Majorana Neutrino: Chirality and Helicity

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Majorana Neutrino Chirality and HelicityTo cite this article V V Dvoeglazov 2012 J Phys Conf Ser 343 012033

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Majorana Neutrino Chirality and Helicity

V V DvoeglazovUniversidad de Zacatecas Ap Postal 636 Suc 3 Cruces Zacatecas 98062 Zac Mexico

E-mail valerifisicauazedumx

Abstract We introduce the Majorana spinors in the momentum representation They obeythe Dirac-like equation with eight components which has been first introduced by MarkovThus the Fock space for corresponding quantum fields is doubled (as shown by Ziino) Theparticular attention has been paid to the questions of chirality and helicity (two concepts whichare frequently confused in the literature) for Dirac and Majorana states

1 The Dirac EquationThe Dirac equation has been considered in detail in a pedagogical way [1 2]

[iγmicropartmicro minusm]Ψ(x) = 0 (1)

At least 3 methods of its derivation exist

bull the Dirac one (the Hamiltonian should be linear in partpartxi and be compatible withE2 minus p2c2 = m2c4)

bull the Sakurai one (based on the equation (E minus σ middot p)(E + σ middot p)φ = m2φ)bull the Ryder one (the relation between 2-spinors at rest is φR(0) = plusmnφL(0))

The γmicro are the Clifford algebra matrices

γmicroγν + γνγmicro = 2gmicroν (2)

Usually everybody uses the following definition of the field operator [3]

Ψ(x) =1

(2π)3sumσ

intd3p2Ep

[uσ(p)aσ(p)eminusipmiddotx + vσ(p)bdaggerσ(p)]e+ipmiddotx] (3)

as given ab initio while the general scheme of its derivation has been presented in [4]I studied in the previous works [5 6 7]

bull σ rarr h (the helicity basis) the motivation was that Berestetskii Lifshitz and Pitaevskii [8]claimed that the helicity eigenspinors are not the parity eigenspinors and vice versabull the modified Sakurai derivation (the additional m2γ

5 term may appear in the Diracequation)

7th International Conference on Quantum Theory and Symmetries (QTS7) IOP PublishingJournal of Physics Conference Series 343 (2012) 012033 doi1010881742-65963431012033

Published under licence by IOP Publishing Ltd 1

bull the derivation of the Barut equation [9] from the first principles namely based on thegeneralized Ryder relation (φh

L(0) = Aφminush lowastL (0) + Bφh lowast

L (0)) In fact we have the secondmass state (micro-meson) from the equation

[iγmicropartmicro minus αpartmicropartmicromminus β]ψ = 0 (4)

due to existence of two parametersbull the selfanti-self charge-conjugate Majorana 4-spinors [10 11] in the momentum

representation

The Wigner rules [12] of the Lorentz transformations for the (0 S) left- φL(p) and the (S 0)right- φR(p) spinors are

(S 0) φR(p) = ΛR(plarr 0)φR(0) = exp(+ S middot ϕ)φR(0) (5)(0 S) φL(p) = ΛL(plarr 0)φL(0) = exp(minusS middot ϕ)φL(0) (6)

with ϕ = nϕ being the boost parameters

cosh(ϕ) = γ =1radic

1minus v2c2 (7)

sinh(ϕ) = βγ =vcradic

1minus v2c2 tanh(ϕ) = vc (8)

They are well known and given eg in [12 13 2]On using the Wigner boost rules and the Ryder relations we can recover the Dirac equation

in the matrix form (∓m 1 p0 + σ middot p

p0 minus σ middot p ∓m 1

)ψ(pmicro) = 0 (9)

or (γ middot pminusm)u(p) = 0 and (γ middot p+m)v(p) = 0 We have used the property [ΛLR(plarr 0)]minus1 =[ΛRL(plarr 0)]dagger above and that both S = σ2 and ΛRL are Hermitian for the finite (S =12 0)oplus (0 S = 12) representation of the Lorentz group Introducing ψ(x) equiv ψ(p) exp(∓ip middotx)and letting pmicro rarr ipartmicro the above equation becomes the Dirac equation (1)

The solutions of the Dirac equation are denoted by u(p) = column(φR(p) φL(p)) andv(p) = γ5u(p) Let me remind that the boosted 4-spinors in the common-used basis (thestandard representation of γ matrices) are

u 12 12

=

radic(E +m)

2m

10

pz(E +m)pr(E +m)

u 12minus 1

2=

radic(E +m)

2m

01

pl(E +m)minuspz(E +m)

(10)

v 12 12

=

radic(E +m)

2m

pz(E +m)pr(E +m)

10

v 12minus 1

2=

radic(E +m)

2m


01

(11)


2

E = +radic

p2 +m2 gt 0 p0 = plusmnE pplusmn = E plusmn pz prl = px plusmn ipy They are the eigenstates ofthe helicity in the case of p3 = |p| only They are the parity eigenstates with the eigenvalues of

plusmn1 In the parity operator the matrix γSR0 =

(1 00 minus1

)was used as usual They also describe

eigenstates of the charge operator Q if at rest

φR(0) = plusmnφL(0) (12)

(otherwise the corresponding physical states are no longer the charge eigenstates) Theirnormalizations conditions are

uσ(p)uσprime(p) = +δσσprime (13)vσ(p)vσprime(p) = minusδσσprime (14)uσ(p)vσprime(p) = 0 (15)

The bar over the 4-spinors signifies the Dirac conjugationThus in the most papers one uses the basis for charged particles in the (S 0) oplus (0 S)

representation (in general)

u+σ(0) = Nσ

100

uσminus1(0) = Nσ

010

vminusσ(0) = Nσ

001

(16)

Sometimes the normalization factor is convenient to choose N(σ) = mσ in order the rest spinorsto vanish in the massless limit

However other constructs are possible in the (12 0)oplus (0 12) representation [9 14 15 1617 18 19]

2 Majorana Spinors in the Momentum RepresentationDuring the 20th century various authors introduced selfanti-self charge-conjugate 4-spinors(including in the momentum representation) see [10 11 17 18] Later many authors[20 5 6 21 22] etc studied these spinors they found corresponding dynamical equations gaugetransformations and other specific features of them The definitions are

