective, and SL(2, C) is the double universal covering group of the SO + (1, 3).
Mapping covariant components x µ of a four vector according to x = σ µ x µ = σ µ x µ onto Herm(2, C), the transformation law corresponding to eq. (34) reads x = (A + ) −1 xA −1 . Then one also has To convince oneself that the homomorphism λ is two-to-one, one observes first that two matrices ±A ∈ SL(2, C) generate the same Lorentz transformation, since AxA + = (−A)x(−A) + . The kernel of λ, i.e., the set of all A ∈ SL(2, C) which fulfill the equation for every Hermitian matrix x, can be determined by first considering the special choice x = 1 0 0 1 , which leads to the condition A = (A + ) −1 for A ∈ ker(λ), and eq. (39) then reduces to xA − Ax = [x, A] = 0 for every Hermitian x. This implies A = α1 2 , α ∈ C. Eventually, from the condition det A = +1 follows A = ±1 2 .
Aside from the important group isomorphism just found above the matrices in SL(2, C) display a further interesting property. Defining the fully anti-symmetric tensor a symplectic (and therefore skew-symmetric) bilinear form u, v = − v, u can be defined on two socalled spinors u and v, which are elements of the two-dimensional complex vector (or spinor) space equipped with the symplectic form ·, · according to In analogy to the SO + (1, 3)-invariance of the metric tensor g, this symplectic form is SL(2, C)-invariant This can be easily demonstrated by a short calculation. With one has such that a further group isomorphism is established Sp(2, C) is the complex symplectic group in two dimensions The accidental isomorphism SO(3, C) ∼ = L ↑ + is mentioned here without proof for the sake of completeness.
To sum up, the trick expressed by eq. (34) serves to construct a real linear representation of the Lie group SL(2, C) by the Lie group SO + (1, 3) with the defining property for representations Eq. (49) can be inverted up to a sign, and in a loose style one may write A(Λ 1 Λ 2 ) = ±A(Λ 1 )A(Λ 2 ), i.e. also a two-valued ray representation of the Lorentz group L ↑ + has been found.
The inversion of eq. (37), which fails for some Λ for topological reasons, reads The derivation of eq. (50) is left to the reader as an exercise.
One may ask whether the two-valued ray representation of the proper orthochronous Lorentz group or the representation of the SL(2, C) by itself is equivalent to the complex conjugate representation. It turns out that no matrix B exists such that for all A ∈ SL(2, C) holds. However, restricting our considerations to the SL(2, C)-subgroup the situation is different, because a special unitary matrix U ∈ SU (2) given by is related to its complex conjugate matrix U * by a similarity transformation expressed by the help of the anti-symmetric tensor = iσ 2 Therewith the ray representation of the rotation group SO(3) by the special unitary group SU (2), obtained from the restriction of the SL(2, C) representation constructed above via the trick in eq. (34) to the subgroup SU (2), turns out to be equivalent to its complex conjugate representation. The SU (2) matrices in the SL(2, C) generate spatial rotations; Hermitian matrices in the SL(2, C) generate the boosts.

Wigner boosts
A Wigner boost is a proper orthochronous Lorentz transformation, which transforms a given four vector q into another four vector p. For illustrative purposes only, the special case with m > 0 shall be investigated here. Since q is a future-directed, timelike vector, p is also contained in the open forward light-cone and one has With q = σ µ q µ = m1 2 and a Wigner boost is given, since The explicit expression for the square root in eq. (56) can be easily verified by a short calculation. For m = 0 the expression becomes useless, signaling fundamental differences between the physics of massive and massless particles.

Topology of the SL(2, C) group manifold
An invertible matrix A ∈ GL(n, C) possesses the polar decomposition where H = H + ∈ Herm(n, C) is a positive definite Hermitian matrix and U = (U −1 ) + ∈ U (n) is unitary. For A ∈ GL(1, C) =Ċ, the polar decomposition reduces to the well-known form Singular matrices can be represented by the product of a unitary and a positive semi-definite Hermitian matrix.
