Labelle P. Supersymmetry DeMYSTiFied

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CHAPTER 4

The Physics of

Weyl, Majorana,

and Dirac Spinors

In Chapter 2 we got acquainted with Weyl spinors and their behavior under Lorentz

transformations. However, as noted before, supersymmetric theories are often writ-

ten in terms of Majorana spinors instead. It is therefore important to understand

the connection among Weyl, Majorana, and Dirac spinors both at the mathematical

level and at the physical level. In addition, we will see how we can go back and

forth between Weyl and Majorana spinors. The ﬁrst step is to understand the oper-

ation of charge conjugation applied to spinors and the connection with antiparticle

states.

Supersymmetry Demystified

4.1 Charge Conjugation and Antiparticles

for Dirac Spinors

Charge conjugation is the operation that turns a ﬁeld describing a certain particle

into a ﬁeld describing the corresponding antiparticle. To study the effect of charge

conjugation, let us consider a Dirac spinor because this probably is more familiar

to most readers. In this chapter and for the remainder of this book we will drop the

L and R labels on the Weyl spinors to alleviate the notation. It will be understood

that η will always represent a right-chiral spinor and χ a left-chiral spinor.

Consider, then, a Dirac spinor corresponding to a certain particle p (which could

be an electron, a neutrino, a quark, etc.):







(4.1)

The antiparticle spinor is obtained by applying the charge-conjugation operator

(which we will write down explicitly in an instant) to the particle state, so we may

write



≡ 

≡





where the c denotes charge conjugation. Here, a bar over the name of a particle

simply will represent the corresponding antiparticle (as in ¯ν for the antineutrino).

It is unrelated to the bar used in Chapters 2 and 3.

Note that in a Dirac spinor, η

and η

are two different right-chiral spinors.

The same comment applies to χ

and χ

; they are different left-chiral spinors. But

clearly, those four spinors are related. To establish the connection, we need to work

out explicitly the charge conjugation.

The operation of charge conjugation is deﬁned as



≡ C

(4.2)

where the bar denotes here the usual Dirac adjoint, and the charge-conjugation

matrix C may be taken to have the following representation:

C =−iγ



iσ

0 −iσ



CHAPTER 4 The Physics of Spinors

With a little work, we may write Eq. (4.2) in a more convenient form:



= C

= C(

†

)

= C(γ

)



†T

= Cγ



†T

≡ C



†T

where we have used (γ

)

= γ

and have introduced

= Cγ

=−iγ



0 iσ

−iσ



Here we have used the fact that γ

= 1. Note that C

= 1 as well.

Written explicitly, the charge-conjugated state is





iσ

†T

−iσ

†T



Using the fact that the charge-conjugated state is the antiparticle state, deﬁned

in Eq. (4.1), we ﬁnally ﬁnd

= iσ

†T

=−iσ

†T

(4.3)

This is an interesting result: These expressions are exactly what we obtained in

Eqs. (2.40) and (2.37)! What Eq. (4.3) tells us is that the right-chiral antiparticle

spinor is the right-chiral quantity built out of the particle left-chiral spinor! Like-

wise, the left-chiral antiparticle state is given by the left-chiral expression built out

of the right-chiral particle spinor. This is, by the way, the reason we introduced fac-

tors of i and −i in Eqs. (2.40) and (2.37): it was in order to make the correspondence

with charge conjugation exact.

It is easy to check that applying charge conjugation twice to a state gives back

the initial state:

(

)

= C

(

)

†T

= C



†T

)

†T

= C

∗



= C



= 

where we have used that the operation †T applied to a matrix (as opposed to a

quantum ﬁeld) is simply a complex conjugation. Applying the charge conjugation

operation to both sides of Eq. (4.2), we ﬁnd



= C







(4.4)

Supersymmetry Demystified

which leads to

= iσ

†T

=−iσ

†T

(4.5)

Of course, this also can be obtained directly by inverting Eq. (4.3).

As expected, the particle left-chiral spinor can be written as the left-chiral quan-

tity built out of the antiparticle right-chiral spinor. Likewise, the particle right-chiral

spinor can be written as the right-chiral expression built out of the antiparticle left-

chiral spinor.

Equations (4.3) and (4.5) teach us an important lesson: We are always free to

express the left- and right-chiral states of an antiparticle in terms of, respectively,

the right- and left-chiral states of the particle ﬁeld (and vice versa). For example,

we may rewrite a Dirac spinor in terms of left-chiral ﬁelds only, at the condition of

introducing the left-chiral antiparticle state:

 =







iσ

†T



(4.6)

This point will be crucial when we build the supersymmetric extension of the

standard model because we will express all fermions in terms of left-chiral states.

