Helicity, chirality, and the Dirac equation in the

non-relativistic limit
Logan Meredith
April 20, 2018

Abstract
The Dirac equation describes spin-1/2 particles with a consideration
for the e↵ects of special relativity. In this paper, we explore two major
emergent results of the Dirac equation. First, we see how the notions of
helicity and chirality arise from the Dirac equation, and exactly corre-
spond to one another in the massless limit. Second, we verify that the
Dirac equation is consistent with the Schrödinger equation in the non-
relativistic limit, both for a free particle and for a charged particle in an
external magnetic field.

1 Introduction
The Dirac equation, named after Paul Dirac, represented an attempt to incor-
porate the e↵ects of special relativity into quantum mechanics, and was intro-
duced in 1928 [2]. The result was a wave equation describing the relativistic
behavior of spin-1/2 particles, such as electrons and neutrinos. A major de-
parture from previous quantum theories, Dirac’s equation describes particles
with 4-component spinors or bispinors, rather than scalar wavefunctions. Fur-
thermore, the Dirac equation predicted the existence of antimatter many years
before it was experimentally observed.
In this paper, we provide descriptions of two results of the Dirac equation.
In the first section, we explore the topics of chirality and helicity. Chirality
is an inherent property of particles, whereas helicity of a particle depends on
the particle’s momentum. However, in the massless limit, the Dirac equation
shows that a particle of positive helicity has positive chirality, and vice versa.
In the second section, we take the non-relativistic limit of the Dirac equation
and show that it reduces to the Schrödinger equation, which describes particles
in the non-relativistic regime.

2 Helicity and chirality

In this section, we define helicity and chirality of particles described by the
Dirac equation. In addition, we use the Dirac equation to show how massless

1
particles of a given chiral handedness express helicity of the same handedness.

2.1 The helicity operator

The Dirac Hamiltonian
H =↵·p+ m (1)
does not necessarily commute with orbital angular momentum or spin angular
momentum, but with total angular momentum. When the particle is at rest,
however, p = 0, and so

[Si , H] = [↵
˜ i /2, H] = i✏ijk ↵j pk = 0. (2)

The Dirac Hamiltonian does commute with p. Furthermore, since p com-

mutes with S, S · p also commutes with H. Normalizing over p, we define the
helicity operator as
S·p
h= , (3)
|p|
which is necessarily a constant of motion. Physically, the helicity operator can
be thought of as a projection operator of spin along the direction of motion.
Note that
(S · p)(S · p) ˜ · p)(↵
(↵ ˜ · p) 1 |p|2 1
h2 = 2
= 2
= I = I. (4)
|p| 4|p| 4 |p|2 4

This implies that the eigenvalues of h are ±1/2.

2.2 The Dirac equation in the massless limit

We will begin our analysis in this section with massive particles and ultimately
look at the massless limit. The free Dirac equation gives
µ
( pµ m)u(p) = 0, (5)

where u is a 4-component spinor. Write

✓ ◆
u1 (p)
u(p) = . (6)
u2 (p)

Then the Dirac equation can be expanded to yield

✓ 0 ◆✓ ◆
(p m)I ·p u1
= 0, (7)
·p (p0 + m)I u2

which, explicitly, gives

p0 u 1 · pu2 = mu1 ,
0
p u2 · pu1 = mu2 .