C = eiθ

0 0 0 minusi0 0 i 00 i 0 0minusi 0 0 0

K = minuseiθγ2K (17)

is the anti-linear operator of the charge conjugation K is the complex conjugation operatorWe define the selfanti-self charge-conjugate 4-spinors in the momentum space

CλSA(p) = plusmnλSA(p) (18)CρSA(p) = plusmnρSA(p) (19)

whereλSA(pmicro) =

(plusmniΘφlowastL(p)φL(p)

) (20)


3

andρSA(p) =

(φR(p)∓iΘφlowastR(p)

) (21)

The Wigner matrix is

Θ[12] = minusiσ2 =(

0 minus11 0

) (22)

and φL φR can be boosted with the ΛLR matrices1

The rest λ and ρ spinors can be defined conforming with (2021) in the analogous way withthe Dirac spinors

λSuarr (0) =

radicm

2

0i10

λSdarr (0) =

radicm

2

minusi001

(24)

λAuarr (0) =

radicm

2

0minusi10

λAdarr (0) =

radicm

2

i001

(25)

ρSuarr (0) =

radicm

2

100minusi

ρSdarr (0) =

radicm

2

01i0

(26)

ρAuarr (0) =

radicm

2

100i

ρAdarr (0) =

radicm

2

01minusi0

(27)

Thus in this basis with the appropriate normalization (the ldquomass dimensionrdquo indeed) theexplicite forms of the 4-spinors of the second kind λSA

uarrdarr (p) and ρSAuarrdarr (p) are

λSuarr (p) =

12radicE +m

ipl

i(pminus +m)pminus +mminuspr

λSdarr (p) =

12radicE +m

minusi(p+ +m)minusipr

minuspl

(p+ +m)

(28)

λAuarr (p) =

12radicE +m

minusipl

minusi(pminus +m)(pminus +m)minuspr

λAdarr (p) =

12radicE +m

i(p+ +m)

ipr

minuspl

(p+ +m)

(29)

1 Such definitions of 4-spinors differ of course from the original Majorana definition in x-representation

ν(x) =1radic2(ΨD(x) + Ψc

D(x)) (23)

Cν(x) = ν(x) that represents the positive real Cminus parity field operator only However the momentum-space Majorana-like spinors open various possibilities for description of neutral particles (with experimentalconsequences see [22]) For instanceldquofor imaginary C parities the neutrino mass can drop out from the single βdecay trace and reappear in 0νββ a curious and in principle experimentally testable signature for a non-trivialimpact of Majorana framework in experiments with polarized sourcesrdquo


4

ρSuarr (p) =

12radicE +m

p+ +mpr

ipl

minusi(p+ +m)

ρSdarr (p) =

12radicE +m

pl

(pminus +m)i(pminus +m)minusipr

(30)

ρAuarr (p) =

12radicE +m

p+ +mpr

minusipl

i(p+ +m)

ρAdarr (p) =

12radicE +m

pl

(pminus +m)minusi(pminus +m)

ipr

(31)

As claimed in [18] λminus and ρminus 4-spinors are not the eigenspinors of the helicity2 Moreoverthe λminus and ρminus are not the eigenspinors of the parity as opposed to the Dirac case (in this

representation P =(

0 11 0

)R where R = (xrarr minusx)) The indices uarrdarr should be referred to the

chiral helicity quantum number introduced in the 60s η = minusγ5h Ref [14] While

Puσ(p) = +uσ(p) Pvσ(p) = minusvσ(p) (32)

we havePλSA(p) = ρAS(p) PρSA(p) = λAS(p) (33)

for the Majorana-like momentum-space 4-spinors on the first quantization level In this basisone has also the relations between the above-defined 4-spinors

ρSuarr (p) = minusiλA

darr (p) ρSdarr (p) = +iλA

uarr (p) (34)

ρAuarr (p) = +iλS

darr (p) ρAdarr (p) = minusiλS

uarr (p) (35)

The normalizations of the spinors λSAuarrdarr (p) and ρSA

uarrdarr (p) are the following ones

λSuarr (p)λS

darr (p) = minusim λSdarr (p)λS

uarr (p) = +im (36)

λAuarr (p)λA

darr (p) = +im λAdarr (p)λA

uarr (p) = minusim (37)

ρSuarr (p)ρS

darr (p) = +im ρSdarr (p)ρS


ρAuarr (p)ρA

darr (p) = minusim ρAdarr (p)ρA

uarr (p) = +im (39)

All other normalization conditions are equal to zeroThe dynamical coordinate-space equations are3

iγmicropartmicroλS(x)minusmρA(x) = 0 (40)

iγmicropartmicroρA(x)minusmλS(x) = 0 (41)

iγmicropartmicroλA(x) +mρS(x) = 0 (42)

iγmicropartmicroρS(x) +mλA(x) = 0 (43)

Neither of them can be regarded as the Dirac equation However they can be written in the8-component form as follows

[iΓmicropartmicro minusm] Ψ(+)

(x) = 0 (44)

[iΓmicropartmicro +m] Ψ(minus)

(x) = 0 (45)

2 See the next Sections for the discussion3 Of course the signs at the mass terms depend on how do we associate the positive- and negative- frequencysolutions with λminus and ρminus spinors


5

withΨ(+)(x) =

(ρA(x)λS(x)

)Ψ(minus)(x) =

(ρS(x)λA(x)

) and Γmicro =

(0 γmicro

γmicro 0

) (46)

One can also re-write the equations into the two-component form Thus one obtains theFeynman-Gell-Mann equations [23] The similar formulations have been presented in Ref [24]and Ref [17] The group-theoretical basis for such doubling has been given in the papersby Gelfand Tsetlin and Sokolik [25] who first presented the theory in the 2-dimensionalrepresentation of the inversion group in 1956 (later called as ldquothe Bargmann-Wightman-Wigner-type quantum field theoryrdquo in 1993)

