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Home , Dirac spinor, Majorana equation, Majorana fermion

2014Fall QFT Final Examination—Majorana

Yao-Chieh Hu1, ∗ 1Department of , National Taiwan University, Taipei 10617, Taiwan (Dated: January 18, 2015) In this short review, I go through the basic formulism of Majorana fermions, discuss related physics and summarize its benefits in different areas.

I. INTRODUCTION complex fields, C−1aˆ† C = ˆb† . (2) Form field theory, specially QED, we know that par- p p ticle excitations of the real scalar field are identical to To have more clearer insight, we start with a classical excitations of the scalar field. Moreover, we field and do the mode expansion: know that the excitations in EM field are identi- Z 3 cal to anti-. Scalar fields, 1 fields(photons) d p 1 −ipx ? ipx  φ(x) = 3 1 aˆp exp +ˆap exp (3) and spin 2 fields(), which are all , can be (2π) /2 (2Ep) /2 described by real fields: φ = φ†. With the knowledge Compare with the case of complex scalar field, of φ† creates a and φ creates an antiparticle, we also expect that these these are identical to their Z d3p 1 ψ(x) = (ˆa exp−ipx +ˆb? expipx)(4). . 3 1 p p (2π) /2 (2Ep) /2 The Dirac theory of the seems to require com- plex fields, and this is the first glance we see field here the of the first term, yield s some- involve distinct particle and antiparticle (fields.) This is thing which is not identical to the second term, i.e. the why and are distinct particles, and particle is, normally, distinct form antiparticle in com- also why even electrically neutral particles, , plex field. The charge conjugation involves changing each are different from anti-neutrons. In this review report, field for its complex conjugation and so a real field φ(x) we are going to make the whole theory more complete will describe particles that are their own antiparticles. by considering another kind of symmetry—containing However, complex field ψ(x) has different behaviours. fermions whose particles and antiparticles are identical. Motivated by above examples, we will going go search Can fermions to be its own antiparticles? That is say, the real solution for . Such a solution al- if we take the charge conjugate of the creation opera- lows the Dirac equation itself is real. This can be seen tor of the fermion, can it return back the same operator from the choice of γ matrices. Recall that there are in- without making any inconsistency? More rigorous, is finite choices of γ as long as they satisfies the Clifford algebra: C−1cˆ†C =c ˆ† (1) {γµ, γν } = 2gµν . (5) possible? This report largely follows [1,2,3], specially [1,2]. In the If we can find a set of γ which are purely imaginary, then second section, we build up our formulism of Majorana the Dirac equation fermions form Dirac solution. In the third section, we (p − m)Ψ = 0 (6) discuss the physics of the Majorana solution, including / mass and charge. In the last section, we go through some will be real and consequently so will its solutions. What discussions of Majorana fermions in the , problems Majorana did is that he do find a set of purely imaginary physicists encountered and future developments. γ:  0 σ2 iσ1 0  γ0 = , γ1 = , (7) II. THE MAJORANA SOLUTION σ2 0 0 iσ1  0 σ2 iσ3 0  γ2 = , γ3 = . (8) Majorana gave us a theory in which a fermion is iden- −σ2 0 0 iσ3 tical to its antiparticle. Such particles are called Majo- rana fermions and are unchanged by the act of charge Using the Dirac equation given by these matrices we will conjugation. find solutions ν(x) which will have the property that In field theory, we know that the charge conjugation ν(x) = ν(x)? which is known as the Majorana con- operator changes particles to antiparticles. In case of dition and reflects the fact that the solutions are iden- tical to their complex, or more precisely, to their charge conjugates. After demonstrating such solution do exist, we would now like to make contact with our previous ∗ [email protected] approach and describe Majoranas solution in the chiral 2 representation, which involves stacking up pairs of two- III. ENTERING THE WORLD OF FIELDS component Weyl to make four-component Dirac, or in this case Majorana, spinors. Notice here we de- At the end of last section, I mentioned that so far we note two-component Weyle spinors as ΨL and ΨR, while are still dealing with complex wave functions but fields. the symbols ψ and ν denote four-component Dirac and To make the theory respectable we should write things in Majorana spinors respectively. terms of field operators. We expect these fields must obey the Majorana condition, which can be expressed in terms of charge conjugation operators as: A. Charge conjugate of a Dirac in chiral representation 0 ? νˆ =ν ˆC = C νˆ . (18) After entering the world of fields, we should be comfort- able with the understanding that herea ˆ† =a ˆ?. −1 0 ? ΨC = C ΨC = C Ψ (9) Now it’s a perfect time to demonstrate the charge con- where C0 = −iγ2. For simplicity, we can use the Dirac jugation of Dirac fields, spinor with only single left/right-handed component as Z d3p 1 an example: Ψˆ = (19) (2π)3/2 (2E )1/2   p ΨL   Ψ = . (10) X s −ipx s ˆ† ipx 0 u (p)ˆas(p) exp +v (p)bs(p) exp .(20) Then take the charge conjugation, s  ? Whose charge conjugation is given as: 2 ? 2 ΨL Ψ(L)C = −iγ Ψ(L) = −iγ (11) 0 Z d3p 1 Ψˆ = C−1Ψˆ C = C0Φˆ ? = (21)  0 −iσ2 Ψ?   0  C 3 1/2 = L = . (12) (2π) /2 (2Ep) iσ2 0 0 iσ2Ψ? L X   − i γ2 us?(p)ˆa†(p) expipx −iγ2vs?(p)ˆb†(p) exp−ipx(22). Similarly, we have the charge conjugation of right-handed s s Wely spinor: s  ?  2 ?  Moreover, one may go through some algebra with explicit 2 0 −iσ ΨR s s Ψ(R)C = −iγ = . (13) form of u (p) and v (p) and show that ΨR 0 Now we have the conjugation of both left- and rigt- −iγ2us?(p) = vs(p), (23) handed spinors, it’s easy to see that conjugating the −iγ2vs?(p) = us(p). (24) charge twice returns the original spinor. With the chi- ral basis , we can build out the Majorana spinors with Then we automatically have the charge conjugated field single left- or right-handed Wely spinor and their charge as: conjugations: Z d3p 1 Ψˆ = C−1Ψˆ C = (25)       C 3 1/2 ΨL 0 ΨL (2π) /2 (2Ep) ν = + 2 ? = 2 ? . (14) 0 iσ ΨL iσ ΨL   X s † ipx s ˆ −ipx for left-handed. And v (p)ˆas(p) exp +u (p)bs(p) exp (26). s  0  −iσ2Ψ?  −iσ2Ψ?  µ = + R = R . (15) ΨR 0 ΨR Noticing that this can be made the same as the original Dirac field if we were to make the replacementsa ˆ†(p) ↔ for right-handed. Then we can define the charge conju- s ˆb†(p) and ˆb (p) ↔ aˆ (p), from which we conclude that gate of a Weyl spinor as Ψ and Ψ . These solution s s s L,C R,C the prescription indeed enacts charge conjugation on the ν and µ obey the all-important property that Dirac field in that it returns a Dirac field with particle νC = ν, (16) operators exchanged for antiparticle operators. µC = µ, (17) With above review of Dirac fields, we can formulate the Majorana field description. In general, when we expand which are the more general expressions of the Majorana a field, say ψ, we write it a : condition above. Notice that a Majorana particle may be built starting with only a left-/right-handed Weyl spinor, ψ = (ˆa − part) + (ˆb† − part) (27) putting its conjugate part into the slot where the right- = (particles) + (antiparticles) (28) /left-handed lives in. Notice that the above discussions are based on the face According to definition of antiparticle, we may further we are treating the spinors as complex wave functions but write it as: fields. This procedure is not actually right but somehow gives us better physical insights. ψ = (particles) + C−1(particles)C. (29) 3