2
Taking the sum and the di↵erence of these equations, we get
(p0 · p)(u1 + u2 ) = m(u1 u2 ),
0
(p + · p)(u1 u2 ) = m(u1 + u2 ).
By defining
1
ul = (u1 u2 ), (8)
2
1
ur = (u1 + u2 ), (9)
2
our equations can be rewritten as
(p0 · p)ur = mul ,
0
(p + · p)ul = mur .
These equations are coupled via the mass term. By letting mass go to zero, we
have the uncoupled equations
p0 u r = · pur , (10a)
0
p ul = · pul . (10b)
Since mass is a Lorentz scalar, these equations are Lorentz covariant. However,
they are not invariant under parity or space reflections. Equations (10) are
known as the Weyl equations [3].
Multiplying both sides of (10) by p0 = · p, we find
2
p0 ur = ( · p)( · p)ur = p2 ur ,
2
p0 ul = p 2 ul .
For a nontrivial solution, we require
2
p0 p2 = 0, (11)
which we recognize as Einstein’s equation for massless particles. It implies that
p0 = ±|p|. For p0 = +|p|, which corresponds to positive energy solutions, we
have
1
· pur = ur , (12a)
|p|
1
· pul = ul . (12b)
|p|
Since /2 is the spin operator for 2-component spinors, (12) imply that ur , ul
are eigenvectors of the helicity operator h with eigenvalues +1/2 and 1/2, re-
spectively. We call the particle described by ur right-handed, and likewise we
call the particle described by ul left-handed. As examples, it has been experi-
mentally determined that electron neutrinos are left-handed particles, whereas
electron antineutrinos are right-handed.

3
2.3 On chirality
The normalized solutions to the massless Dirac equation
p
/u(p) = 0 = p
/v(p) (13)
are given by
r ✓ ◆ r ✓ ◆
E ũ |p| ũ
u(p) = 1 = 1 , (14a)
2 E · pũ 2 |p| · pũ
r ✓1 ◆ r ✓ 1 ◆
E E · pṽ |p| |p| · pṽ
v(p) = = , (14b)
2 ṽ 2 ṽ
where u denotes the positive energy solutions and v denotes the negative energy
solutions. Evidently, if u is a solution to (13), so is 5 u, where we define
5 ⌘i 0 1 2 3. (15)
Since 2
5 = I, the eigenvalues of 5 are ±1. We say that eigenvectors of 5 with
eigenvalue +1 have right-handed or positive chirality. Likewise, we say that
eigenvectors with eigenvalue 1 have left-handed or negative chirality.
For a general spinor u, we can recover the right-handed and left-handed
chiral components by using the chiral projection operators
I+ 5
PR ⌘ , (16a)
2
I 5
PL ⌘ . (16b)
2
These projection operators have the following properties:
PR2 = PR , PL2 = PL ,
PR PL = PL PR = 0, PR + PL = I.
Together, these properties imply that any 4-component spinor can be uniquely
decomposed into right-handed and left-handed components.
The right-handed and left-handed chiral components of the normalized so-
lutions (14) can be explicitly written as
r 0 ⇣ ⌘ 1
·p
|p| @ 2 ⇣I + |p| ⌘ ũA
1
u R = PR u = ·p
, (17a)
2
2 I + |p| ũ
1

r 0 ⇣ ⌘ 1
·p
|p| @ 2 ⇣I
1
|p| ⌘ũ A
u L = PL u = ·p
, (17b)
2
2 I
1
|p| ũ
r 0 ⇣ ⌘ 1
·p
|p| @ 2 ⇣I + |p| ⌘ ṽ A
1
v R = PR v = ·p
, (17c)
2
2 I + |p| ṽ
1

r 0 ⇣ ⌘ 1
·p
|p| @ 2⇣ I
1
|p|⌘ ṽ A
v L = PL v = ·p
. (17d)
2
2 I
1
|p| ṽ

4
Note that 2-component spinors of the form
✓ ◆
(±) 1 ·p
= I± ˜
2 |p|
have definite helicity, so that
✓ ◆
(±) ·p1 ·p
h = I± ˜
2|p| 2 |p|
✓ ◆
1 ·p
=± I± ˜
4 |p|
1
= ± (±) .
2
This means that 4-component spinors with positive chirality are defined by
2-component spinors with positive helicity, and similarly, 4-component spinors
with negative chirality are defined by 2-component spinors with negative helicity.
Hence we can write
r ✓ ◆
|p| u(+)
uR = , (18a)
2 u(+)
r ✓ ◆
|p| u( )
uL = , (18b)
2 u( )
r ✓ ◆
|p| v (+)
vR = , (18c)
2 v (+)
r ✓ ( )◆
|p| v
vL = . (18d)
2 v( )

Recall the notation

µ
= (I, ), ˜ µ = (I, ),

and define ✓ ◆
p
p̂µ = (1, p̂) = 1, .
|p|
Now define the helical projection operators
1 µ
P (+) = ˜ p̂µ , (19a)
2
1 µ
P( )
= p̂µ . (19b)
2
These projection operators have the following properties:
2
P (±) = P (±) , P (+) + P ( )
= I,
(+) ( ) ( ) (+)
P P =P P = 0.