The existence of the polar decomposition for matrices follows from the observation that the matrix H = AA + is Hermitian, since (AA + ) + = A ++ A + = AA + . Obviously,H is also invertible when A is invertible. If v is an eigenvector of AA + with a corresponding eigenvalue λ, then due to Hermiticity λ is real and from one even has λ > 0. Since det A = 0, λ 1,...n > 0 holds for all n eigenvalues λ 1 , . . . λ n of AA + . Diagonalizing AA + by a unitary matrixŨ ∈ U (n) leads to the real diagonal matrix accordinglyH =Ũ −1 DŨ also holds. Now, choosing H and U as follows directly leads to the desired polar decomposition, since For the special case A ∈ SL(2, C) ⊂ GL(2, C) follows a unique decomposition with det H = 1 and U ∈ SU (2), since The representation where h ∈ R 3 can be chosen in an arbitrary manner, implies implying the homeomorphism since SU (2) ∼ = S 3 . Since both manifolds R 3 and S 3 are simply connected, the same observation follows for the group manifold SL(2, C) as a topological product space. Whereas the group manifold of the SU (2) is homeomorphic to the compact three-dimensional sphere S 3 , the Lorentz group manifold is non-compact due to the non-compact factor R 3 in eq. (67). Is is assumed here that the reader is well acquainted with the basic topological facts concerning the manifolds and matrix Lie groups equipped with their standard topologies discussed so far.
All four components of the Lorentz group O(1, 3) like the SO + (1, 3) are not simply connected, but the covering group SL(2, C) of SO + (1, 3) is. This topological difference and the related two-to-one surjective homomorphism of the SL(2, C) onto the proper orthochronous Lorentz group discussed above is the origin of spinor physics, a fact which is deeply related to Wigner's theorem where physical states are related to rays in a Hilbert space [7]. However, we will not dwell any further on the specific aspects of Wigner's theorem in this paper.
and the transformation (34) we now use the fact that a two-to-one correspondence Λ(±A) ↔ ±A(Λ) exists between proper orthochronous Lorentz transformations Λ ∈ SO + (1, 3) = L ↑ + and two corresponding elements ±A of the special linear group SL(2, C) in order to construct the most fundamental spinor wave equations.
Obviously, eq. (69) implies In analogy to the construction given by eq. (25), one defines the matrix-valued differential operator [8] E.g., this operator can act by a formal matrix multiplication from the left on a two-component wave function Ignoring the trivial case where the two components of the wave function individually behave as scalar wave functions the spinor components of the wave function ψ(x) passively transform equivalently to a irreducible fundamental ray transformation ( 1 2 , 0) or (0, 1 2 ) in the following sense with fixed invertible matrices S, S ∈ GL(2, C), such that one has of course SAS −1 ∈ SL(2, C) An observer in an inertial system IS' using coordinates x µ will use the operator (71) according ∂ = σ µ ∂ µ . With eq. (70), one immediately notes the corresponding transformation law The following simple differential equation for a two-component, necessarily complex spinor wave shall serve now as a first Ansatz for a relativistically invariant wave equation. In this context, relativistic invariance means that the wave equation (78) holds in all inertial systems, a fact that is readily verified. From ∂ψ(x) = 0 trivially follows A∂ψ(x) = 0. Here, A is an SL(2, C)-matrix associated with a Lorentz transformation Λ. In wise foresight, one requests that ψ(x) obeys the manifestly covariant transformation law in accordance with eq. (76) The special property of the matrix A ∈ SL(2, C) = Sp(2, C) Obviously, the 'natural law' ∂ψ = 0 is valid in a manifestly Lorentz-invariant form in every inertial system. Manifest Lorentz invariance of a formalism provides great advantages from the calculational point of view, but this certainly does not imply that more involved formalisms depending on a specific frame of reference may play a rôle in theoretical physics.
Spinors transforming according to eq. (76) are called left-chiral spinors. One has to mention that this chirality (from the greek χειρ, 'hand') should not be confused with the helicity (from the greek ελιξ, the 'twisted') of the particles described by the wave functions discussed here. Helicity is defined via the direction of the momentum and the angular momentum of a particle, and it is not a Lorentz-invariant property for massive particles. In the massless case however, helicity can be linked directly to chirality.