The Lagrangian is

L =i

2

[λSγmicropartmicroλ

S minus (partmicroλS)γmicroλS + ρAγmicropartmicroρ

A minus (partmicroρA)γmicroρA+

+λAγmicropartmicroλA minus (partmicroλ

A)γmicroλA + ρSγmicropartmicroρS minus (partmicroρ

S)γmicroρSminus

minusm(λSρA + λSρA minus λSρA minus λSρA)] (47)

The connections with the Dirac spinors have been found [5 22]4 For instanceλSuarr (p)λSdarr (p)λAuarr (p)λAdarr (p)

=12

1 i minus1 iminusi 1 minusi minus11 minusi minus1 minusii 1 i minus1

u+12(p)uminus12(p)v+12(p)vminus12(p)

(48)

See also Refs [25 17] and the discussion below Thus we can see that the two 4-spinor setsare connected by the unitary transformations and this represents itself the rotation of thespin-parity basis

The sets of λminus spinors and of ρminus spinors are claimed to be the bi-orthonormal sets each inthe mathematical sense [18] provided that overall phase factors of 2-spinors θ1 + θ2 = 0 or πFor instance on the classical level λS

uarrλSdarr = 2iN2 cos(θ1 + θ2)5

Several remarks have been given in the previous works

bull While in the massive case there are four λ-type spinors two λS and two λA (the ρminus spinorsare connected by certain relations with the λminus spinors for any spin case) in the masslesscase λS

uarr and λAuarr may identically vanish provided that one takes into account that φplusmn12

Lmay be the eigenspinors of (σ middot n)2 the 2times 2 helicity operatorbull The possibility exists for generalizations of the concept of the Fock space which lead to the

ldquodoublingrdquo Fock space [25 17]

The covariant derivative (and hence the interaction) was shown [5] to be introduced in thisconstruct in the following way

partmicro rarr nablamicro = partmicro minus ig L5Amicro (49)

where L5 = diag(γ5 minusγ5) the 8times8 matrix In other words with respect to the transformations

λprime(x)rarr (cosαminus iγ5 sinα)λ(x) (50)

λprime(x)rarr λ(x)(cosαminus iγ5 sinα) (51)ρprime(x)rarr (cosα+ iγ5 sinα)ρ(x) (52)ρ prime(x)rarr ρ(x)(cosα+ iγ5 sinα) (53)

4 I also acknowledge personal communications from D V Ahluwalia on these matters5 We used above θ1 = θ2 = 0


6

the spinors retain their properties to be selfanti-self charge conjugate spinors and the proposedLagrangian [5] remains to be invariant This fact tells us that while the selfanti-self chargeconjugate states have zero eigenvalues of the ordinary (scalar) charge operator but they canpossess the axial charge (cf with the discussion of [17] and with the old idea of R E Marshakndash they claimed the same)

In fact from this consideration one can recover the Feynman-Gell-Mann equation (and itscharge-conjugate equation) Our equations can be re-written in the two-component form [23]

[πminusmicro π

microminus minusm2 minus g

2σmicroνFmicroν

]χ(x) = 0 [

π+micro π

micro + minusm2 +g

2σmicroνFmicroν

]φ(x) = 0

(54)

where already one has πplusmnmicro = ipartmicro plusmn gAmicro σ0i = minusσ0i = iσi σij = σij = εijkσk

Next because the transformations

λprimeS(p) =(

Ξ 00 Ξ

)λS(p) equiv λlowastA(p) (55)

λprimeprimeS(p) =(iΞ 00 minusiΞ

)λS(p) equiv minusiλlowastS(p) (56)

λprimeprimeprimeS (p) =(

0 iΞiΞ 0

)λS(p) equiv iγ0λlowastA(p) (57)

λIVS (p) =

(0 ΞminusΞ 0

)λS(p) equiv γ0λlowastS(p) (58)

with the 2times 2 matrix Ξ defined as (φ is the azimuthal angle related with p)

Ξ =(eiφ 00 eminusiφ

) ΞΛRL(plarr 0)Ξminus1 = Λlowast

RL(plarr 0) (59)

and corresponding transformations for λA do not change the properties of bispinors to be inthe selfanti-self charge-conjugate spaces the Majorana-like field operator (bdagger equiv adagger) admitsadditional phase (and in general normalization) transformations

νML prime(xmicro) = [c0 + i(τ middot c)] νML dagger(xmicro) (60)

where cα are arbitrary parameters The τ matrices are defined over the field of 2 times 2 matricesand the Hermitian conjugation operation is assumed to act on the c- numbers as the complexconjugation One can parametrize c0 = cosφ and c = n sinφ and thus define the SU(2) groupof phase transformations One can select the Lagrangian which is composed from the both fieldoperators (with λminus spinors and ρminus spinors) and which remains to be invariant with respect tothis kind of transformations The conclusion is it is permitted the non-Abelian construct whichis based on the spinors of the Lorentz group only (cf with the old ideas of T W Kibble andR Utiyama) This is not surprising because both the SU(2) group and U(1) group are thesub-groups of the extended Poincare group (cf [2])

The Dirac-like and Majorana-like field operators can be built from both λSA(p) and ρSA(p)or their combinations For instance

ν(xmicro) equivint

d3p(2π)3

12Ep

sumη

[λS

η (p) aη(p) exp(minusip middot x) + λAη (p) bdaggerη(p) exp(+ip middot x)

] (61)


7

The anticommutation relations are the following ones (due to the bi-orthonormality)

[aηprime(pprime) adaggerη(p)]+ = (2π)32Epδ(pminus pprime)δηminusηprime (62)

and

[bηprime(pprime) bdaggerη(p)]+ = (2π)32Epδ(pminus pprime)δηminusηprime (63)

Other anticommutators are equal to zero ([aηprime(pprime) bdaggerη(p)]+ = 0)Finally it is interesting to note that

[ν

ML(xmicro) + CνML dagger

(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0

)aη(pmicro)eminusipmiddotx+

+(

0φη

L(pmicro)

)adaggerη(pmicro)eipmiddotx

] (64)[

νML

(xmicro)minus CνML dagger(xmicro)

]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)


+(minusiΘφlowast η

L(pmicro)

0


] (65)

thus naturally leading to the Ziino-Barut scheme of massive chiral fields Ref [17]The content of this Section is mainly based on the previous works of the 90s by

