The Two-Component Majorana Equation-Novel Derivations and Known Symmetries

Home , Majorana equation

Journal of Modern Physics, 2011, 2, 1109-1114 doi:10.4236/jmp.2011.210137 Published Online October 2011 (http://www.SciRP.org/journal/jmp)

Eckart Marsch Max-Planck-Institut für Sonnensystemforschung, Katlenburg-Lindau, Germany E-mail: [email protected] Received May 26, 2011; revised July 10, 2011; accepted July 25, 2011

Abstract

We revisit the two-component Majorana equation and derive it in a new form by linearizing the relativistic dispersion relation of a massive particle, in a way similar to that used to derive the Dirac equation. We are using thereby the Pauli spin matrices, corresponding to an irreducible representation of the Lorentz group, and a lucid and transparent algebraic approach exploiting the newly introduced spin-flip operator. Thus we can readily build up the Majorana version of the Dirac equation in its chiral representation. The Lor- entz-invariant complex conjugation operation involves the spin-flip operator, and its connection to chiral symmetry is discussed. The eigenfunctions of the Majorana equation are calculated in a concise way.

Keywords: Dirac Equation, Majorana Equation, Charge Conjugation, Chiral Symmetry

1. Introduction algebraic nomenclature for the Dirac equation, and in a lucid way present its classical symmetries such as parity, The so called two-component Majorana equation [1,2] time reversal and charge exchange. permitting a finite mass term has been used in modern quantum field theory for the description of massive 2. Dirac Equation in a Concise Algebraic neutrinos, ever since convincing empirical evidence [3] Form for their finite masses and associated oscillations has been found in the past decade. Yet, in the standard model In this somewhat tutorial subsection, we consider the of elementary particle physics (see, e.g., text books [3,4]) Dirac equation in the standard and chiral representations. the fermions involved still are assumed to be massless, The derivation of the Dirac equation can be found in any and thus obey chiral symmetry. The corresponding textbook on quantum field theory (e.g., the ones of Kaku neutrinos are conventionally described by the massless [5] or Das [9]). We use standard symbols, notations and Weyl [5] equation involving two-component Pauli [6] definitions, and conventionally use units of  ==1c . spinors. As is well known, the Weyl equation corres- The Dirac Equation [8] generally reads ponds to two possible irreducible representations of the  p=i =m  (1) Lorentz group. 

In this paper we present among other topics a novel with pp=, 0  px = iit = ,  being the four- derivation of the two-component Majorana equation with momentum operator acting on the spinor wave function a mass term, whereby we linearize the relativistic dis-   x,t . The particle mass is m , and the four-vector persion relation of a massive particle in a way similar to named as usually   is composed of the four Dirac that used in deriving the Dirac equation. Moreover, a matrices. They can concisely be expressed in terms of consistent interpretation for the Dirac equation is given the spin operator σ and the three operators ,  and concerning its property of covariant complex conjugation  to be defined in their matrix form below. The three [7]. It emerges as an intrinsic basic symmetry of the Pauli matrices, to which we may add the 22 unit reducible Dirac equation with its four-component spinors, matrix denoted as  0 , have standard form, and includes of course the charge exchange operation for 01  0i   1 0 the Dirac equation [8] when being coupled to a gauge xy=,=,=    z   (2) field like the electromagnetic field. We use a concise 10 i 0   0 1