In the non-interacting case, a spinor ψ L (x) as in eq. (82) obeys the so-called left-chiral Weyl equation which has been put in a form explicitly containing the nabla operator Of course, the right-chiral case is missing so far in the present discussion. Therefore, one considers the operator where the easily verifiable identites σ −1 = − σ T = − σ * have been used. Then, the right-chiral Weyl equation for a right-chiral spinor ψ R reads and again one can check for the manifest Lorentz invariance of this equation. The transformation law (77) implies for the operator ∂ the transformation law and postulating the simple transformation law correponding to the right-chiral (0, 1 2 )-representation for the right-chiral field ψ R (x) directly leads to the desired result Together, the operators ∂ and ∂ possess the interesting property which can be expressed in a more elegant manner by exploiting the analytic symmetry Eq. (91) implies that both field components of a left-or right-chiral Weyl field fulfill the Klein-Gordon equation. From a group theoretical perspective, the differential operators ∂ and ∂ are of a more fundamental significance that the wave operator , since the two two-component spinor operators, which are related to the (1/2, 0)and (0, 1/2)-representations of the proper orthochronous Lorentz group, allow for the construction of wave equation for higher-spin fields with more involved transformation properties. The Klein-Gordon wave operator, which is linked to the trivial representation of the Lorentz group, does not contain this group theoretical information.
The Weyl equations do not describe a parity invariant world. Introducing a passive parity transformation and considering an observer describing the dynamics of a Weyl field by ψ(x) and a point reflected observer describing the same Weyl field in 'his own words' by ψ (x ), one must have a linear transformation law connecting the mathematical entities used by the two observers with an appropriate 2 × 2-matrix A P which makes it possible to translate theoretical or experimental aspects related to the Weyl field from one observer to the other. It is a simple exercise to show that no matrix A P exists such that both ψ(x) and ψ (x ) fulfill the left-chiral (or right-chiral) Weyl equation at the same time. In fact, a parity transformation transforms a left-chiral field into a right-chiral field and vice versa. Of course one may wonder how it is possible to mirror an observer. Anyway, it is much easier to boost or to rotate a person or a measuring device.

Two-component Majorana equations
The Weyl equations suffer from the disadvantage that they do not describe massive particles. Modifying, e.g., the left-chiral Weyl equation by a naive mass term according to with an arbitrary but non-vanishing 2 × 2 mass matrixm, the wave equation turns out to be non-Lorentz invariant. A solution ψ (x ) of the left-chiral Weyl equation in an inertial system IS' does not fulfill the Weyl equation in a different inertial system IS since In 1937, Ettore Majorana found an unconventional way out of this disturbing situation by coupling a field with its complex conjugate field [4]. The left-chiral Majorana equation obeys the desired transformation law, The mass term m must be a scalar in order to commute with every possible spinor Lorentz transformation matrix A in eq. (97).
In complete analogy to the considerations above, one may write down the right-chiral Majorana equation. Since the mass term can be equipped with a so-called Majorana phase in both the left-and the right-chiral case, it is common usage in the literature to formulate the field equations and the corresponding transformation laws with in the following manner (m L,R ∈ R, |η| = 1): Majorana fields play an important rôle as fundamental theoretical building blocks in supersymmetric quantum field theories.
In the non-interacting case the phases η L,R have no physical significance and can be removed by a redefinition of the fields by the help of a global gauge transformation. With ψ L (x) → ψ L (x) = ψ L (x)e −iδ L /2 and η L = e iδ L one has, e.g., in the left-chiral case i.e. ψ L (x) fulfills a phase-free Majorana equation.
For left-handed Majorana particles one obviously has due to ∇ψ L (x) = 0 Differentiating the left equation above with respect to time and using the complex conjugate equation at the right,ψ * L,2 = −iη * L m L ψ L,1 , leads tö Therefore, ψ L,1 is a linear combination of e −imx 0 -and e +imx 0 -terms, and particles with their spin directed parallel or anti-parallel to the 3or (z-)axis are described by the wave functions (ψ 2 = − i m ηψ * 1 ) The wave functions given by eqns. (104) and (105) can be sped up, e.g., by Wigner boosts. Both the left-and the right-chiral Majorana fields ψ L,R describe one species of particles in the following sense: all plane wave solutions of the corresponding left-or right-chiral Majorana equations can be transformed into each other by appropriate Poincaré transformations, i.e. by Lorentz transformations and spacetime translations. Starting from the idealized, improper state of a particle at rest with a given spin direction, all other states of the particle with sharp momentum can be generated by boosts and rotations.