D V Ahluwalia and by me (V V Dvoeglazov) dedicated to the Majorana-like momentum-representation 4-spinors However recently the interest to this model raised again [21]

3 Chirality and Helicity31 Historybull Ahluwalia in Ref [18] claimed ldquoIncompatibility of Self-Charge Conjugation with Helicity

Eignestates and Gauge Interactionsrdquo I showed that the gauge interactions of λminus and ρminus4-spinors are different As for the selfanti-self charge-conjugate states and their relationsto the helicity eigenstates the question is much more difficult see below Either we shouldaccept that the rotations would have physical significance or due to some reasons weshould not apply the equivalence transformation to the discrete symmetry operators Asfar as I understood his paper [18]6 the latter standpoint is precisely his standpointbull Z-Q Shi and G J Ni promote a very extreme standpoint Namely ldquothe spin states the

helicity states and the chirality states of fermions in Relativistic Quantum Mechanics areentirely different a spin state is helicity degenerate a helicity state can be expanded asa linear combination of the chirality states the polarization of fermions in flight must bedescribed by the helicity statesrdquo (see also the Conclusion Section of the second paper [27])In fact they showed experimental consequences of their statement ldquothe lifetime of RHpolarized fermions is always greater than of LH ones with the same speed in flightrdquoHowever we showed that the helicity chiral helicity and chirality operators are connectedby the unitary transformations Do rotations have physical significance in their opinion

6 He claimed [18] ldquoJust as the operator of parity in the (j 0)oplus(0 j) representation space is independent of whichwave equation is under study similarly the operations of charge conjugation and time reversal do not depend ona specific wave equation Within the context of the logical framework of the present paper without this beingtrue we would not even know how to define self-anti self conjugate (j 0)oplus (0 j) spinorsrdquo


8

bull M Markov wrote two Dirac equations with the opposite signs at the mass term [24] longago

[iγmicropartmicro minusm] Ψ1(x) = 0 (66)[iγmicropartmicro +m] Ψ2(x) = 0 (67)

In fact he studied all properties of this relativistic quantum model (while he did not knowyet the quantum field theory in 1937) Next he added and subtracted these equationsWhat did he obtain

iγmicropartmicroφ(x)minusmχ(x) = 0 (68)iγmicropartmicroχ(x)minusmφ(x) = 0 (69)

thus φ and χ solutions can be presented as some superpositions of the Dirac 4-spinors uminusand vminus These equations of course can be identified with the equations for λminus and ρminusspinors we presented above As he wrote himself he was expecting ldquonew physicsrdquo fromthese equationsbull Sen Gupta [14] and others claimed that the solutions of the equation (which follows from

the general Sakurai method of the derivation of relativistic quantum equations)[iγmicropartmicro minusm1 minusm2γ

5]

Ψ = 0 (70)

are not the eigenstates of chiral [helicity] operator γ0(γ middot p)p in the massless limit Theequation may describe both massive and massless (m1 = plusmnm2) states However in themassive case the equation (70) has been obtained by the equivalence transformation of γmatricesbull Barut and Ziino [17] proposed yet another model They considered γ5 operator as the

operator of the charge conjugation Thus in their opinion the charge-conjugated Diracequation has the different sign comparing with the ordinary formulation

[iγmicropartmicro +m]ΨcBZ = 0 (71)

and the so-defined charge conjugation applies to the whole system fermion+electromagneticfield erarr minuse in the covariant derivative The concept of the doubling of the Fock space hasbeen developed in the Ziino works (cf [25 26]) In their case the charge conjugate statesare at the same time the eigenstates of the chirality

Let us discuss the above statements

bull The helicity operator is

h =(

(σ middot p) 00 (σ middot p)

)(72)

However we can do the equivalence transformation of the helicity h-operator by the unitarymatrix It is well known [28] that one can obtain

U1(σ middot a)Uminus11 = σ3|a| (73)

In the case of the momentum vector one has

U1 =(

1 pl(p+ p3)minuspr(p+ p3) 1

)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)

Then applying other unitary matrix U31 0 0 00 0 1 00 1 0 00 0 0 1

(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)

we transform to the basis where the helicity is equal to γ5 the chirality operatorbull In Ref [14] the chiral helicity η = minusγ5h was introduced It is equal (within the sign) to the

well-known matrix α multiplied by n Again

U1(α middot n)Uminus11 = |n|

1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)

with the same matrix U1 And applying the second unitary transformation

U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)

we again come to the γ5 matrix The determinats are DetU1 6= 0 DetU23 = minus1 6= 0Thus the helicity the chirality and the chiral helicity are connected by the unitarytransformationsbull It is not surprising to have such a situation because the different helicity 2-spinors can

be connected apart the anti-linear transformation [2 18] ξh = (minus1)12minusheiαhΘ[12]Kξminushby the unitary transformation too For isntance when we parametrize the 2-spinors as inRef [19]

ξuarr = N eiα(

cos (θ2)sin (θ2) ei φ

) (80)

ξdarr = N eiβ(

sin (θ2)minus cos (θ2) ei φ

) (81)

we obtainξdarr = Uξuarr = ei(βminusα)

(0 eminusiφ

minuseiφ 0

)ξuarr (82)

andξuarr = U daggerξdarr = ei(αminusβ)

(0 minuseminusiφ

eiφ 0

)ξdarr (83)

To say that the 4-spinor is the eigenspinor of the chiral helicity and at the same time it isnot () the eigenspinor of the helicity operator (and that the physical results would depend onthis) signifies the same as to say that rotations have physical significance on the fundamentallevel


10

32 Non-commutativityI say also a couple of words on the unitarity transformations in the context of the ldquonon-commutativerdquo physics

Recently we discussed the Sakurai-van der Waerden method of the derivation of the Diracequation (and the derivation of the higher-spin equations as well) see Ref [29] We can startfrom

(EI(2) minus σ middot p)(EI(2) + σ middot p)Ψ(2) = m2Ψ(2) (84)

and we obtain[iγmicropart

micro minusm1 minusm2γ5]Ψ(x) = 0 (85)