And introducing the Majorana condition— where we have, (particle) = (antiparticle), we have:     φL 0 −1 † † ΦL = , ΦR = . (39) C (ˆas(p))C =a ˆs(p), (30) 0 φR C−1(us(p)ˆa†(p))C = −iγ2us?aˆ†(p). (31) s s Here is a difficulty—according to what we discussed be- So, it’s clear all we need to do is replace the antiparti- fore, Majorana solutions may be written in terms of Weyl cle part of Dirac field with what we got from Majorana spinors of a single (L- or R-handed) one might condition: wonder whether it is possible to identify a massive Ma- jorana field. This worry looks to be valid since a mass s ˆ† 2 s? † v (p)bs(p) ←→ −iγ u (p)ˆas(p). (32) term in the Lagrangian of the formνν ¯ vanishes if we treat the fields in the Lagrangian as complex, as we do Finally we get the general Majorana field: for the Dirac case. Surprisingly, it turns out that we actually are able to define massive Majorana fields, as Z d3p 1 vˆ = (33) long as they anticommute. In this case we still can 3 1/2 (2π) /2 (2Ep) have massive Majorana fields with building blocks—left-   handed Weyl spinors only, with mass mL, or Majorana X s −ipx 2 s? † ipx u (p)ˆas(p) exp −iγ u (p)ˆas(p) exp (34). fields built from right-handed Weyl fields only, with mass s mR. Then we can write down our Lorentz-invariant mass One might easily check the above equation enjoys the term of Lagrangian as: nice property—Majorana condition, 1 LM = − (N¯ L,C MNL + N¯ LMNL,C ), (40) Z d3p 1 2 vˆ = C−1vCˆ = (35) C 3 1/2 (2π) /2 (2Ep) where,   X 2 s? † ipx s −ipx       − iγ u (p)ˆa (p) exp +u (p)ˆas(p) exp (36) νL νL,C mL mD s NL = , NL,C = , M = (41). s νR,C νR mD mR =v. ˆ (37) And where, As a short conclusion at the end of this section, what     φL 0 we have went through is very formal, the key-point we νL = , νL,C = 2 ? , (42) should keep in mind—we do build a Majorana field whose 0 iσ φL   excitations are Majorana fermions. Namely, it is possible 0 2 ?  νR = , νR,C = −iσ φR (43) to find a kind of fermions whose antiparticles are identical φR to themselves. There are three kinds of mass involved in this Lagrangian—mD: Dirac mass, mL: left-handed Majo- IV. PHYSICS OF MAJORANA PARTICLES rana mass and mR: right-handed Majorana mass. As an short example, considering Majorana field built form only left-handed Wely spinor, we can set m = We will now discuss the physics of these particles— D m = 0 and m 6= 0, Majorana mass and Majorana charge. R L 1 L = LM + ν¯pν/ (44) 2 A. Majorana Mass 1 1 = ν¯pν/ − (¯νL,C mLνL +ν ¯LmLνL,C ), (45) 2 2 First we consider the mass of the particle excitations of where ν = νL + νL,C . Applying the variational principle the Majorana field. However, recall what we’ve learned † from QED that Weyl spinors are necessarily massless. In with respect to φL, we get the for the Dirac theory, the mass term of Lagrangian is a mixing of field φL, Ψ and Ψ . Consequently, massive Dirac spinors must L R iσ¯µ∂ φ − m iσ2φ? = 0. (46) contain independent left- and right-handed parts and the µ L L L particles may be thought of as oscillating between the The above equation can be rearrange to the form of Dirac two. equation, The Lorentz-invariant mass term in the Dirac La- grangian may be written in terms of Weyl spinors (p/ − mL)ν = 0. (47) as[1,2,3], One may repeat the above procedure for right-handed LM = mD(Φ¯RΦL + Φ¯LΦR), (38) fields without any difficulties. 4

