2014Fall QFT Final Examination—Majorana Fermions
Yao-Chieh Hu1, ∗
1Department of Physics, 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
Form field theory, specially QED, we know that par- ticle excitations of the real scalar field are identical to antiparticle excitations of the scalar field. Moreover, we know that the photon excitations in EM field are identi- cal to anti-photons. Scalar fields, spin 1 fields(photons) and spin 2 fields(gravitons), which are all bosons, can be described by real fields: φ = φ†. With the knowledge of φ† creates a particle and φ creates an antiparticle, we also expect that these these particles are identical to their antiparticles.
The Dirac theory of the electron seems to require com- plex fields, and this is the first glance we see fermion field involve distinct particle and antiparticle (fields.) This is why electrons and positrons are distinct particles, and also why even electrically neutral particles, neutrons, are different from anti-neutrons. In this review report, we are going to make the whole theory more complete by considering another kind of symmetry—containing fermions whose particles and antiparticles are identical.
Can fermions to be its own antiparticles? That is say, if we take the charge conjugate of the creation opera- tor of the fermion, can it return back the same operator without making any inconsistency? More rigorous, is
C−1cˆ†C = ˆc† (1) possible?
This report largely follows [1,2,3], specially [1,2]. In the second section, we build up our formulism of Majorana fermions form Dirac solution. In the third section, we discuss the physics of the Majorana solution, including mass and charge. In the last section, we go through some discussions of Majorana fermions in the Nature, problems physicists encountered and future developments.
II. THE MAJORANA SOLUTION
Majorana gave us a theory in which a fermion is iden- tical to its antiparticle. Such particles are called Majo- rana fermions and are unchanged by the act of charge conjugation.
In field theory, we know that the charge conjugation operator changes particles to antiparticles. In case of
complex fields,
C−1ˆa†pC = ˆb†p. (2) To have more clearer insight, we start with a classical field and do the mode expansion:
φ(x) =
Z d3p (2π)3/2
1
(2Ep)1/2 ˆapexp−ipx+ˆa?pexpipx (3) Compare with the case of complex scalar field, ψ(x) =
Z d3p (2π)3/2
1
(2Ep)1/2(ˆapexp−ipx+ˆb?pexpipx).(4) here the complex conjugate of the first term, yield s some- thing which is not identical to the second term, i.e. the particle is, normally, distinct form antiparticle in com- plex field. The charge conjugation involves changing each field for its complex conjugation and so a real field φ(x) will describe particles that are their own antiparticles.
However, complex field ψ(x) has different behaviours.
Motivated by above examples, we will going go search the real solution for Dirac equation. Such a solution al- lows the Dirac equation itself is real. This can be seen from the choice of γ matrices. Recall that there are in- finite choices of γ as long as they satisfies the Clifford algebra:
{γµ, γν} = 2gµν. (5) If we can find a set of γ which are purely imaginary, then the Dirac equation
(/p − m)Ψ = 0 (6)
will be real and consequently so will its solutions. What Majorana did is that he do find a set of purely imaginary γ:
γ0= 0 σ2 σ2 0
, γ1=iσ1 0 0 iσ1
, (7)
γ2=
0 σ2
−σ2 0
, γ3=iσ3 0 0 iσ3
. (8)
Using the Dirac equation given by these matrices we will find solutions ν(x) which will have the property that ν(x) = ν(x)? which is known as the Majorana con- 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 approach and describe Majoranas solution in the chiral
2 representation, which involves stacking up pairs of two-
component Weyl spinors to make four-component Dirac, or in this case Majorana, spinors. Notice here we de- note two-component Weyle spinors as ΨL and ΨR, while the symbols ψ and ν denote four-component Dirac and Majorana spinors respectively.
A. Charge conjugate of a Dirac spinor in chiral representation
ΨC= C−1ΨC = C0Ψ? (9) where C0 = −iγ2. For simplicity, we can use the Dirac spinor with only single left/right-handed component as an example:
Ψ =ΨL
0
. (10)
Then take the charge conjugation,
Ψ(L)C= −iγ2Ψ(L)?= −iγ2ΨL
0
?
(11)
=
0 −iσ2 iσ2 0
Ψ?L 0
=
0
iσ2Ψ?L
. (12) Similarly, we have the charge conjugation of right-handed Wely spinor:
Ψ(R)C = −iγ2
0 ΨR
?
=−iσ2Ψ?R 0
. (13) Now we have the conjugation of both left- and rigt- handed spinors, it’s easy to see that conjugating the charge twice returns the original spinor. With the chi- ral basis , we can build out the Majorana spinors with single left- or right-handed Wely spinor and their charge conjugations:
ν =ΨL
0
+
0
iσ2Ψ?L
=
ΨL
iσ2Ψ?L
. (14) for left-handed. And
µ =
0 ΨR
+−iσ2Ψ?R 0
=−iσ2Ψ?R ΨR
. (15) for right-handed. Then we can define the charge conju- gate of a Weyl spinor as ΨL,C and ΨR,C. These solution ν and µ obey the all-important property that
νC= ν, (16)
µC= µ, (17)
which are the more general expressions of the Majorana condition above. Notice that a Majorana particle may be built starting with only a left-/right -handed Weyl spinor, putting its conjugate part into the slot where the right- /left -handed lives in.