Alternatively(EI(4) + α middot p +mβ)(EI(4) minus α middot pminusmβ)Ψ(4) = 0 (86)

Of course as in the original Dirac work we have

β2 = 1 αiβ + βαi = 0 αiαj + αjαi = 2δij (87)

For instance their explicite forms can be chosen

αi =(σi 00 minusσi

) β =

(0 12times2

12times2 0

) (88)

where σi are the ordinary Pauli 2times 2 matricesWe can also postulate the non-commutativity for the sake of general consideraton (as in

Ref [30])[Epi]minus = Θ0i = θi (89)

Therefore the equation (86) does not lead to the well-known equation E2 minus p2 = m2 Insteadwe have

E2 minus E(α middot p) + (α middot p)E minus p2 minusm2 minus iσ times I(2)[ptimes p]

Ψ(4) = 0 (90)

For the sake of simplicity we may assume the last term to be zero Thus we come toE2 minus p2 minusm2 minus (α middot θ)

Ψ(4) = 0 (91)

However let us perform the unitary transformation with U1 matrix7 For α matrices we cometo

U1(α middot θ)Uminus11 = |θ|

1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)

And applying the second unitary transformation with U2 matrix as before we have

U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)

The final equation is[E2 minus p2 minusm2 minus γ5

chiral|θ|]Ψprime(4) = 0 (94)

In the physical sense this implies that the mass splitting exists for a Dirac particle over thenon-commutative space This procedure may be attractive for explanation of the mass creationand the mass splitting for fermions7 Of course the certain relations for the components θ should be assumed Moreover in our case θ should notdepend on E and p Otherwise we must take the noncommutativity [Epi]minus again


11

4 ConclusionsWe presented a pedagogical review of the formalism for the momentum-space Majorana-likeparticles in the (12 0) oplus (0 12) representation of the Lorentz Group They satisfy the 8-component analogue of the Dirac equation Apart they have different gauge transformationscomparing with the usual Dirac 4-spinors Their helicity chirality and chiral helicity propertieshave been investigated in detail These operators are connected by the given unitarytransformations At the same time we showed that the Majorana-like 4-spinors can be obtainedby the rotation of the spin-parity basis Meanwhile several authors have claimed that thephysical results would be different on using calculations with these Majorana-like spinors

Thus the (S 0) oplus (0 S) representation space (even in the case of S = 12) has additionalmathematical structures leading to deep physical consequences which have not yet been exploredbefore

However several claims made by other researchers concerning with chirality helicity chiralhelicity should not be considered to be true until the time when experiments confirm themUsually it is considered that the rotations (unitary transformations) have no any physicalconsequences on the level of the Lorentz-covariant theories

References[1] Sakurai J J 1967 Advanced Quantum Mechanics (Addison-Wesley)[2] Ryder L H 1985 Quantum Field Theory (Cambridge Cambridge University Press Cambridge)[3] Itzykson C and Zuber J B 1980 Quantum Field Theory (McGraw-Hill Book Co) p 156[4] Bogoliubov N N and Shirkov D V 1976 Introduction to the Theory of Quantized Fields 3rd Edition (Moscow

Nauka)[5] Dvoeglazov V V 1995 Int J Theor Phys 34 2467 1995 Nuovo Cim A 108 1467 1997 Hadronic J 20 435

1998 Acta Phys Polon B29 619[6] Dvoeglazov V V 1997 Mod Phys Lett A12 2741[7] Dvoeglazov V V 2004 Int J Theor Phys 43 1287[8] Berestetskii V B Lifshitz E M and Pitaevskii L P 1982 Quantum ElectrodynamicsThe IV Volume of the

Landau Course of Theoretical Physics 2nd Edition (Butterworth-Heinemann)[9] Barut A O 1978 Phys Lett B 73 310 1979 Phys Rev Lett 42 1251

[10] Majorana E 1937 Nuovo Cimento 14 171[11] Bilenky S M and Pontekorvo B M 1978 Phys Repts 42 224[12] Wigner E P 1939 Ann Math 40 149 Weinberg S 1964 Phys Rev 133 B1318[13] Faustov R N 1971 Relativistic Transformations Preprint ITF-71-117P Kiev[14] Sen Gupta N D 1967 Nucl Phys B4 147[15] Tokuoka Z 1967 Prog Theor Phys 37 581 ibid 603[16] Raspini A 1996 Fizika B5 159[17] Barut A and Ziino G 1993 Mod Phys Lett A8 1099 Ziino G 1996 Int J Mod Phys A 11 2081[18] Ahluwalia D V 1996 Int J Mod Phys A 11 1855 1994 Incompatibility of Self-Charge Conjugation with

Helicity Eignestates and Gauge Interactions Preprint LANL UR-94-1252 Los Alamos[19] Dvoeglazov V V 1997 Fizika B6 111[20] Lounesto P 2002 Clifford Algebras and Spinors (Cambridge Cambridge University Press) Ch 11 and 12[21] Da Rocha R and Rodrigues Jr W 2006 Where are Elko Spinor Fields in Lounesto Spinor Field Classification

Mod Phys Lett A 21 65 (Preprint math-ph0506075)[22] Kirchbach M Compean C and Noriega L 2004 Beta Decay with Momentum-Space Majorana Spinors Eur

Phys J A 22 149[23] Feynman R P and Gell-Mann M 1958 Phys Rev 109 193[24] Markov M 1937 ZhETF 7 579 ibid 603 1964 Nucl Phys 55 130[25] Gelfand I M and Tsetlin M L 1956 ZhETF 31 1107 Sokolik G A 1957 ZhETF 33 1515[26] Dvoeglazov V V 1998 Int J Theor Phys 37 1915[27] Shi Z Q and Ni G J 2002 Chin Phys Lett 19 1427 Shi Z Q 2011 Preprint arXiv11010481v1[28] Berg R A 1966 Nuovo Cim 42 148[29] Dvoeglazov V V 2003 Rev Mex Fis Supl 49 99 (Proceedings of the DGFM-SMF School Huatulco 2000)[30] Dvoeglazov V V 2003 Proceedings of ICSSUR-VIII June 9-13 2003 Puebla Mexico (Princeton Rinton