and build the three-vector σ =( xyz , , ), by help 3. New Derivation of the Majorana Equation which we can introduce the two four-vector forms of the Pauli matrices, which we may refer to as Weyl matrices. In this section a novel derivation of the two-component They are defined by the four vectors Majorana equation [8] term is presented, by linearizing the standard dispersion relation of a massive relativistic  =,σ (3)  0  particle. For this purpose, we have at the outset to define  and obey   =    . Similarly to the Pauli matrices, we an important operator, which is not of pure algebraic introduce the three real matrices nature but involves the complex-conjugation operator named as C . This antisymmetric operator called S is 01 1 0 0 1 =,=,=  (4) defined as SiC=  y , and obeys the relations 10 0 1 10  = SS †1 = S and S 2 1= . The operation of S on Note that  =  , and that all three matrices mu- the spin vector σ leads to its inversion, i.e., the 1 tually anti-commute with each other, just like the Pauli operation SSσ= σ yields a spin flip. Also remem- T* matrices do. All what we need in the following are the ber from Equation (2) that σ = σ , where the super- algebraic properties of the operators involved, like script T indicates the transposed matrix. As S flips the  =0,  =0, and    0= , and that spin, we have   S = S  , and Si = iS because of  22 1== and  2 1= . Similarly, the spin operator the action of C . Therefore, S also anti-commutes with the momentum four vector p , and thus we have components obey  xy yx=0,  xz zx=0,  2 2 2 Sp p S =0and ppSS = 0 . and   zy    yz 0= , and it is x  y ==  z 1= . 00 Furthermore, for any three vector v the relation Making use of the spin-flip operator S one can ()=σ vv22 holds. In terms of all these operators we derive the Majorana equation with a mass term in new can express the gamma four-vector as a direct product and straightforward way from the relativistic energy and and thus rewrite it in the standard Dirac representation as momentum relation which in fact is constitutive for all 222  field equations, Em=  P , where the four-momentum =,  σ (5)  0  of the particle is PE=( ,P). The energy-momentum Finally, we can concisely quote the Dirac equation relation is usually written as mass-shell condition, with mass term in conventional Hamiltonian form as  2 PP = m (10)  itmσ x,=  x ,t (6) Following Dirac [8], his Equation (1) results from a t x linearization of (10), which requires to go beyond the Subsequently, we shall also work with the Dirac algebra of complex numbers and needs the introduction equation in its chiral representation, and therefore quote of his famous matrices =ασ and  , yielding the the relevant matrices here, which are given instead of (4) Hamiltonian H = m  α  p . As shown in text books in the form [4,5], one requires four-dimensional matrices to obtain the algebra of the Dirac matrices defined in Equations 10 01 0 1 =,=,= (7) (2-5). The Pauli matrices (2) alone only suffice if there is 01 10 10 no mass term, in which case one obtains the Weyl Here we used to ease the notation the same symbols as equation [5]. in the standard representation. There is another important However, recalling the above properties of the spin 5 matrix named  , which (see for example the textbook flip operator, SiC=  y , we can overcome this problem of Kaku [4]) is defined as  5012= i3. Inserting the and still use the two-component representation given by above matrices and the Pauli matrices we obtain, the Pauli matrices (2). Namely, by help of S we can go a mathematical step beyond pure matrix algebra and 5 =i  0 xyz =  0 (8) define the linear but non-hermitian operator H = mS σ p (11) Here we used the properties that  xyz= i 0 , 2 =  , and  =  . By means of  5 the chiral 0 0 When squaring this equation and multiplying it out, projection operators, P =  )1/2(1 , can be written 2  the key properties S =1 , SSσσ =0, and Si iS =0 10  00 must be exploited. Because of this last relation, p and PP=,= (9) 0   p also anti-commute with S . Thus with p = H we 00  01 0 obtain We are subsequently going to consider the symmetry 222 properties of the Dirac Equation (1), including chirality. mSp==000 σ p Spσ p pσ p (12)