It must be kept in mind here that even though only left-chiral ﬁelds appear, the

upper two components still transform as a right-chiral spinor, so the four-component

spinor always has the structure:

 =



quantity that transforms like a right-chiral spinor

quantity that transforms like a left-chiral spinor



Of course, we could as well have written the Dirac spinor in terms of right-chiral

ﬁelds only:

 =







−iσ

†T



4.2 CPT Invariance

A short but crucial point needs to be mentioned here. Realistic relativistic quantum

ﬁeld theories must be CPT-invariant (some authors write TCP or PCT); i.e., they

must be invariant under the combined operations of charge conjugation C, parity P,

CHAPTER 4 The Physics of Spinors

CPT

Figure 4.1 Two states related by CPT.

and time reversal T. A consequence of this requirement is that lagrangian densities

must be “real” in the sense that they must satisfy

†

= L (4.7)

This shows right away why we cannot have a theory containing a left-chiral

spinor χ without including as well its hermitian conjugate χ

†

because a lagrangian

containing only one of the two obviously could not respect Eq. (4.7).

Now, from Eq. (4.3) we see that χ

†

can be written in terms of the right-chiral

antiparticle spinor η

. This means that the states χ

and η

always appear together

in a CPT invariant ﬁeld theory, which we illustrate in Figure 4.1.

In physical terms, this implies that if a theory contains a left-chiral spinor, it

also necessarily contains the corresponding antiparticle, which will be right-chiral.

When promoted to the status of quantum ﬁeld operator, χ

creates the right-chiral

antiparticle and annihilates the left-chiral particle.

For similar reasons, the pair of ﬁelds shown in Figure 4.2 also must always

appear together in a CPT-invariant quantum ﬁeld theory.

In other words, if there is a right-chiral particle state, there also must be a

corresponding left-chiral antiparticle state.

Actually, the two ﬁgures are totally equivalent because it is completely arbitrary

what we call the particle and what we call the antiparticle. So we are free to switch

the labels p and

p in the ﬁrst ﬁgure, which then becomes the second ﬁgure. What

we have learned is that if a particle of a certain chirality is present in a theory, there

will necessarily be a corresponding antiparticle of opposite chirality. If the particle

is massless, chirality and helicity can be used interchangeably.

Note that CPT invariance does not constrain us to necessarily have all four states

,η

,χ

, and η

present. It is possible to build a completely consistent theory out

of χ

and η

alone or out of χ

and η

alone. We will discuss this key observation

more fully in the next few sections.

¯p

CPT

Figure 4.2 Two other states related by CPT.

Supersymmetry Demystified

4.3 A Massless Weyl Spinor

Let us start with the simplest fermion theory that we could possibly imagine: the

theory of a noninteracting massless two-component Weyl spinor. We will take it to

be left-chiral. Then our lagrangian contains only the kinetic term given by

kin

= χ

†

¯σ

i∂

χ (4.8)

EXERCISE 4.1

Show that the lagrangian (4.8) is real in the sense of Eq. (4.7).

From what we learned in the preceding section, we know that one would observe

in the lab a left-chiral particle and its corresponding right-chiral antiparticle (again,

we could exchange the particle and antiparticle labels, but we choose to call the

left-chiral state the particle). It is instructive to make a count of the degrees of

freedom involved. A two-component spinor contains two complex variables, so

there are four “off-shell” degrees of freedom (i.e., four degrees of freedom before

the equations of motions are imposed). Physical massless spinors must obey the

Weyl equations Eq. (2.29), which provide two constraints, leaving two degrees of

freedom and therefore two types of observable states.

Since we know that the particles observed would be the left-chiral particle and the

corresponding right-chiral antiparticle, we should be able to write the lagrangian in

terms of χ

and η

instead of χ

and χ

†

. Indeed, this can be done using the second

relation of Eq. (4.5):

=−iσ

†T

so that

†

= η

(−iσ

)

†

= η

(iσ

)

The lagrangian then takes the form

kin

= η

iσ

¯σ

i∂

Note that this is a completely viable theory; there is no need to include the states

and χ

. They are not needed to build a consistent theory. Of course, we could

choose to add them in, but then they would be totally independent and therefore

would correspond to a new particle in its own right.

CHAPTER 4 The Physics of Spinors

4.4 Adding a Mass: General Considerations

Let’s start again with a left-chiral Weyl spinor χ

, but now we will make it massive.