Press) p 125


12

Majorana Neutrino Chirality and Helicity

V V DvoeglazovUniversidad de Zacatecas Ap Postal 636 Suc 3 Cruces Zacatecas 98062 Zac Mexico

E-mail valerifisicauazedumx

Abstract We introduce the Majorana spinors in the momentum representation They obeythe Dirac-like equation with eight components which has been first introduced by MarkovThus the Fock space for corresponding quantum fields is doubled (as shown by Ziino) Theparticular attention has been paid to the questions of chirality and helicity (two concepts whichare frequently confused in the literature) for Dirac and Majorana states

1 The Dirac EquationThe Dirac equation has been considered in detail in a pedagogical way [1 2]

[iγmicropartmicro minusm]Ψ(x) = 0 (1)

At least 3 methods of its derivation exist

bull the Dirac one (the Hamiltonian should be linear in partpartxi and be compatible withE2 minus p2c2 = m2c4)

bull the Sakurai one (based on the equation (E minus σ middot p)(E + σ middot p)φ = m2φ)bull the Ryder one (the relation between 2-spinors at rest is φR(0) = plusmnφL(0))

The γmicro are the Clifford algebra matrices

γmicroγν + γνγmicro = 2gmicroν (2)

Usually everybody uses the following definition of the field operator [3]

Ψ(x) =1

(2π)3sumσ

intd3p2Ep

[uσ(p)aσ(p)eminusipmiddotx + vσ(p)bdaggerσ(p)]e+ipmiddotx] (3)

as given ab initio while the general scheme of its derivation has been presented in [4]I studied in the previous works [5 6 7]

bull σ rarr h (the helicity basis) the motivation was that Berestetskii Lifshitz and Pitaevskii [8]claimed that the helicity eigenspinors are not the parity eigenspinors and vice versabull the modified Sakurai derivation (the additional m2γ

5 term may appear in the Diracequation)


Published under licence by IOP Publishing Ltd 1






representation





1minus v2c2 (7)






)ψ(pmicro) = 0 (9)



u 12 12

=

radic(E +m)

2m

10

pz(E +m)pr(E +m)

u 12minus 1

2=

radic(E +m)

2m

01


(10)

v 12 12

=

radic(E +m)

2m

pz(E +m)pr(E +m)

10

v 12minus 1

2=

radic(E +m)

2m


01

(11)


2

E = +radic



(1 00 minus1








u+σ(0) = Nσ

100

uσminus1(0) = Nσ

010

vminusσ(0) = Nσ

001

(16)




C = eiθ

0 0 0 minusi0 0 i 00 i 0 0minusi 0 0 0




whereλSA(pmicro) =


) (20)


3

andρSA(p) =


) (21)



0 minus11 0

) (22)



λSuarr (0) =

radicm

2

0i10

λSdarr (0) =

radicm

2

minusi001

(24)

λAuarr (0) =

radicm

2

0minusi10

λAdarr (0) =

radicm

2

i001

(25)

ρSuarr (0) =

radicm

2

100minusi

ρSdarr (0) =

radicm

2

01i0

(26)

ρAuarr (0) =

radicm

2

100i

ρAdarr (0) =

radicm

2

01minusi0

(27)



λSuarr (p) =

12radicE +m

ipl


λSdarr (p) =

12radicE +m


minuspl

(p+ +m)

(28)

λAuarr (p) =

12radicE +m

minusipl


λAdarr (p) =

12radicE +m

i(p+ +m)

ipr

minuspl

(p+ +m)

(29)



D(x)) (23)



4

ρSuarr (p) =

12radicE +m

p+ +mpr

ipl

minusi(p+ +m)

ρSdarr (p) =

12radicE +m

pl


(30)

ρAuarr (p) =

12radicE +m

p+ +mpr

minusipl

i(p+ +m)

ρAdarr (p) =

12radicE +m

pl


ipr

(31)


representation P =(

0 11 0








uarr (p) (34)

ρAuarr (p) = +iλS


uarr (p) (35)



λSuarr (p)λS


uarr (p) = +im (36)

λAuarr (p)λA



ρSuarr (p)ρS



ρAuarr (p)ρA


uarr (p) = +im (39)








(x) = 0 (44)


(x) = 0 (45)



5

withΨ(+)(x) =

(ρA(x)λS(x)

)Ψ(minus)(x) =

(ρS(x)λA(x)

) and Γmicro =

(0 γmicro

γmicro 0

) (46)


The Lagrangian is

L =i

2






S)γmicroρSminus



=12



(48)
















6



[πminusmicro π


2σmicroνFmicroν

]χ(x) = 0 [

π+micro π

micro + minusm2 +g

2σmicroνFmicroν

]φ(x) = 0

(54)



λprimeS(p) =(

Ξ 00 Ξ





0 iΞiΞ 0


λIVS (p) =

(0 ΞminusΞ 0





RL(plarr 0) (59)





ν(xmicro) equivint

d3p(2π)3

12Ep

sumη

[λS


] (61)


7



and



[ν


(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0


+(

0φη

L(pmicro)


] (64)[

νML


]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)



L(pmicro)

0


] (65)








8







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12






representation





1minus v2c2 (7)






)ψ(pmicro) = 0 (9)



u 12 12

=

radic(E +m)

2m

10

pz(E +m)pr(E +m)

u 12minus 1

2=

radic(E +m)

2m

01


(10)

v 12 12

=

radic(E +m)

2m

pz(E +m)pr(E +m)

10

v 12minus 1

2=

radic(E +m)

2m


01

(11)


2

E = +radic



(1 00 minus1








u+σ(0) = Nσ

100

uσminus1(0) = Nσ

010

vminusσ(0) = Nσ

001

(16)




C = eiθ

0 0 0 minusi0 0 i 00 i 0 0minusi 0 0 0




whereλSA(pmicro) =


) (20)


3

andρSA(p) =


) (21)



0 minus11 0

) (22)



λSuarr (0) =

radicm

2

0i10

λSdarr (0) =

radicm

2

minusi001

(24)

λAuarr (0) =

radicm

2

0minusi10

λAdarr (0) =

radicm

2

i001

(25)