The coupled system of equations (106) and (107) is manifestly Lorentz invariant, since and introducing the so-called gamma matrices in chiral representatioñ one obtains the Dirac equation in its chiral representation with It is straightforward to check that the gamma matrices fulfill the anti-commutation relations e.g., one has Historically, the relations (114) were found in 1928 by Paul Adrien Maurice Dirac in his Ansatz [3] to 'linearize' the Klein-Gordon equation according to which led him to conditions for the coefficients γ µ = g µν γ ν enforcing the introduction of a Clifford algebra of gamma matrices γ 0 , γ 1 , γ 2 and γ 3 , which can be represented in the lowest-dimensional case by 4 × 4-matrices.
It turns out that the gamma matrices can be represented in different ways. The anti-commutation relations (114) are invariant with respect to a similarity transformation with a non-singular matrix in GL(4, C), and apart from the chiral representation the literature tends to use a standard representation with matrices γ µ called the Dirac representation which is linked to the chiral representation bỹ In the sequel, chiral gamma matrices shall be denoted byγ µ and (standard) Dirac matrices by γ µ . The standard Dirac matrices are explicitly given by and for many purposes, it is convenient to define a matrix γ 5 in a representation-independent manner It is well-known that the solutions of the Dirac equation describe spin-1 2 particles together with their antiparticles with the same mass. The Dirac matrices γ µ are especially well-suited for investigations of the low-energy limit of the Dirac equation.
The matrix U in the eqns. (118) is unitary; as a matter of fact, all representations of the gamma matrices which are unitarily equivalent to the chiral or Dirac representation exhibit the following (anti-)-Hermiticity relations respectively which provide some advantages for the discussion of energy and momentum observables.
An important result of the theory of Clifford algebras states that each set of four 4 × 4-matrices fulfilling the anticommutation relations (114) can be brought into the chiral or Dirac form by a similarity transformation of the kind (118), where U is invertible but not necessarily unitary. This nice feature enables theoretical physicists working in different solar systems to compare their calculations by some simple conversions. In this sense, the Dirac equation is universal.
Applying a similarity transformation to the Dirac matrices according tô (123) with an invertible matrix B, then the transformed Dirac spinorΨ = BΨ fulfills the Dirac equation with the new gamma matricesγ µ again, since from (iγ µ ∂ µ − m)Ψ(x) = 0 follows From now on, spacetime arguments will be omitted for the sake of notational brevity. In the eqns. (106) and (107), solely one single real mass term coupling a left-and a right-chiral two-component field shows up. Indeed, a more general Ansatz i∂ψ R −m D,L ψ L = 0 (126) with complex chiral mass termsm D,R andm D,L is conceivable. Acting with the operator −i∂ on eq. (125) and using eq. (126) yields Hence, the left-chiral part ψ L respects a Klein-Gordon-type equation, and the same follows in complete analogy for the right-chiral part ψ R +m D,RmD,L ψ R = 0. However, for the correct energy-momentum relation to hold true, one must requirem The degenerate case m 2 = 0 is not particularly interesting. E.g., for Only one real mass term m D is relevant for the present theory from the physical point of view. Of course, one may argue about the physical relevance of parameters in non-interacting theories. Eventually, the phase factors e ±iϕ D can be trivially eliminated by a chiral phase transformation with β = −ϕ D , so that the fieldsψ L,R fulfill the phase-free Dirac equation These phase transformations do not represent a gauge transformation of the four-component Dirac spinor, merely one has to state that the same physical information is encoded in the fieldsψ L,R as in ψ L and ψ R . Thus, the transformation trick above does not imply that phases in interacting theories are not related to measurable quantities. A gauge transformation would leave e iϕ D unchanged.
Actually, purely imaginary representations of the gamma matrices which are unitarily equivalent to the standard Dirac matrices exist. Using such matrices in a so-called Majorana representation, the Dirac equation becomes a purely real differential equation.

Real four-component Majorana equation
Decomposing the two complex components of a left-chiral spinor according to one obtains from the two-component Majorana equation (for the sake of simplicity a trivial Majorana phase such that η L = 1 shall be used for the forthcoming considerations) (142) after a separation into real and imaginary parts, the real linear system of first order differential equations By the help of the purely imaginary Majorana (gamma) matrices (γ M µ = g µν γ ν M ) A further, purely imaginary representation of the Majorana matrices spread in the literature is given byγ (149) This representation can be obtained from the original representation (148) by the unitary transformatioñ where det(Ũ ) = −1.
the condition that the Majorana-Dirac equation should describe neutral particles becomes therefore the neutrality condition for the transformed four-components Majorana spinor now readŝ For real, i.e. orthogonal U ∈ O(4) ⊂ U (4) one has U U T = 1 4 , and so againΨ * M =Ψ M .