Copyright © 2011 SciRes. JMP E. MARSCH 1111 which when inserting the differential operators explicitly treated as real numbers. yields the Klein-Gordon equation. Similarly, we obtain Returning finally to Equation (13) again, we now from (11) with p0 = H a linear wave equation (named derive its eigenfunctions. In order to solve (13), we make after Weyl without the mass term), which involves only the usual plane-wave ansatz, the Pauli matrix operators acting on a two-component   xPx,=exptu  iEti vexp  iEti Px. The spinor named  and reads as follows: resulting linked eigenspinor equations are  EuEmSvσ PP  ,= P ,E (17) itmσ x,=S x ,t (13) t x EvEmSuσ PP  ,=  P ,E (18) This equation is nothing but the two-component Majorana equation [2], as it can be derived from Dirac’s By insertion of the first into the second equation, or equation in chiral form, if one imposes on it the con- vice versa, the relativistic dispersion relation yields the dition of Lorentz-covariant complex conjugation, a pro- two eigenvalues cedure which is clearly described in a recent tutorial EmP= 2P2 (19) paper of Pal [7]. 1,2 Obviously, the Majorana Equation (13) is very sensi- which are obtained from the requirement for nontrivial tive to the operation S . Namely, when we apply it from solutions of the spinors u and v to exist. The nega- the left side we find that tive root in (19) cannot be neglected, since as usually it is related to antiparticles. We can solve (17) for v and  iStmSStmσ xx,=    , =x ,t insert it back into the above ansatz for  x,t to obtain t x finally the solutions of (13) and (14) in the concise forms (14) E σ P Consequently, if  solves the Equation (13) with the  xP,=1tSi expEtiux (20) m plus sign in front of σ , then S solves it with the minus sign, but with a minus sign also at the mass term. E σ P Therefore, Equations (13) and (14) are closely connected, StS xP,=expiEtixu (21) m and represent only one independent complex spinor field, which can either be  or S . Instead of its two To validate these solutions by direct differentiation, complex component fields we may consider four real careful attention must be paid to the anticommutation fields, which correspond to the real and imaginary parts rules between S and i , respectively σ . We are free of the two components of  , and hence the name to choose for the eigenspinor u the two standard spin † † Majorana equation for (13), because it is equivalent to up and down eigenfunctions: u1 (1,0)= and u2 (0,1)= . the real four-component representation of the Dirac Similar solutions like that of Equations (20) and (21) are spinor as obtained from (1) by a unitary transformation, a obtained for the negative energy root in (19), yielding the result which is explicitly shown in [3], for example. antiparticle wavefunctions. Superposition of all these Making again use of the spin-flip operator S , one can Fourier modes and their summation over the momentum derive the Majorana equation with a mass term in yet variable P leads finally to the general solution of the another way from the relativistic energy-momentum Majorana fields, whose quantization then readily follows mass-shell condition (10). The algebra of the newly from the canonical rules [2-4,9].  defined four-vector SS=, 0 σ permits a representation that is equivalent to the Clifford algebra of the   4. Symmetries of the Dirac Equation matrices. Namely, we find a decomposition of the metric Revisited tensor g  in the form, Let us now consider the symmetries of the Dirac SS ,= SSSS =2 g (15)  equation, in particular the charge exchange C , parity which is validated by using the anti-commutator P , and time reversal T operations. Generally speaking the Dirac equation is invariant under the symmetry S,=σ 0, and that S 2 =1 . Note also that SS =4 .   operation O , if the spinor

Consequently, we can write the mass shell condition (10) O also in the form:  = O (22) PP== g PP SSPP = m2 (16) also fulfils the Dirac equation. Therefore, when applying  the operation O from the left and its inverse O1 Here the momentum four-vector components were from the right, whereby the unit operator is given by the