Notice that the above discussions are based on the face we are treating the spinors as complex wave functions but fields. This procedure is not actually right but somehow gives us better physical insights.
III. ENTERING THE WORLD OF FIELDS
At the end of last section, I mentioned that so far we are still dealing with complex wave functions but fields.
To make the theory respectable we should write things in terms of field operators. We expect these fields must obey the Majorana condition, which can be expressed in terms of charge conjugation operators as:
ˆ
ν = ˆνC= C0νˆ?. (18) After entering the world of fields, we should be comfort- able with the understanding that here ˆa†= ˆa?.
Now it’s a perfect time to demonstrate the charge con- jugation of Dirac fields,
Ψ =ˆ
Z d3p (2π)3/2
1
(2Ep)1/2 (19)
X
s
us(p)ˆas(p) exp−ipx+vs(p)ˆb†s(p) expipx
.(20)
Whose charge conjugation is given as:
ΨˆC= C−1ΨC = Cˆ 0Φˆ?=
Z d3p (2π)3/2
1
(2Ep)1/2 (21) X
s
− i γ2us?(p)ˆa†s(p) expipx−iγ2vs?(p)ˆb†s(p) exp−ipx
. (22)
Moreover, one may go through some algebra with explicit form of us(p) and vs(p) and show that
−iγ2us?(p) = vs(p), (23)
−iγ2vs?(p) = us(p). (24) Then we automatically have the charge conjugated field as:
ΨˆC = C−1ΨC =ˆ
Z d3p (2π)3/2
1
(2Ep)1/2 (25) X
s
vs(p)ˆa†s(p) expipx+us(p)ˆbs(p) exp−ipx
.(26)
Noticing that this can be made the same as the original Dirac field if we were to make the replacements ˆa†s(p) ↔ ˆb†s(p) and ˆbs(p) ↔ ˆas(p), from which we conclude that the prescription indeed enacts charge conjugation on the Dirac field in that it returns a Dirac field with particle operators exchanged for antiparticle operators.
With above review of Dirac fields, we can formulate the Majorana field description. In general, when we expand a field, say ψ, we write it a :
ψ = (ˆa − part) + (ˆb†− part) (27)
= (particles) + (antiparticles) (28) According to definition of antiparticle, we may further write it as:
ψ = (particles) + C−1(particles)C. (29)
3 And introducing the Majorana condition—
(particle) = (antiparticle), we have:
C−1(ˆa†s(p))C = ˆa†s(p), (30) C−1(us(p)ˆa†s(p))C = −iγ2us?ˆa†s(p). (31) So, it’s clear all we need to do is replace the antiparti- cle part of Dirac field with what we got from Majorana condition:
vs(p)ˆb†s(p) ←→ −iγ2us?(p)ˆa†s(p). (32) Finally we get the general Majorana field:
ˆ v =
Z d3p (2π)3/2
1
(2Ep)1/2 (33)
X
s
us(p)ˆas(p) exp−ipx−iγ2us?(p)ˆa†s(p) expipx
. (34)
One might easily check the above equation enjoys the nice property—Majorana condition,
ˆ
vC= C−1ˆvC =
Z d3p (2π)3/2
1
(2Ep)1/2 (35) X
s
− iγ2us?(p)ˆa†s(p) expipx+us(p)ˆas(p) exp−ipx
(36)
= ˆv. (37)
As a short conclusion at the end of this section, what we have went through is very formal, the key-point we should keep in mind—we do build a Majorana field whose excitations are Majorana fermions. Namely, it is possible to find a kind of fermions whose antiparticles are identical to themselves.
IV. PHYSICS OF MAJORANA PARTICLES
We will now discuss the physics of these particles—
Majorana mass and Majorana charge.
A. Majorana Mass
First we consider the mass of the particle excitations of the Majorana field. However, recall what we’ve learned from QED that Weyl spinors are necessarily massless. In Dirac theory, the mass term of Lagrangian is a mixing of ΨL and ΨR. Consequently, massive Dirac spinors must contain independent left- and right-handed parts and the particles may be thought of as oscillating between the two.