Before proceeding, it’s important to point out that some references restrict the

term Weyl spinors to massless spinors, in which case Weyl spinors are always

eigenstates of the helicity operator. Others deﬁne Weyl spinors as being the chirality

eigenstates, which are irreducible representations of the Lorentz group. With the

ﬁrst deﬁnition, it is helicity that is used as the deﬁning property of Weyl spinors,

whereas in the second convention, it is behavior under Lorentz transformations that

is judged as being more fundamental. For the proponents of the ﬁrst deﬁnition,

massive Weyl fermions is an oxymoron, whereas the other camp has no problem

with that. Of course, in the massless case, the two schools of thought live in perfect

harmony; it is just when discussing massive particles that things get ugly.

In this book we will follow the second convention; i.e., we will use the term Weyl

spinors to denote the eigenstates of the chirality operator that have the Lorentz

transformations given in Eq. (2.32). Therefore, with our convention, it does make

sense to talk about massive Weyl spinors. The down side is that our massive Weyl

spinors do not have a well-deﬁned helicity, but this is a small price to pay because

helicity will play very little role in this book.

Let’s get back to the physics of massive fermions. Things are much more in-

teresting than in the massless case. As we argued in Section 2.3, it is possible to

use a Lorentz transformation to change the helicity of a massive spinor. This im-

plies that we need both helicities to describe a general massive particle state. From

Eqs. (2.30) and (2.31) and the discussion preceding them, we must conclude that if

we need both helicity states to describe a massive particle, we need both chiralities

as well. Therefore, for a massive fermion, both chiralities of the particle must be

present in the theory.

We can summarize the situation by saying that if we start with a left-chiral

particle state χ

, Lorentz transformations (LT) necessitate the introduction of a

right-chiral spinor η

, as illustrated schematically in Figure 4.3. This ﬁgure is not to

Figure 4.3 Two states of a massive spin 1/2 fermion required by Lorentz

transformations.

Supersymmetry Demystified

¯p

CPT

Figure 4.4 States associated to a massive left-chiral spinor through Lorentz and CPT

transformations.

be interpreted as saying that a Lorentz transformation changes a left-chiral state

into a right-chiral state—it is the helicity that is ﬂipped by a Lorentz boost—but as

meaning that Lorentz transformations require the existence of both chiral states for

a massive fermion.

Now comes the interesting part. As we saw in the preceding section, CPT in-

variance forces us to introduce the charge-conjugate state η

, which is right-chiral.

On the other hand, as we have just seen, for a massive particle, Lorentz invariance

also forces us to introduce a right-chiral state η

. The situation is summarized in

Figure 4.4.

The questions that come immediately to mind are: Can we take those two right-

chiral states forced on us to be the same? Can we take them to be different physical

states? In other words, do we have a choice?

The answer to all these questions is yes! We do have a choice, and this leads to

deﬁning two different types of four-component spinors.

4.5 Adding a Mass: Dirac Spinors

First, let’s consider the case where we choose the two right-chiral states to represent

different physical states, η

= η

. In that case, there are four physical (on-shell)

distinct states: the left- and right-chiral states of the particle χ

and η

as well as

the left- and right-chiral states of the antiparticle χ

and η

. This is illustrated in

Figure 4.5.

We can combine the left- and right-chiral states of the particle into a four-

component spinor that is the familiar Dirac spinor that we all know and love and

which we encountered in the ﬁrst chapter:







CHAPTER 4 The Physics of Spinors

¯p

CPT

¯p

Figure 4.5 The four states of a massive Dirac fermion.

Since we assume the antiparticle states χ

and η

to be distinct from χ

and η

we have



= 

Let us write the mass term for such a Dirac spinor. From the Dirac lagrangian

Eq. (2.2), we know that the mass is

−m



=−m

†



=−m



†

+ χ

†



(4.9)

This shows clearly that both χ

and η

are necessary in order to write down a Dirac

mass term.

As noted earlier, we also could write the Dirac spinor in terms of left-chiral

spinors only, using Eq. (4.6). The mass term then is

−m



=−m



(−iσ

)χ

+ χ

†

iσ

†T



(4.10)

which, using the spinor dot products deﬁned in Eq. (2.50), can be written in the

elegant form

−m



=−m(χ

· χ

+ ¯χ

· ¯χ

) (4.11)

EXERCISE 4.2

Show that the mass term in Eq. (4.11) is real in the sense of Eq. (4.7).