The discussion above illustrates the complete equivalence of the four-component and the two-component Majorana formalism in the literature. The four-component field is related to a irreducible fourdimensional real spinor representation of the Lorentz group, whereas the two-component formalism is based on the two fundamental complex spinor representations.

Weyl-Majorana-Dirac formalism
Considering now the most general free field case, the left-and right-chiral fields can be coupled via linear and anti-linear terms according to the following "Weyl-Majorana-Dirac equation" with phases ϕ D,L , ϕ D,R , ϕ L , ϕ R ∈ (−π, π] can also be used for notational convenience. When all mass terms vanish, the eqns. (154) and (155) trivially describe a left-and a right-chiral field. But in the following, we consider the non-trivial Majorana-Dirac case where none of the mass terms above vanishes.
One may note first that using where the operator K denotes complex conjugation. Hence, eqns. (154) and (155) are fully equivalent to (keeping in mind that (iσ 2 ) 2 = 2 = −1 2 , Kηϕ = η * Kϕ, and Ki = −iK) i.e., the left-chiral field ψ L is physically equivalent to a right-chiral field ψ * L , whereas the right-chiral field ψ R is equivalent to the left-chiral field ψ * R .
Working with the original eqns. (154) and (155), which can be cast into the form yields the compact representation Acting with (−iγ ν ∂ ν −m) on eq. (161) leads to where (beware that Ki = −iK) Using K∂ = ∂ K and K∂ = ∂ K again, the expression above reduces to We remember now that it is in fact possible to rescale, e.g., the right-chiral field ψ R according to eq. (133) in order to obtain field equations where the modulus of the Dirac mass terms fulfills m D,R = m D,L . Additionally, introducing a phase-transformed left-chiral field ψ L according to eq. (136) eqns. (154) and (155) can be written as A phase transformation of the right-chiral field ψ R only Observing that the left-chiral Dirac mass termm D,L picks up the opposite phase compared to the rightchiral mass termm D,R under a phase transformation and considering the effect of rescaling one of the fields ψ L,R shows that one could also start with an equivalent field theory wherem D,R =m D,L = m D = 0. Additionally, the phase ofm D can be chosen to fulfill Form D =m D,R =m D,L , the operator (iγ ν ∂ ν )m −m(iγ µ ∂ µ ) in eq. (166) vanishes and the Majorana-Dirac equation (160) is equivalent, after appropriate rescaling and phase transformation of the corresponding fields, to the field equation where any superscripts due to aforegoing scaling and phase transformations have been omitted, and the fields obey the Klein-Gordon equation with generalized mass terms However, there is still the freedom to perform a gauge transformation leaving the Dirac massm D invariant but changingm K andm R by a common phase. This freedom can be used to redefine the fields and correspondingly rotate the phases ofm L andm R in order to obtain and every Majorana-Dirac equation with suitably redefined fields leads to the Klein-Gordon equation withμ =m * Dm L +m DmR . This Klein-Gordon equation can be written in a manifestly real form by introducing the real spinor Φ = (ψ 1 , ψ 2 , . . .
with eight real components given by Obviously, the mass operatorM 2 must be positive semi-definite in order to exclude time-asymmetric complex mass solutions or even tachyonic solutions of the Majorana-Dirac equation, and it must be Hermitian in order to generate a unitary dynamics of the single particle states described by the wave function Φ. Only then the solutions of eq. (180) describe well-behaved normalizable single particle states as part of a stable theory which are eigenstates of the energy-momentum squared Casimir operator of the double covering group of the inhomogeneous Poincaré groupP ↑ + T 1,3 SL(2, C), which is the semi-direct product of the time-space translation group T 1,3 and the universal cover SL(2, C) of the proper orthochronous Lorentz group SO + (1,3). This condition restricts the admissible mass terms, as discussed in the following.
Using the abbreviationsμ = µ 1 + iµ 2 andm 2 D = ν 1 + iν 2 , the mass operatorM 2 readŝ (181) Some straightforward algebra results in the following four doubly degenerate eigenvalues ofM 2 The Hermiticity ofM 2 implies the conditions ν 2 = 0 and µ 2 = 0. From ν 2 follows thatm 2 D is real; considering additionally that κ is real and alsoμ must be real,m D becomes a real and positive parameter with relation (173). Then, the eigenvalues ofM 2 become with m =m L +m R real. This is the origin of the Majorana phase ϕ M graphically depicted in Fig. (2). Writingm the sine and the cosine law imply together with For the special case ϕ M = 0, the two Majorana masses become The explicit expressions for the mass eigenvalues illustrate how the presence of Majorana mass term split a Dirac field into a couple of two Majorana fields.