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Hence any 2-component spinor can be uniquely decomposed into components
of these operators, which project into spaces of positive and negative helicity.
This notion can be generalized to 4-component spinors by defining the following
helical projection operator:
✓ (±) ◆
(±) P 0
P4⇥4 = . (20)
0 P (±)
It is trivial to check that
(±)
[PR,L , P4⇥4 ] = 0,
which implies that spinors can be simultaneous eigenstates of chirality and he-
licity in the massless limit.
We have seen that, in the massless limit, particles of positive chirality have
positive helicity, and likewise, particles of negative chirality have negative helic-
ity. But this is only true for massless particles. Chirality can be thought of as
an inherent trait of particles, whereas helicity depends on the momentum of a
particle. For massive particles, it is possible to Lorentz boost to di↵erent frames
of reference to change helicity. But this is not true for massless particles; hence
chirality corresponds exactly to helicity for massless particles.

2.4 Properties of eigenstates of the helicity projections

In this section, we will derive properties of right-handed spinors and state the
analogous properties of left-handed spinors, since their derivation is nearly iden-
tical.
Write positive and negative energy solutions to the massless Dirac equation
with right-handed chirality as
1
u(+) = (I + · p̂)ũ,
2
(+) 1
v = (I + · p̂)ṽ.
2
Choose ✓ ◆ ✓ ◆
1 0
ũ = , ṽ = .
0 1
Then the positive and negative energy solutions can be written explicitly as
✓ ◆
(+) 1 |p| + p3
u (p) = p ,
2|p|(|p| + p3 ) p1 + ip2
✓ ◆
1 p1 ip2
v (+) (p) = p .
2|p|(|p| p3 ) |p| p3
It is simple to check that these spinors satisfy
u(+)† u(+) = v (+)† v (+) = 1,
u(+)† (p)v (+) ( p) = v (+)† ( p)u(+) (p) = 0,
1
u(+) u(+)† = v (+) v (+)† = (I + · p̂).
2

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The 4-component spinors defined by (18) further satisfy

u†R uR = vR
†
vR = |p|, (21a)
u†R (p)vR ( p) = vR
†
( p)uR (p) = 0, (21b)
✓ ◆
|p| P (+) P (+)
uR u†R = vR vR
†
= . (21c)
2 P (+) P (+)
Note that, with p0 = |p|, we can write
✓ ◆✓ ◆✓ ◆
1 0 1 p0 ·p I 0 I I
p
/ (I + 5 ) =
4 4 ·p p0 0 I I I
✓ ◆✓ ◆
|p| I · p̂ I I
=
4 · p̂ I I I
✓ (+) (+)
◆
|p| P P
= .
2 P (+) P (+)
Hence (21c) can be rewritten as
1
uR u†R = vR vR
†
= p
/
0
(I + 5 ). (22)
4
The analogous relation for left-handed spinors, which we will not derive here, is
given by
1 0
uL u†L = vL vL
†
= p / (I 5 ). (23)
4

3 Reduction to the Schrödinger equation

In this section, we consider the free Dirac equation and the Dirac equation for
a charged particle in an external magnetic field and show that they reduce to
Schrödinger equations describing the same physical systems.