1 decomposition OO =1, we obtain the result 0 S  C =  (27) S 0 11  OOiiσ  mOOOx,=0 t t x In the standard representation (4) we simply get the (23) transposed matrix. Consequently, C22==S 1, as it We still need to define the time and space coordinate should be. Traditionally, the operator C is associated inversion operations T and P on a spinor  by with charge conjugation, because the effect it has on the Dirac equation when coupled to a gauge field, like the Ttxx,=  ,t (24) electromagnetic four-vector potential A , is simply that the sign of the charge is changed. As the minimal Ptx,= x ,t (25) coupling [4,9] to the field (the particle charge is e ) and also recall the complex conjugation operation C , yields the replacement, which gives CiC1 =  i , and yields   ieA (28) * Ctx,= x ,t (26) the effect of C will, solely due to the action of C , be the substitution of e by e . However, what then is the Here the asterisk denotes as usually the complex meaning of C for the free Dirac equation, when not conjugate number. In what follows we do not always being coupled to a gauge field? quote explicitly the unit matrix  , or the unit matrix 0 simply named 1, which is associated with the matrices in (4) and (7). With all these preparations in mind, it is 5. Chiral Symmetry and Complex easy to see which operators provide the requested Conjugation symmetry operations. We compose them in the Table 1. To complete the operator algebra, it is important to note Again, what is the genuine symmetry the operation C the following commutation relations. Of course, the is related with? The answer has been given in a most coordinate reversal operators T and P commute with lucid way by Pal [7] in his recent paper. The operation ,  and  , and alike with the three Pauli matrices. C in fact is the Lorentz-covariant complex conjugation, These in turn anti-commute with S , but obey which is an operation that of course makes sense without [, σσ]=[, ]=[ ,σ ]=0 , with the commutator denoted even considering gauge fields. However, the operator σ by [,] . plays a key role not only in this issue, but also for the Let us first consider in (23) the time reversal, two irreducible representations of the Lorentz group TO == ST . Apparently, it does not affect the mass related with two-component Pauli spinors. We recall the term with  , and also leaves the first term as well as operator S causes the spin σ to flip and inverts its T * direction. The parity operation does not change the sign iσ and α unchanged. Therefore, also =,iy  x t solves the Dirac Equation (23). Similarly, the parity of σ being a pseudo vector, which is consistent with operation OP== P commutes with the mass term the required invariance of the spin angular momentum and does not affect the first term in (23). But it also commutator, σσ =2i σ, under parity. So the intrinsic leaves the momentum term invariant since  and x symmetry described by C is closely connected with the both change signs together. Therefore, also degeneracy of the Dirac equation (with its four- P =x,t solves the Dirac equation. Finally, we component spinors) with respect to a spin flip, which introduce charge conjugation as OC== S , which corresponds to interchanging the two irreducible changes the signs of all three terms in (23) with no net representations of the Lorentz group. As a consequence, effect, yet thereby leaves the kinetic term σ invariant we can combine the two interrelated Majorana Equations C * as well. Therefore, also =ity x, solves the (13) and (14), and thus construct from them the four- Dirac equation. In conclusion, the symmetry operations component spinor of Table 1 work in a transparent and simple way on the C  Dirac equation. We finally quote the charge conjugation  =  (29) operator explicitly in chiral matrix form (7) as S

C Table 1. Symmetry operations. How does this state  transform under the operation C ? We apply the charge-conjugation or covariant Operation Time reversal Parity Charge conjugation complex conjugation operation in its matrix form in (27) Operator T = ST P =  P C = S on this special spinor and then obtain

