3 The Dirac equation (II)

Quantum Field Theory-I Prof. G. Isidori

UZH and ETH, HS-2017 Revised version: October 15, 2017

http://www.physik.uzh.ch/en/teaching/PHY551/HS2017.html

3 The Dirac equation (II)

3.4 The Dirac Lagrangian

In order to derive the Dirac equation from a Lagrangian density we need to construct a Lorentz scalar (i.e. the Lagrangian) starting from ψ. To this purpose, it is worth noting that

S(1 + ω)

= 1

− i

2 ω

J ˆ

(31)

S(1 + ω)

= 1

+ i

2 ω

( ˆ J

)

=

γ

S(1 + ω)

γ

rs

(32)

where the last identity follows from (γ

)

= γ

γ

γ

and the explicit expression of J ˆ

. It is then easy to verify that, defining

ψ ¯ = ψ

γ

(33)

the product ¯ ψψ is a Lorentz scalar. The Lagrangian density that allows us to derive the Dirac equation is then

L

= ¯ ψ(iγ

∂

− m)ψ (34)

whose equation of motions (treating ψ and ¯ ψ as independent fields) are (i∂ / − m)ψ = 0 and ψ(i ¯ ← −

∂ / + m) = 0 (35) Decomposing the Dirac spinor into Weyl spinors, ψ

= P

ψ, the Lagrangian in Eq. (34) assumes the form

L

= ¯ ψ

(i∂ /)ψ

+ ¯ ψ

(i∂ /)ψ

− m(ψ

ψ

+ ψ

ψ

) (36)

where it becomes manifest that the mass term in the Dirac Lagrangian mix the two

irreducible representation of the Lorentz group.

3.4.1 Bilinear currents

So far we have analyzed the transformation properties of the spinors under proper Lorentz transformations. In order to find a representation of the (discrete) parity transformation, we need to find a unitary operator ˆ P , such that ˆ P

= 1, that anticommutes with the spatial components of γ

and commutes with γ

, as expected if γ

transform as a four-vector:

P ˆ

γ

P ˆ = ˆ P γ

P ˆ = −γ

P γ ˆ

P ˆ = γ

(37) It is easy to verify that ˆ P = +γ

or ˆ P = −γ

fulfill this goal. By looking at the explicit structure of γ

in the chiral representation, one can see that the parity transformation connects the two irreducible (

, 0) and (0,

) representations of the proper Lorentz group. On the other hand, the parity operator is diagonal in the standard representation of the γ matrices.

Using the transformation properties of ψ and ¯ ψ under the full Lorentz group it is the possible to deduce the following properties of the bilinear currents ¯ ψ(x)Γψ (x):

ψ ¯

(x

)ψ

(x

) = ψ(x)ψ(x) ¯ (scalar)

ψ ¯

(x

)γ

ψ

(x

) = det(Λ) ¯ ψ(x)γ

ψ(x) (pseudo − scalar) ψ ¯

(x

)γ

ψ

(x

) = Λ

ψ(x)γ ¯

ψ(x) (vector)

ψ ¯

(x

)γ

γ

ψ

(x

) = det(Λ)Λ

ψ(x)γ ¯

γ

ψ(x) (pseudo − vector) ψ ¯

(x

)σ

ψ

(x

) = Λ

Λ

ψ(x)σ ¯

ψ(x) (tensor)

(38)

3.4.2 Coupling to the electromagnetic field

In the case of the non-relativistic Schr¨ odinger equation, the wave equation of a spin- less particle of charge e, interacting with an electromagnetic field, is obtained by the replacement

H

= m + p

2m = m − ∇

2m −→ H

+ H

= m − ( ∇ ~ + ie ~ A)

2m + eΦ (39)

where Φ and A ~ are the scalar and the vector electromagnetic potential, respectively.

We have explicitly added a constant mass term, on both sides of (39), to facilitate the interpretation of this transformation as the non-relativistic limit of an appropriate covariant treatment. The natural relativistic generalization of the transformation (39) is the so called minimal substitution

∂

−→ D

= ∂

+ ieA

(40)

where Φ and A ~ are combined in A

= (Φ, ~ A). Using this transformation, the Dirac Lagrangian assumes the form

L

= ¯ ψ(iγ

D

− mψ) = L

− eA

J

(41)

where J

= ¯ ψγ

ψ is nothing but the Noether current associated to the global phase transformation

ψ(x) → e

ψ(x) (42)

It is worth nothing that in the m = 0 case the Dirac Lagrangian has two inde- pendent conserved currents,

J

= ¯ ψ

γ

ψ

and J

= ¯ ψ

γ

ψ

(43) associated to the two independent global phase transformations

ψ

(x) → e

ψ

(x) and ψ

(x) → e

ψ

(x) (44) These currents play an important role in the theory of weak interactions.

3.5 Solutions of the Dirac equation

Since the solutions of the Dirac equation satisfy also the Klein-Gordon equation, we should be able to write the most general solution of the former as a superposition of this two types of plane waves

ψ

(x) = e

u

(p) and ψ

(x) = e

v

(p) (45) where, in both cases, p

= √

~

p

+ m

.

