Dirac and Majorana field operators with self/antiself charge-conjugate states
aa r X i v : . [ phy s i c s . g e n - ph ] A p r Astron. Nachr. / AN , No. , 1 – 4 () /
DOI
Dirac and Majorana Field Operators with Self/Anti-Self Charge Con-jugate States
Valeriy V. Dvoeglazov ⋆ UAF, Universidad Aut´onoma de Zacatecas, Ap. Postal 636, Suc. 3, Zacatecas 98061 Zac., M´exicoThe dates of receipt and acceptance should be inserted later
Key words
Field Operators, QFT, Dirac, Majorana, Neutral ParticlesWe discuss relations between Dirac and Majorana-like field operators with self/anti-self charge conjugate states. Theconnections with recent models of several authors have been found. c (cid:13) WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim
In Refs. (Dvoeglazov 2003,2006,2009,2011,2013,2015,2016) we considered the procedure of construction of thefield operators ab initio (including for neutral particles).The Bogoliubov-Shirkov method has been used, Ref. (Bo-goliubov & Shirkov 1984).In the present article we investigate the spin-1/2 case forself/anti-self charge conjugate states. We look for interre-lations between the Dirac field operator and the Majoranafield operator. It seems that the calculations give mathemat-ically and physically reasonable results in the helicity basisonly.We write the charge conjugation operator into the form: C = e iθ c − i i i − i K = − e iθ c γ K . (1)It is the anti-linear operator of charge conjugation. K is thecomplex conjugation operator. We define the self/anti-self charge-conjugate 4-spinors in the momentum space(Ahluwalia 1996): Cλ S,A ( p ) = ± λ S,A ( p ) , (2) Cρ S,A ( p ) = ± ρ S,A ( p ) . (3)Thus, λ S,A ( p µ ) = (cid:18) ± i Θ φ ∗ L ( p ) φ L ( p ) (cid:19) , (4)and ρ S,A ( p ) = (cid:18) φ R ( p ) ∓ i Θ φ ∗ R ( p ) (cid:19) . (5) φ L , φ R can be boosted with the Lorentz transformation Λ L,R matrices. ⋆ Corresponding author: e-mail: valeri@fisica.uaz.edu.mx Such definitions of 4-spinors differ, of course, from the original Ma-jorana definition in x-representation: ν ( x ) = 1 √ D ( x ) + Ψ cD ( x )) , (6) Cν ( x ) = ν ( x ) that represents the positive real C − parity field oper-ator. However, the momentum-space Majorana-like spinors open variouspossibilities for description of neutral particles (with experimental conse-quences, see (Kirchbach & Compean & Noriega 2004). The rest λ − and ρ − spinors are: λ S ↑ ( ) = r m i , λ S ↓ ( ) = r m − i , (7) λ A ↑ ( ) = r m − i , λ A ↓ ( ) = r m i , (8) ρ S ↑ ( ) = r m − i , ρ S ↓ ( ) = r m i , (9) ρ A ↑ ( ) = r m i , ρ A ↓ ( ) = r m − i . (10)Thus, in this basis the explicit forms of the 4-spinors of thesecond kind λ S,A ↑↓ ( p ) and ρ S,A ↑↓ ( p ) are: λ S ↑ ( p ) = 12 p E p + m ip l i ( p − + m ) p − + m − p r , (11) λ S ↓ ( p ) = 12 p E p + m − i ( p + + m ) − ip r − p l ( p + + m ) ,λ A ↑ ( p ) = 12 p E p + m − ip l − i ( p − + m )( p − + m ) − p r ,λ A ↓ ( p ) = 12 p E p + m i ( p + + m ) ip r − p l ( p + + m ) , The choice of the helicity parametrization for p → is doubtful inRef. (Ahluwalia & Grumiller 2005), and it leads to unremovable contra-dictions, in my opinion. c (cid:13) WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim
V. V. Dvoeglazov: Dirac and Majorana ρ S ↑ ( p ) = 12 p E p + m p + + mp r ip l − i ( p + + m ) , (12) ρ S ↓ ( p ) = 12 p E p + m p l ( p − + m ) i ( p − + m ) − ip r ,ρ A ↑ ( p ) = 12 p E p + m p + + mp r − ip l i ( p + + m ) ,ρ A ↓ ( p ) = 12 p E p + m p l ( p − + m ) − i ( p − + m ) ip r . As we showed λ − and ρ − λ − and ρ − arenot (if we use the parity matrix P = (cid:18) (cid:19) R ) the eigenspinors of the parity, as op-posed to the Dirac case. The indices ↑↓ should be referredto the chiral helicity quantum number introduced in the 60s, η = − γ h , for λ spinors. While P u σ ( p ) = + u σ ( p ) , P v σ ( p ) = − v σ ( p ) , (13)we have P λ
S,A ( p ) = ρ A,S ( p ) , P ρ S,A ( p ) = λ A,S ( p ) (14)for the Majorana-like momentum-space 4-spinors onthe first quantization level. In this basis one has ρ S ↑ ( p ) = − iλ A ↓ ( p ) , ρ S ↓ ( p ) = + iλ A ↑ ( p ) , (15) ρ A ↑ ( p ) = + iλ S ↓ ( p ) , ρ A ↓ ( p ) = − iλ S ↑ ( p ) . (16)The analogs of the spinor normalizations (for λ S,A ↑↓ ( p ) and ρ S,A ↑↓ ( p ) ) are the following ones: λ S ↑ ( p ) λ S ↓ ( p ) = − im , λ S ↓ ( p ) λ S ↑ ( p ) = + im , (17) λ A ↑ ( p ) λ A ↓ ( p ) = + im , λ A ↓ ( p ) λ A ↑ ( p ) = − im , (18) ρ S ↑ ( p ) ρ S ↓ ( p ) = + im , ρ S ↓ ( p ) ρ S ↑ ( p ) = − im , (19) ρ A ↑ ( p ) ρ A ↓ ( p ) = − im , ρ A ↓ ( p ) ρ A ↑ ( p ) = + im . (20)All other conditions are equal to zero.The λ − and ρ − spinors are connected with the u − and v − spinors by the following formula: λ S ↑ ( p ) λ S ↓ ( p ) λ A ↑ ( p ) λ A ↓ ( p ) = 12 i − i − i − i − − i − − ii i − u +1 / ( p ) u − / ( p ) v +1 / ( p ) v − / ( p ) (21)provided that the 4-spinors have the same physical dimen-sion. The change of the mass dimension of the field operator has no suffi-cient foundations because the Lagrangian can be constructed on using thecoupled Dirac equations, see Ref. (Dvoeglazov 1995). After that one canplay with √ m to reproduce all possible mathematical results, which may(or may not) answer to the physical reality. We construct the field operators on using the Bogoliubov-Shirkov procedure with λ Sη ( p ) : Ψ( x ) = 1(2 π ) Z d p δ ( p − m ) e − ip · x Ψ( p ) == 1(2 π ) X η = ↑↓ Z d p δ ( p − E p ) e − ip · x √ m [ λ Sη ( p , p ) c η ( p , p )] = (22) = √ m (2 π ) Z d p E p [ δ ( p − E p ) + δ ( p + E p )][ θ ( p ) + θ ( − p )] e − ip · x X η = ↑↓ λ Sη ( p ) c η ( p )= √ m (2 π ) X η = ↑↓ Z d p E p [ δ ( p − E p ) + δ ( p + E p )] (cid:2) θ ( p )( p ) λ Sη ( p ) c η ( p ) e − ip · x ++ θ ( p ) λ Sη ( − p ) c η ( − p ) e + ip · x (cid:3) = √ m (2 π ) X η = ↑↓ Z d p E p θ ( p ) h λ Sη ( p ) c η ( p ) | p = E p e − i ( E p t − p · x ) ++ λ Sη ( − p ) c η ( − p ) | p = E p e + i ( E p t − p · x ) i Thus, comparing with the Dirac field operator we have1) instead of u h ( ± p ) we have λ Sη ( ± p ) ; 2) possible changeof the annihilation operators, a h → c η . Apart, one can makecorresponding changes due to normalization factors. Thus,we should have X η = ↑↓ λ Aη ( p ) d † η ( p ) = X η = ↑↓ λ Sη ( − p ) c η ( − p ) . (23)Multiplying by λ A − κ ( p ) or λ S − κ ( − p ) , respectively, we findsurprisingly: d † κ ( p ) = − ip y p σ yκτ c τ ( − p ) , (24) c κ ( − p ) = − ip y p σ yκτ d † τ ( p ) . (25)The above-mentioned contradiction may be related to thepossibility of the conjugation which is different from thatof Dirac. Both in the Dirac-like case and the Majorana-likecase ( c η ( p ) = e − iϕ d η ( p ) ) we have difficulties in the con-struction of field operators (Dvoeglazov 2018b).The bi-orthogonal anticommutation relations are givenin Ref. (Ahluwalia 1996). See other details inRef. (Dvoeglazov 1995a, 1997). Concerning with the P , C and T properties of the corresponding statessee Ref. (Dvoeglazov 2011) in this model.Similar formulations have been presented inRefs. (Markov 1937), and (Barut & Ziino 1993). Namely,the reflection properties are different for some solutions ofrelativistic equations therein. Two opposite signs at the massterms have been taken into account. The group-theoreticalbasis for such doubling has been given in the papers by c (cid:13) WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim stron. Nachr. / AN () 3
Gelfand, Tsetlin (1957) and Sokolik (1957), who first pre-sented the theory of 5-dimensional spinors (or, the one inthe 2-dimensional projective representation of the inversiongroup) in 1956 (later called as “the Bargmann-Wightman-Wigner-type quantum field theory” in 1993).The corresponding connection with the time reversion hasbeen clarified therein. It was one of the first attempts to ex-plain the K -meson decays. M. Markov proposed two Diracequations with opposite signs at the mass term (Markov1937) to be taken into account: [ iγ µ ∂ µ − m ] Ψ ( x ) = 0 , (26) [ iγ µ ∂ µ + m ] Ψ ( x ) = 0 . (27)In fact, he studied all properties of this relativistic quan-tum model (while the quantum field theory has not yet beencompleted in 1937). Next, he added and subtracted theseequations. What did he obtain? iγ µ ∂ µ ϕ ( x ) − mχ ( x ) = 0 , (28) iγ µ ∂ µ χ ( x ) − mϕ ( x ) = 0 . (29)Thus, the corresponding ϕ and χ solutions can be presentedas some superpositions of the Dirac 4-spinors u − and v − .These equations, of course, can be identified with the equa-tions for the Majorana-like λ − and ρ − , which we presentedin Ref. (Dvoeglazov 1995b). iγ µ ∂ µ λ S ( x ) − mρ A ( x ) = 0 , (30) iγ µ ∂ µ ρ A ( x ) − mλ S ( x ) = 0 , (31) iγ µ ∂ µ λ A ( x ) + mρ S ( x ) = 0 , (32) iγ µ ∂ µ ρ S ( x ) + mλ A ( x ) = 0 . (33)Neither of them can be regarded as the Dirac equation. How-ever, they can be written in the 8-component form as fol-lows: [ i Γ µ ∂ µ − m ] Ψ (+) ( x ) = 0 , (34) [ i Γ µ ∂ µ + m ] Ψ ( − ) ( x ) = 0 , (35)with Ψ (+) ( x ) = (cid:18) ρ A ( x ) λ S ( x ) (cid:19) , Ψ ( − ) ( x ) = (cid:18) ρ S ( x ) λ A ( x ) (cid:19) , (36) Γ µ = (cid:18) γ µ γ µ (cid:19) . (37)It is possible to find the corresponding Lagrangian, projec-tion operators, and the Feynman-Dyson-Stueckelberg prop-agator. For example, L = i (+) Γ µ ∂ µ Ψ (+) − ( ∂ µ Ψ (+) )Γ µ Ψ (+) ++ Ψ ( − ) Γ µ ∂ µ Ψ ( − ) − ( ∂ µ Ψ ( − ) )Γ µ Ψ ( − ) (cid:3) −− m [Ψ (+) Ψ (+) − Ψ ( − ) Ψ ( − ) ] . (38)The projection operator P + can be easily found, as usual, P + = Γ µ p µ + m m . (39) Of course, the signs at the mass terms depend on, how do we associatethe positive- or negative- frequency solutions with λ and ρ . However, due to the fact that P − satisfies the Dirac equa-tion with the opposite sign, we cannot have P + + P − =1 . This is not surprising because the corresponding states Ψ ± do not form the complete system of the 8-dimensionalspace. One should consider the states Γ Ψ ± ( p ) too. Seealso (Dvoeglazov 