The discussion of the rather trivial cases where one or several of the Majorana or Dirac mass terms are absent or degenerate cases where some masses have the same modulus and special phase relations is left to the reader as an interesting exercise. In order to understand the findings of the last section from a more general perspective, we finally leave the restricted framework of the fundamental representations of the proper orthochronous Lorentz group and shortly revisit the most important results from the theory of its real and complex finite-dimensional representations [10]. Such a discussion fits nicely into the considerations exposed so far for spin-1 2 fields only and will clarify the group theoretical background of the results obtained in the last section. It is assumed below that the reader is well acquainted with the basic notions of representation theory. relates complex conjugate representations which are not equivalent for j = j . The representations ϑ j,j are real, i.e. they can be represented by real (2j +1) 2 ×(2j +1) 2 -matrices. Only the trivial representation ϑ 0,0 of the Lorentz group is unitary. All other unitary irreps of the Lorentz group are infinite-dimensional and are commonly constructed by the help of wave function spaces.
The following fields, transforming according to the lowest-dimensional, not necessarily irreducible (ray) representations ϑ j,j of the proper orthochronous Lorentz group, play the most important rôles in relativistic (quantum) field theory in 3 + 1-dimensional Minkowski spacetime: • (j, j ) = (0, 0): Real or complex scalar field ϕ(x).
As a reminiscence to the literature using dotted and undotted spinor indices according to varying conventions, two types of spinor indices were used above to distinguish between the two fundamental SL(2, C)-representations.
, and the transformation law following for V αβ (x) under the direct product of the representations ϑ 1 2 ,0 and ϑ 0, 1 can be cast into an interesting form by using the generalized Pauli matrices as a basis of the complex vector space of the 2 × 2-matrices M at(2, C) in order to define the four fields V 0 , V 1 , V 2 , and V 3 according to Indeed, V µ is a vector field and transforms like the spacetime coordinates. The V µ are not necessarily complex, as one knows from the relativistic four-potential A µ in electrodynamics or the (massive) classical Proca field Z µ used to describe the classical Z-boson. In the complex case, the vector field may be used to describe charged fields W * µ = W µ and associated particles like the W -bosons.
• (j, j ) = (1, 0) oder (0, 1): Complex Riemann-Silberstein vector fields The direct sum of the representations (1, 0) ⊕ (0, 1) can be used to construct a six-dimensional real representation of the Lorentz group which is linked to the Lorentz transformation properties of the electric and magnetic field E(x) and B(x), respectively.
• ( 1 2 , 0) ⊕ (0, 1 2 ): Dirac spinors Ψ(x). Dirac spinors are used to describe the Standard Model spin-1 2 particles, i.e. leptons and quarks. The representation ( 1 2 , 0) ⊕ (0, 1 2 ) can be restricted to four real dimensions and leads to the concept of four-component Majorana fields. This observation is one of the main subjects of this paper and will be elucidated below in further detail.

Real (ray) representations of the proper orthochronous Lorentz group
Complex half-integer representations ϑ j,j with j + j = 1 2 , 3 2 , 5 2 , . . . are called spinor ray representations of the proper orthochronous Lorentz group, integer representations ϑ j,j with j + j ∈ N are tensor representations. Spinor representations are faithful representations of the SL(2, C). The reduction formula (193) explicitly holds in the case of the complex representation theory of the groups SL(2, C) and SO + (1, 3). However, also real irreducible representations play a crucial rôle in quantum mechanics in connection with the description of neutral fields like, e.g., the Higgs field in the Standard Model, the real antisymmetric field strength tensor in electrodynamics or the gravitational field. Of course, these classical entities lead to states in corresponding (Fock-) Hilbert spaces after second quantization, and these states can be superposed according to the manifest complex structure of quantum mechanics.
The real irreducible SL(2, C)-representations can be classified into two types [11]: is obtained from restricting a complex representation ϑ j,j acting on C (2j+1) 2 C to a real subspace which is isomorphic to R (2j+1) 2 R . A more suggestive notation used below for such representations obtained from the complex irreps (j, j) is (j, j) R .