B. Majorana Charge ””: a spin-1/2 , which we consider as the fermionic corresponding of photon. Starting with pure curiosity, we search for the electri- If the photino corresponds to the photon then it must cal charge of the Majorana fields and it turns out to be be its own antiparticle. This implies that the photino nothing—no electrical charge for Majorana fields. This is a , as will be the ”” and is, of course, necessary for the particle excitations of various types of ””. this field to satisfy the Majorana condition—be identi- cal to the antiparticles. It can also be seen from the field B. Mysterious equations by noting that if ν = νL + νL,C , then if we α make the transformation νL ← exp νL, we must have −α The very first moment we met is in QED— νL,C ← exp νL,C because of the complex conjugations seem to be well described as solutions to Weyls equation. between νL and νL,C . To get charges for such a the- ory, we may start from introducing an U(1) symmetry. In elementary [2,4], mysteriously, all neu- Unfortunately, it is impossible to do a U(1) transforma- trinos are left-handed massless particles with negative he- tion for Majorana spinors which simultaneously provides licity whereas all antineutrinos are left-handed massless both upper and lower slots with the same phase factor particles with positive helicity. We used to expect neutri- expα(they behave like a singlet under this symmetry.) nos to be massless particles. However, the discovery that That is—the Majorana equation can not be made invari- neutrinos emitted from the Sun with one flavour may ant under local U(1) transformations. In other words, a be detected with a different flavour. This suggests that particle carrying the conserved U(1) charge cannot be a these particles actually possess a nonzero mass!(however Majorana particle! still very small.) This leads us to the wonder—whether there is a Dirac mass. One may also wonder that is there exists the possibility that neutrinos are actually Majo- rana particles with Majorana mass. V. AMAZING NATURE OF MAJORANA

Having formulated this theory, we now ask what are C. Condensed the benefits? Indeed, for many years it seemed to be an interesting solution in need of a problem. However, In , emergent in recent years, physicists encounters several interesting give us an ideal playground for searching for exotic exci- phenomenons. tations such as Majorana fermions. In a semiconductor or a metal, electrons and holes look different because they are oppositely charged, and so it does not seem possible that the particles (electrons) and antiparticles (holes) can A. SUSY be symmetrically related. However, for superconduc- tor, on the other hand, the distinction between electrons Roughly speaking, the basic idea of supersym- and holes disappears and so seems like a possible envi- mertry(SUSY) is the symmetry between bosons and ronment for realizing Majorana fermions. Another bene- fermions. Considering the possibility that for every fit from superconductors is that they screen electric and species of in the Universe(no matter how distant magnetic fields, and so charge is not a good observable. are they) there exists a corresponding species of fermion Furthermore, it is possible that the quasiparticles of a (and vice versa) with the same mass. Although SUSY is superconducting system are Majorana fermions. An ex- on the way of developing, we can still make use of it. In a ample of such a circumstance involves a superconductor supersymmetric Universe we should expect the existence in the presence of vortices, which changes the equations of the of motion of the electrons and can lead to the trapping ”selectron”: a spin-0 particle with the mass of an of electronhole pairs which can be described as Majorana electron, which we consider as the bosonic corresponding fermions. of electron; and the

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