2018a) for the methods of obtaining thepropagators in the non-trivial cases.In the previous papers I explained: the connection withthe Dirac spinors has been found (Dvoeglazov 1995b;Kirchbach & Compean & Noriega 2004) through the uni-tary matrix, provided that the 4-spinors have the same phys-ical dimension. Thus, this represents itself the rotation ofthe spin-parity basis. However, it is usually assumed thatthe λ − and ρ − spinors describe the neutral particles, mean-while, the u − and v − spinors describe the charged parti-cles. Kirchbach, Compean and Noriega (2004) found theamplitudes for neutrinoless double beta decay ( νβ ) inthis scheme. It is obvious from (21) that there are some ad-ditional terms comparing with the standard calculations ofthose amplitudes. One can also re-write the above equationsinto the two-component forms. Thus, one obtains the Feyn-man and Gell-Mann (1958) equations.Barut and Ziino (1993) proposed yet another model.They considered γ operator as the operator of the chargeconjugation. In their case the self/anti-self charge conjugatestates are, at the same time, the eigenstates of the chirality.Thus, the charge-conjugated Dirac equation has a differentsign compared with the ordinary formulation: [ iγ µ ∂ µ + m ]Ψ cBZ = 0 , (40)and the so-defined charge conjugation applies to the wholesystem, fermion + electromagnetic field, e → − e in the co-variant derivative. The superpositions of the Ψ BZ and Ψ cBZ also give us the “doubled Dirac equation”, as the equationsfor λ − and ρ − spinors. The concept of the doubling ofthe Fock space has been developed in the Ziino works, cf.(Gelfand & Tsetlin 1957; Sokolik 1957; Dvoeglazov 1998)in the framework of the quantum field theory (Ziino 1996).Next, it is interesting to note that we have for the Majorana-like field operators ( a η ( p ) = b η ( p ) ): h ν ML ( x µ ) + C ν ML † ( x µ ) i / Z d p (2 π ) E p (41) X η (cid:20)(cid:18) i Θ φ ∗ η L ( p µ )0 (cid:19) a η ( p µ ) e − ip · x ++ (cid:18) φ ηL ( p µ ) (cid:19) a † η ( p µ ) e ip · x (cid:21) , h ν ML ( x µ ) − C ν ML † ( x µ ) i / Z d p (2 π ) E p (42) X η (cid:20)(cid:18) φ η L ( p µ ) (cid:19) a η ( p µ ) e − ip · x ++ (cid:18) − i Θ φ ∗ η L ( p µ )0 (cid:19) a † η ( p µ ) e ip · x (cid:21) . The reasons of the change of the fermion mass dimension are unclearin the recent works on elko . c (cid:13) WILEY-VCH Verlag GmbH&Co. KGaA, Weinheim
V. V. Dvoeglazov: Dirac and Majorana
This naturally leads to the Ziino-Barut scheme of massivechiral fields. See, however, the recent paper (Dvoeglazov2018b) which deals with the problems of the Majorana fieldoperator.
Acknowledgements.
I acknowledge discussions with colleagues atrecent conferences. I am grateful to the Zacatecas University forprofessorship.
References
Ahluwalia D.V.: 1996, Int. J. Mod. Phys. A11, 1855.Ahluwalia D.V., Grumiller D.: 2005, JCAP 0507, 012.Barut A., Ziino G.: 1993, Mod. Phys. Lett. A8, 1099.Bogoliubov N.N., Shirkov D.V.: 1984,
Introduction to the Theoryof Quantized Fields.
Beyond the Standard Model. Proceedings ofthe Vigier Symposium. (Noetic Press, Orinda, USA), p. 388.Dvoeglazov V.V.: 2011, J. Phys. Conf. Ser. 284, 012024.Dvoeglazov V.V.: 2013, Bled Workshops 14-2, 199.Dvoeglazov V.V.: 2015, in
Einstein and Others: Unification.