• Type 2: ϑ R j,k ϑ R k,j with j = k is obtained from restricting the direct sum ϑ j,k ⊕ ϑ k,j of an SL(2, C) irrep and its complex conjugate to the real subspace R From a 'complex point of view', such representations are reducible, but they are not reducible in the real sense. These representations shall be denoted below by (j, k) R (k, j) R . Having projected out such a real representation from ϑ j,k ⊕ ϑ k,j , there remains a second equivalent real representation with, of course, the same dimension; the total dimension of both real representations is then 4(2j + 1)(2k + 1) = dim R C 2(2j+1)(2k+1) .
Since the second type is directly linked to the group theory of Dirac and Majorana fields, this case shall be investigated in the following pedestrian way. The real irreducible representations contained in the complex reducible SL(2, C)-representation ϑ j,k ⊕ ϑ k,j can be isolated by the following explicit calculations. Let R and I denote the real and the imaginary part of the n × n-representation matrix D(A) = R(A) + iI(A) with A ∈ SL(2, C) and n = (2j + 1)(2k + 1), corresponding to a given representation (j, k). Then, a 2n × 2n-representation matrixD of the direct sum (j, k) ⊕ (k, j) = (j, k) ⊕ (j, k) * can be written asD As a complex representation matrix,D acts on complex 2n-component column vectors in C 2n C . However, we now focus on the real 2n-dimensional subspace spanned by vectors which can be represented in the form Such vectors are real linear combinations of the 2n basis vectors WhenD acts as a linear operator on such a vector v, one obtains This result can be immediately translated into a real representation defined by real matricesD which display the multiplicative (homomorphism) representation property of their complex counterparts: becomes in the real casê D 2,1 =D 2D1 = R 2 −I 2 I 2 R 2 R 1 −I 1 I 1 R 1 = R 2 R 1 − I 2 I 1 −R 2 I 1 − I 2 R 1 R 2 I 1 + I 2 R 1 R 2 R 1 − I 2 I 1 .
(214) The antisymmetric tensor field F µν is given by 6 real spacetime-dependent field components transforming under the (1, 0) R -representation; the traceless symmetric tensor field H µν contains 9 independent real field components (→ (1, 1) R ), and the component in T µν which is proportional to the inverse metric tensor is related to the real scalar field ϕ(x) (→ (0, 0) R ), everything in accordance with the decomposition displayed in (211).
In matrix notation, the transformation (212) can be expressed by T = ΛT Λ T , and the trace T µ µ becomes tr (T g). Obviously, due to the defining property Λ T gΛ = g of the matrices in O(1, 3) tr (T g) = tr(ΛT Λ T g) = tr(T Λ T gΛ) = tr (T g) holds, i.e., the trace of a second rank tensor is a Lorentz invariant scalar.
Having all this group theoretical tools in our backpack, the obervations elaborated in the last section by explicit calculations now receive a simple explanation. Coupling a left-and a right-handed chiral (Weyl) spinor field by mass terms as performed in eqns. (154) and (155) imposes an additional dynamics on the total field of four complex or eight real field components. In the Dirac case, the mass spectrum is degenerate and the structure of the field equations remains complex such that the field components transform according to the reducible complex representation ( 1 2 , 0) ⊕ (0, 1 2 ); complex linear superpositions of solutions of the field equations are still solutions. In the Majorana case, the representation splits up into two equivalent real four-dimensional representations ( 1 2 , 0) R of type 2, with representation spaces which are invariant under the proper orthochronous Lorentz group, containing two real four-component Majorana fields with independent dynamics imposed by the equations of motion and independent Majorana masses.

Conclusions
In this paper, a comprehensive derivation and concise discussion of the free field wave equations governing the dynamics of the fundamental two-component and four-component spin- 1 2 matter fields in Minkowski spacetime is presented. The discussion is solely based on first principles like Lorentz symmetry, locality, causality, and unitarity which result in the hyperbolic differential equations describing Weyl, Dirac or Majorana fields. Coupling a fundamental two-component left-chiral field with a twocomponent right-chiral field in the most general non-trivial way leads to Dirac fields or Majorana fields and an emergent Majorana phase. A pure matrix-based formalism is used, avoiding an explicit van der Waerden notation [9] with dotted and undotted spinor indices sometimes confusing researchers from different fields which are more familiar with a notation inspired from linear algebra.