= E

. Within this decomposition, the Dirac equation is equivalent to

(p / − m)u(p) = 0 (p / + m)v(p) = 0 (46)

Let’s consider first the case m 6= 0. In such case these two equations can easily

be solved in the Lorentz frame where ~ p = 0 (choice of frame that is always possible

if m 6= 0). Within the original interpretation of the Dirac equation as a one-particle

state wave equation, this is nothing but the particle rest frame.

Employing the standard decomposition of the γ matrices (where γ

is diagonal), a convenient basis for the ~ p = 0 spinors is

u

(m, 0) ∝











1 0 0 0











= . ξ

0

!

u

(m, 0) ∝











0 1 0 0











= . ξ

0

!

v

(m, 0) ∝











0 0 1 0











= . 0 η

!

v

(m, 0) ∝











0 0 0 1











= . 0 η

!

(47)

where ξ

and η

are two-component spinors of the type 1 0

!

and 0

1

!

.

The ~ p 6= 0 basis could be obtained from the above basis by means of an appro- priate Lorentz boost. Then, similarly to Klein-Gordon case, the general solution of the Dirac equation then takes the form

ψ

(x) =

Z

d

~ p (2π)

2E

X

s=1,2

h

e

a

(~ p)u

(p) + e

b

(~ p)

v

(p)

(48) where a

(~ p) and b

(~ p) are arbitrary complex coefficients

and p = (E

, ~ p).

Instead of performing explicitly the Lorentz boost of the rest-frame solutions, we can deduce the form of the spinors u

(p) and v

(p) in generic rest frames noting that the operators (p / − m) and (p / + m) act as projectors on u-v basis. Thus using the basis of four simple spinors introduced above, we can write

u

(p) ∝ (p / + m)u

(m, 0) (i = 1, 2)

v

(p) ∝ (−p / + m)v

(m, 0) (i = 1, 2) (49) It is easy to verify that the u

(p) and v

(p) thus defined satisfy Eq. (46). Intro- ducing the (non-spinorial) normalization factor N (E

), their explicit expression in the standard representation is

u

(p)

stand.

= N (E

)

"

(E

+ m) ξ

~ σ~ p ξ

#

(50) v

(p)

stand.

= N (E

)

"

~ σ~ p η

(E

+ m) η

#

(51)

3We are not yet interpretingψas a quantum field. . .

that reduces to Eq. (47) in the ~ p → 0 limit, for m 6= 0, but is well-defined also for m = 0. A convenient choice for the normalization factor is

N (E

) = [m + E

]

(52)

With such choice the four solutions (49) satisfy the following normalization condi- tions

¯

u

(p)u

(p) = 2mδ

u ¯

(p)v

(p) = 0

¯

v

(p)v

(p) = −2mδ

v ¯

(p)u

(p) = 0 (53) The expressions of u

(p) and v

(p) are quite convenient in the non-relativistic limit, where upper and lower components of the four-dimensional Dirac spinor are associated to the positive and negative frequency modes. Exactly like the particle and anti-particle solutions of the Klein Gordon field, these two frequency modes decouple for |~ p| m.

Using the above explicit expressions of u

(p) and v

(p) it is easy to verify that the following spin-sum relations (see exercise set n. 4):

X

a=1,2

u

(p)¯ u

(p) = p / + m

X

a=1,2

v

(p)¯ v

(p) = p / − m (54)

3.5.1 Solutions in the chiral representation for m = 0

In the fully relativist limit (i.e. for m → 0), the chiral representation turns out to be more useful. In this representation, the expressions of u

(p) and v

(p) for m = 0 assume the form

u

(p)

chiral

= 1

√ 2

1 −1

1 1

!

u

(p)

stand.

=

2|~ p|

"

P

ξ

P

ξ

#

(55) v

(p)

chiral

= − 1

√ 2

1 −1

1 1

!

v

(p)

stand.

=

2|~ p|

"

P

η

−P

η

#

(56) where we have introduced the helicity projection operators

P

= 1

2 1 ± ~ σ · ~ p

|~ p|

!

(57)

4 The minus sign in front of (59) is an arbitrary choice

The helicity (or the projection of the spin along the direction of motion) is a con- served quantity for a massless particles.

The appearance of the projection operators in (59) makes clear that upper and lower comments of the spinors, in the chiral representation, correspond to states with different chirality. Recalling that in such representation upper and lower comments of the spinors are selected by the the P

projectors, we deduce that the latter are equivalent to helicity projection operators.

Working with helicity eigenstates, or defining the two component spinors ξ

and η

, such that

~ σ · ~ p

|~ p| ξ

= ±ξ

~ σ · ~ p

|~ p| η

= ±η

(58) we can write the basis of four spinors in the chiral representation as

u

(p) .

= P

u(p)|

=

2|~ p|

"

ξ

0

#

v

(p)|

=

2|~ p|

"

η

0

#

u

(p) .

= P

u(p)|

=

2|~ p|

"

0 ξ

#

v

(p)|

=

2|~ p|

"

0

−η

#

(59)