ρSuarr (0) =

radicm

2

100minusi

ρSdarr (0) =

radicm

2

01i0

(26)

ρAuarr (0) =

radicm

2

100i

ρAdarr (0) =

radicm

2

01minusi0

(27)



λSuarr (p) =

12radicE +m

ipl


λSdarr (p) =

12radicE +m


minuspl

(p+ +m)

(28)

λAuarr (p) =

12radicE +m

minusipl


λAdarr (p) =

12radicE +m

i(p+ +m)

ipr

minuspl

(p+ +m)

(29)



D(x)) (23)



4

ρSuarr (p) =

12radicE +m

p+ +mpr

ipl

minusi(p+ +m)

ρSdarr (p) =

12radicE +m

pl


(30)

ρAuarr (p) =

12radicE +m

p+ +mpr

minusipl

i(p+ +m)

ρAdarr (p) =

12radicE +m

pl


ipr

(31)


representation P =(

0 11 0








uarr (p) (34)

ρAuarr (p) = +iλS


uarr (p) (35)



λSuarr (p)λS


uarr (p) = +im (36)

λAuarr (p)λA



ρSuarr (p)ρS



ρAuarr (p)ρA


uarr (p) = +im (39)








(x) = 0 (44)


(x) = 0 (45)



5

withΨ(+)(x) =

(ρA(x)λS(x)

)Ψ(minus)(x) =

(ρS(x)λA(x)

) and Γmicro =

(0 γmicro

γmicro 0

) (46)


The Lagrangian is

L =i

2






S)γmicroρSminus



=12



(48)
















6



[πminusmicro π


2σmicroνFmicroν

]χ(x) = 0 [

π+micro π

micro + minusm2 +g

2σmicroνFmicroν

]φ(x) = 0

(54)



λprimeS(p) =(

Ξ 00 Ξ





0 iΞiΞ 0


λIVS (p) =

(0 ΞminusΞ 0





RL(plarr 0) (59)





ν(xmicro) equivint

d3p(2π)3

12Ep

sumη

[λS


] (61)


7



and



[ν


(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0


+(

0φη

L(pmicro)


] (64)[

νML


]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)



L(pmicro)

0


] (65)








8







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12

E = +radic



(1 00 minus1








u+σ(0) = Nσ

100

uσminus1(0) = Nσ

010

vminusσ(0) = Nσ

001

(16)




C = eiθ

0 0 0 minusi0 0 i 00 i 0 0minusi 0 0 0




whereλSA(pmicro) =


) (20)


3

andρSA(p) =


) (21)



0 minus11 0

) (22)



λSuarr (0) =

radicm

2

0i10

λSdarr (0) =

radicm

2

minusi001

(24)

λAuarr (0) =

radicm

2

0minusi10

λAdarr (0) =

radicm

2

i001

(25)

ρSuarr (0) =

radicm

2

100minusi

ρSdarr (0) =

radicm

2

01i0

(26)

ρAuarr (0) =

radicm

2

100i

ρAdarr (0) =

radicm

2

01minusi0

(27)



λSuarr (p) =

12radicE +m

ipl


λSdarr (p) =

12radicE +m


minuspl

(p+ +m)

(28)

λAuarr (p) =

12radicE +m

minusipl


λAdarr (p) =

12radicE +m

i(p+ +m)

ipr

minuspl

(p+ +m)

(29)



D(x)) (23)



4

ρSuarr (p) =

12radicE +m

p+ +mpr

ipl

minusi(p+ +m)

ρSdarr (p) =

12radicE +m

pl


(30)

ρAuarr (p) =

12radicE +m

p+ +mpr

minusipl

i(p+ +m)

ρAdarr (p) =

12radicE +m

pl


ipr

(31)


representation P =(

0 11 0








uarr (p) (34)

ρAuarr (p) = +iλS


uarr (p) (35)



λSuarr (p)λS


uarr (p) = +im (36)

λAuarr (p)λA



ρSuarr (p)ρS



ρAuarr (p)ρA


uarr (p) = +im (39)








(x) = 0 (44)


(x) = 0 (45)



5

withΨ(+)(x) =

(ρA(x)λS(x)

)Ψ(minus)(x) =

(ρS(x)λA(x)

) and Γmicro =

(0 γmicro

γmicro 0

) (46)


The Lagrangian is

L =i

2






S)γmicroρSminus



=12



(48)
















6



[πminusmicro π


2σmicroνFmicroν

]χ(x) = 0 [

π+micro π

micro + minusm2 +g

2σmicroνFmicroν

]φ(x) = 0

(54)



λprimeS(p) =(

Ξ 00 Ξ





0 iΞiΞ 0


λIVS (p) =

(0 ΞminusΞ 0





RL(plarr 0) (59)





ν(xmicro) equivint

d3p(2π)3

12Ep

sumη

[λS


] (61)


7



and



[ν


(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0


+(

0φη

L(pmicro)


] (64)[

νML


]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)



L(pmicro)

0


] (65)








8







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12

andρSA(p) =


) (21)



0 minus11 0

) (22)



λSuarr (0) =

radicm

2

0i10

λSdarr (0) =

radicm

2

minusi001

(24)

λAuarr (0) =

radicm

2

0minusi10

λAdarr (0) =

radicm

2

i001

(25)

ρSuarr (0) =

radicm

2

100minusi

ρSdarr (0) =

radicm

2

01i0

(26)

ρAuarr (0) =

radicm

2

100i

ρAdarr (0) =

radicm

2

01minusi0

(27)



λSuarr (p) =

12radicE +m

ipl


λSdarr (p) =

12radicE +m


minuspl

(p+ +m)

(28)

λAuarr (p) =

12radicE +m

minusipl


λAdarr (p) =

12radicE +m

i(p+ +m)

ipr

minuspl

(p+ +m)

(29)



D(x)) (23)



4

ρSuarr (p) =

12radicE +m

p+ +mpr

ipl

minusi(p+ +m)

ρSdarr (p) =

12radicE +m

pl


(30)

ρAuarr (p) =

12radicE +m

p+ +mpr

minusipl

i(p+ +m)