3.1 Free particles

Recall that positive and negative energy solutions to the massive Dirac equation
have the form
r ✓ ◆ ✓ ◆
E+m ũ uL
u= ·p ⌘ , (24a)
2m E+m ũ uS
r ✓ ·p ◆ ✓ ◆
E+m E+m ṽ vS
v= ⌘ . (24b)
2m ṽ vL
Note that
·p
uS = uL , (25a)
E+m
·p
vS = vL . (25b)
E+m

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In the non-relativistic limit, |p| ⌧ m, and so uS ⌧ uL and vS ⌧ vL . Hence
the subscript S refers to the small component and the subscript L refers to the
large component.
The positive energy solutions satisfy
✓ ◆✓ ◆ ✓ ◆
mI ·p uL uL
=E , (26)
· p mI uS uS

which can be expanded to yield

· puS = (E m)uL , (27a)

· puL = (E + m)uS . (27b)

By substituting (25a) into (27a), we find that

·p
( · p) uL = (E m)uL . (28)
E+m
But since |p| ⌧ m, E ⇡ m, and so we have

p2
uL = (E m)uL = ENR uL , (29)
2m
where we use the fact that ENR ⌘ E m is the non-relativistic energy of the
particle. Note that (29) is the time-independent Schrödinger equation for a
free particle. We require the Dirac equation, a relativistic theory, to reduce to
a non-relativistic theory in the non-relativistic theory, and so this result is as
expected.

3.2 Charged particles in an external magnetic field

We shall couple a charged particle to a field through a minimal coupling. Take

pµ ! pµ eAµ ,

where e is the charge and A is the 4-potential. Since the coordinate representa-
tion of the momentum is pµ = i@µ , we have

@µ ! @µ + ieAµ .

The positive energy Dirac equation under this transformation is therefore

(↵ · (p eA) + m)u = Eu,

which can be expanded to yield

✓ ◆✓ ◆ ✓ ◆
mI · (p eA) uL uL
=E . (30)
· (p eA) mI uS uS

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This equation results in two equations given by
· (p eA)uS = (E m)uL , (31a)
· (p eA)uL = (E + m)uS . (31b)
From (31b), we have
· (p eA) · (p eA)
uS = uL ⇡ uL , (32)
E+m 2m
in the non-relativistic limit. Substituting (32) into (31a) yields
· (p eA)
[ · (p eA)] uL ⇡ (E m)uL . (33)
2m
Let us compute the products that appear in (33) to reduce notation. Note
that
2
[ · (p eA)] = (p eA)2 + i · [(p eA) ⇥ (p eA)]
2
= (p eA) ie · (p ⇥ A + A ⇥ p).
Furthermore, we calculate
(p ⇥ A + A ⇥ p)i = ✏ijk (pj Ak + Aj pk )
= ✏ijk (pj Ak Ak p j )
= ✏ijk [pj , Ak ]
= i✏ijk [rj , Ak ]
= i(r ⇥ A)i
= iBi ,
where B is the magnetic field. Hence (33) can be rewritten as
1 ⇥ ⇤
(p eA)2 e · B uL = ENR uL . (34)
2m
We recognize this as the Schrödinger equation for interaction with a magnetic
dipole. The magnetic dipole moment operator is
e e
µ= =g S, (35)
2m 2m
where g = 2 since S = /2.
Since this is a minimal coupling, we ignore higher-order e↵ects. These are
considered further in quantum electrodynamics. These e↵ects tend to slightly
change the g-factor for elecrons. Other particles that are not pointlike, such
as protons, can have wildly di↵ering g-factors, resulting in anomalous magnetic
dipole moments. These moments can be accomodated through a non-minimal
coupling with an interaction Hamiltonian
e µ⌫
HI = Fµ⌫ ,
2m
where Fµ⌫ = @µ A⌫ @⌫ Aµ is the electromagnetic field tensor and  is the
anomalous magnetic dipole moment. This is known as the Pauli interaction.

9
Notes
This paper represents a reproduction of my lecture notes on the properties of
the Dirac equation, which are derived from Section 3 of Das’ book on quantum
field theory [1]. I have cut down much of the material and added some useful
references to get a better picture of the utility and historical context of the
content. Consider the text of this paper to be a script and the equations to be
things that should be drawn on the chalkboard during the lecture.

References
[1] Ashok Das. Lectures on quantum field theory. World Scientific, 2008.

[2] Paul AM Dirac. The quantum theory of the electron. Proc. R. Soc. Lond.
A, 117(778):610–624, 1928.
[3] Hermann Weyl. Gravitation and the electron. Proceedings of the National
Academy of Sciences, 15(4):323–334, 1929.

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