The Lorentz-invariant mass term in the Dirac La- grangian may be written in terms of Weyl spinors as[1,2,3],
LM= mD( ¯ΦRΦL+ ¯ΦLΦR), (38)
where we have,
ΦL=φL
0
, ΦR= 0 φR
. (39)
Here is a difficulty—according to what we discussed be- fore, Majorana solutions may be written in terms of Weyl spinors of a single chirality(L- or R-handed) one might wonder whether it is possible to identify a massive Ma- jorana field. This worry looks to be valid since a mass term in the Lagrangian of the form ¯νν vanishes if we treat the fields in the Lagrangian as complex, as we do for the Dirac case. Surprisingly, it turns out that we actually are able to define massive Majorana fields, as long as they anticommute. In this case we still can have massive Majorana fields with building blocks—left- handed Weyl spinors only, with mass mL, or Majorana fields built from right-handed Weyl fields only, with mass mR. Then we can write down our Lorentz-invariant mass term of Lagrangian as:
LM= −1
2( ¯NL,CMNL+ ¯NLMNL,C), (40) where,
NL =
νL νR,C
, NL,C =νL,C
νR
, M = mL mD mD mR
. (41) And where,
νL=φL
0
, νL,C =
0 iσ2φ?L
, (42)
νR= 0 φR
, νR,C = −iσ2φ?R
(43) There are three kinds of mass involved in this Lagrangian—mD: Dirac mass, mL: left-handed Majo- 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 mD = mR= 0 and mL 6= 0,
L = LM+1
2ν/pν¯ (44)
=1
2¯ν/pν −1
2(¯νL,CmLνL+ ¯νLmLνL,C), (45) where ν = νL+ νL,C. Applying the variational principle with respect to φ†L, we get the Majorana equation for the field φL,
i¯σµ∂µφL− mLiσ2φ?L= 0. (46) The above equation can be rearrange to the form of Dirac equation,
(/p − mL)ν = 0. (47)
One may repeat the above procedure for right-handed fields without any difficulties.
4 B. Majorana Charge
Starting with pure curiosity, we search for the electri- cal charge of the Majorana fields and it turns out to be nothing—no electrical charge for Majorana fields. This is, of course, necessary for the particle excitations of this field to satisfy the Majorana condition—be identi- cal to the antiparticles. It can also be seen from the field equations by noting that if ν = νL+ νL,C, then if we make the transformation νL ← expανL, we must have νL,C ← exp−ανL,C because of the complex conjugations between νL and νL,C. To get charges for such a the- ory, we may start from introducing an U(1) symmetry.
Unfortunately, it is impossible to do a U(1) transforma- tion for Majorana spinors which simultaneously provides both upper and lower slots with the same phase factor expα(they behave like a singlet under this symmetry.) That is—the Majorana equation can not be made invari- ant under local U(1) transformations. In other words, a particle carrying the conserved U(1) charge cannot be a Majorana particle!
V. AMAZING NATURE OF MAJORANA
Having formulated this theory, we now ask what are the benefits? Indeed, for many years it seemed to be an interesting solution in need of a problem. However, in recent years, physicists encounters several interesting phenomenons.
A. SUSY
Roughly speaking, the basic idea of supersym- mertry(SUSY) is the symmetry between bosons and fermions. Considering the possibility that for every species of boson in the Universe(no matter how distant are they) there exists a corresponding species of fermion (and vice versa) with the same mass. Although SUSY is on the way of developing, we can still make use of it. In a supersymmetric Universe we should expect the existence of the
”selectron”: a spin-0 particle with the mass of an electron, which we consider as the bosonic corresponding of electron; and the
”photino”: a spin-1/2 massless particle, which we consider as the fermionic corresponding of photon.
If the photino corresponds to the photon then it must be its own antiparticle. This implies that the photino is a Majorana fermion, as will be the ”Higgsino” and various types of ”gaugino”.
B. Mysterious Neutrino
The very first moment we met is in QED—neutrinos seem to be well described as solutions to Weyls equation.
In elementary particle physics[2,4], mysteriously, all neu- trinos are left-handed massless particles with negative he- licity whereas all antineutrinos are left-handed massless particles with positive helicity. We used to expect neutri- nos to be massless particles. However, the discovery that neutrinos emitted from the Sun with one flavour may be detected with a different flavour. This suggests that these particles actually possess a nonzero mass!(however 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.
C. Condensed Matter
In condensed matter physics, emergent quasiparticles give us an ideal playground for searching for exotic exci- 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 be symmetrically related. However, for superconduc- tor, on the other hand, the distinction between electrons and holes disappears and so seems like a possible envi- ronment for realizing Majorana fermions. Another bene- fit from superconductors is that they screen electric and magnetic fields, and so charge is not a good observable.
Furthermore, it is possible that the quasiparticles of a superconducting system are Majorana fermions. An ex- ample of such a circumstance involves a superconductor in the presence of vortices, which changes the equations of motion of the electrons and can lead to the trapping of electronhole pairs which can be described as Majorana fermions.
[1] Quantum Field Theory for the Gifted Amateur, T.Lancaster and S.J. Blundell, 2014.
[2] An Intro. to Quantum Field Theory, M. Peskin and D.
Schroeder, 1995.
[3] QUANTUM FIELD THEORY and the STANDARD MODEL, M. Schwartz, 2013.
[4] Modern Particle Physics, M. Jackson, 2013.