ρAdarr (p) =

12radicE +m

pl


ipr

(31)


representation P =(

0 11 0








uarr (p) (34)

ρAuarr (p) = +iλS


uarr (p) (35)



λSuarr (p)λS


uarr (p) = +im (36)

λAuarr (p)λA



ρSuarr (p)ρS



ρAuarr (p)ρA


uarr (p) = +im (39)








(x) = 0 (44)


(x) = 0 (45)



5

withΨ(+)(x) =

(ρA(x)λS(x)

)Ψ(minus)(x) =

(ρS(x)λA(x)

) and Γmicro =

(0 γmicro

γmicro 0

) (46)


The Lagrangian is

L =i

2






S)γmicroρSminus



=12



(48)
















6



[πminusmicro π


2σmicroνFmicroν

]χ(x) = 0 [

π+micro π

micro + minusm2 +g

2σmicroνFmicroν

]φ(x) = 0

(54)



λprimeS(p) =(

Ξ 00 Ξ





0 iΞiΞ 0


λIVS (p) =

(0 ΞminusΞ 0





RL(plarr 0) (59)





ν(xmicro) equivint

d3p(2π)3

12Ep

sumη

[λS


] (61)


7



and



[ν


(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0


+(

0φη

L(pmicro)


] (64)[

νML


]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)



L(pmicro)

0


] (65)








8







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12

ρSuarr (p) =

12radicE +m

p+ +mpr

ipl

minusi(p+ +m)

ρSdarr (p) =

12radicE +m

pl


(30)

ρAuarr (p) =

12radicE +m

p+ +mpr

minusipl

i(p+ +m)

ρAdarr (p) =

12radicE +m

pl


ipr

(31)


representation P =(

0 11 0








uarr (p) (34)

ρAuarr (p) = +iλS


uarr (p) (35)



λSuarr (p)λS


uarr (p) = +im (36)

λAuarr (p)λA



ρSuarr (p)ρS



ρAuarr (p)ρA


uarr (p) = +im (39)








(x) = 0 (44)


(x) = 0 (45)



5

withΨ(+)(x) =

(ρA(x)λS(x)

)Ψ(minus)(x) =

(ρS(x)λA(x)

) and Γmicro =

(0 γmicro

γmicro 0

) (46)


The Lagrangian is

L =i

2






S)γmicroρSminus



=12



(48)
















6



[πminusmicro π


2σmicroνFmicroν

]χ(x) = 0 [

π+micro π

micro + minusm2 +g

2σmicroνFmicroν

]φ(x) = 0

(54)



λprimeS(p) =(

Ξ 00 Ξ





0 iΞiΞ 0


λIVS (p) =

(0 ΞminusΞ 0





RL(plarr 0) (59)





ν(xmicro) equivint

d3p(2π)3

12Ep

sumη

[λS


] (61)


7



and



[ν


(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0


+(

0φη

L(pmicro)


] (64)[

νML


]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)



L(pmicro)

0


] (65)








8







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12

withΨ(+)(x) =

(ρA(x)λS(x)

)Ψ(minus)(x) =

(ρS(x)λA(x)

) and Γmicro =

(0 γmicro

γmicro 0

) (46)


The Lagrangian is

L =i

2






S)γmicroρSminus



=12



(48)
















6



[πminusmicro π


2σmicroνFmicroν

]χ(x) = 0 [

π+micro π

micro + minusm2 +g

2σmicroνFmicroν

]φ(x) = 0

(54)



λprimeS(p) =(

Ξ 00 Ξ





0 iΞiΞ 0


λIVS (p) =

(0 ΞminusΞ 0





RL(plarr 0) (59)





ν(xmicro) equivint

d3p(2π)3

12Ep

sumη

[λS


] (61)


7



and



[ν


(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0


+(

0φη

L(pmicro)


] (64)[

νML


]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)



L(pmicro)

0


] (65)








8







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12



[πminusmicro π


2σmicroνFmicroν

]χ(x) = 0 [

π+micro π

micro + minusm2 +g

2σmicroνFmicroν

]φ(x) = 0

(54)



λprimeS(p) =(

Ξ 00 Ξ





0 iΞiΞ 0


λIVS (p) =

(0 ΞminusΞ 0





RL(plarr 0) (59)





ν(xmicro) equivint

d3p(2π)3

12Ep

sumη

[λS


] (61)


7



and



[ν


(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0


+(

0φη

L(pmicro)


] (64)[

νML


]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)



L(pmicro)

0


] (65)








8







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12



and



[ν


(xmicro)]2 =

intd3p

(2π)31

2Ep

sumη

[(iΘφlowast η

L(pmicro)

0


+(

0φη

L(pmicro)


] (64)[

νML


]2 =

intd3p

(2π)31

2Ep

sumη

[(0

φηL(pmicro)



L(pmicro)

0


] (65)








8







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12







5]

Ψ = 0 (70)







h =(


)(72)




U1 =(


)(74)


9

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12

andU1 =

(U1 00 U1

) (75)

Thus we obtain

U1hUminus11 = |n|

(σ3 00 σ3

)(76)


(σ3 00 σ3

) 1 0 0 00 0 1 00 1 0 00 0 0 1

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (77)




1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|n| (78)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

=

= γ5chiral (79)



ξuarr = N eiα(


) (80)

ξdarr = N eiβ(


) (81)


(0 eminusiφ

minuseiφ 0

)ξuarr (82)


(0 minuseminusiφ

eiφ 0

)ξdarr (83)



10











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12











) β =

(0 12times2

12times2 0

) (88)





Ψ(4) = 0 (90)


Ψ(4) = 0 (91)



1 0 0 00 minus1 0 00 0 minus1 00 0 0 1

= α3|θ| (92)


U2α3Udagger2 =

1 0 0 00 0 0 10 0 1 00 1 0 0

α3

1 0 0 00 0 0 10 0 1 00 1 0 0

=

1 0 0 00 1 0 00 0 minus1 00 0 0 minus1

(93)





11












Press) p 125


12












Press) p 125


12

Majorana Neutrino: Chirality and Helicity

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Transcript of Majorana Neutrino: Chirality and Helicity