C C0 S Cin a four-component spinor that is again called  . C==  = (30) How does this state transform under the operation SSS0  C = S ? We apply the charge-conjugation operation in Consequently, the charge-conjugation operator leaves its matrix form based on (7) on the above defined spinor, C the special state  unchanged, which therefore is an and by using S 2 =1 we obtain eigenfunction with eigenvalue unity. We recall here Dirac’s Equation (6) in its chiral matrix CC0 S 1 S   C== (33) form: S 0 1 S  0   Again, the charge-conjugation operator leaves the  pi=  =m (31)  eigen spinor  C in the standard representation of the   0 Dirac equation unchanged. Upon insertion of  C in the Apparently, any chiral Dirac spinor can be decom- standard Dirac equation, we obtain posed into two-component Pauli spinors such that †††  =,, where the plus sign indicates the iS iσ  SmS=  (34) right-chiral and minus sign the left-chiral components of t x the original Dirac spinor. We recall the chiral projection  operators defined in (9), which yield the spinors iiSSmσ   =  S (35) † † t x R ==P,0 and L ==0,P. C Now by comparison with equation (29) we have the Thus it is readily verified that  solves (6), since  and S by definition fulfill individually the respective obvious connections,  =  and  = S . Therefore, a comparison with the Majorana Equations (13) and (14) Majorana Equation (13) and (14). shows that  C fulfils the chiral Dirac equation, but it is special in so far as it also is an eigenfunction of the 6. Summary and Conclusions charge-conjugation operator, which is to say the cova- We reconsidered the two-component Majorana equation. riant complex conjugation operator C . Because of this By making use of the spin-flip operator, we derived this constraint the spinor (29) is determined fully by the equation with a mass term in a novel and manifestly two-component spinor  that obeys the Majorana Equa- covariant way, which revealed its intimate relations to tion (13). If we then apply C separately to the right- or the Dirac equation in its chiral as well as standard form. left-chiral fields we obtain The eigenfunctions of the two-component Majorana CC CC equation were also calculated. CC=and=   (32) RL    L R The new mathematical approach employed here demonstrates for self-complex-conjugate states the com- Thus this operator exchanges the chirality of the chiral plete equivalence of the complex Majorana equation with projections of the complex-self-conjugate spinor  C , its two-component spinor  and the Dirac equation, and it transforms left- into right-handed states and vice when being constrained to its four-component spinor versa, a virtue which is essentially associated with the C solutions  which can be built up by the two Majorana spin flip operator S defining C . In conclusion, in this spinors  and S . However, the complex Majorana special situation the operator C acts as chirality con- equation does not guarantee, but in fact breaks one of the jugation or reversal. While being composed of the two important basic field symmetries, namely charge irreducible Majorana equations, the reducible chiral exchange (represented here by the operator S ), and thus Dirac equation has as an intrinsic symmetry this chirality for the CPT-invariant description of massive charged conjugation. The same chirality reversal is obtained for fermions the unconstrained complex Dirac equation is of the real four-component Majorana equation, in which course required. complex conjugation of the chiral eigenspinors changes The two-component Majorana equation, coming in the left- into right-chiral spinors and vice versa. two related forms (13) and (14) that are connected by a When we use the standard Dirac representation with spin flip, represents the simplest possible covariant the matrices of (4), the standard Dirac spinor can also be relativistic wave equation for a massive but uncharged decomposed into two-component Pauli spinors such that fermion. These two forms represent the two irreducible ††† =,12, and by comparison of the Majorana representations of the Lorentz group in terms of Pauli equations with the chiral representation we have the spinors and matrices. Thus the Majorana equation can connections,  =212 , and directly be derived by linearization of the quadratic rela-

1,2 ==1S . We insert these components tivistic energy-momentum relation, a procedure which

Copyright © 2011 SciRes. JMP 1114 E. MARSCH leads to (11) without explicit recourse to the Dirac Physik A Hadrons and Nuclei, Vol. 56, No. 5-6, 1929, pp. equation. 330-352. doi:10.1007/BF01339504 [6] W. Pauli, “Zur Quantenmechanik des Magnetischen 7. References Elektrons,” Zeitschrift für Physik A Hadrons and Nuclei, Vol. 43, No. 9-10, 1927, pp. 601-623. doi:10.1007/BF01397326 [1] E. Majorana, “Teoria Simmetrica Dell’ Elettrone E Del Positrone,” Il Nuovo Cimento (1924-1942), Vol. 14, No. [7] P. B. Pal, “Dirac, Majorana and Weyl Fermions,” 4, 1937, pp. 171-184. doi:10.1007/BF02961314 arXiv:1006.1718v2 [hep-ph], 2010. [2] R. N. Mohapatra and P. B. Pal, “Massive Neutrinos in [8] P. M. A. Dirac, “The Quantum Theory of the Electron,” Physics and Astrophysics,” World Scientific, Singapore, Proceedings of the Royal Society of London. Series A, 2004. doi:10.1142/9789812562203 Containing Papers of a Mathematical and Physical Character, Vol. 117, No. 778, 1928, pp. 610-624. [3] M. Fukugita and T. Yanagida, “Physics of Neutrinos and Applications to Astrophysics,” Springer, Berlin, 2003. [9] A. Das, “Lectures on Quantum Field Theory,” World Scientific, Singapore, 2008. [4] M. Kaku, “Quantum Field Theory, A Modern Introduc- doi:10.1142/9789812832870 tion,” Oxford University Press, New York, 1993.

[5] H. Weyl, “Elektron und Gravitation I,” Zeitschrift für