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Quantum Electrodynamics (QED), Winter 2015/16
Fermions - Reference
Dirac Algebra
The Dirac algebra is introduced to define the 4-dimensional representation appropriate for fields carrying spin-1/2.
{γ µ , γ ν } = γ µ γ ν + γ ν γ µ = 2 g µν 1 ,
where g µν is the Minkowski metric and 1 is the 4×4 identity for the matrix representation and
!
!
i
0
1
0
σ
γ0 =
γi =
1 0
−σ i 0
with Pauli sigma matrices σ i .
The 4-dimensional representation of the Lorentz group is given by S µν =
generators
!
0
i σi
0i
,
S =−
2 0 −σ i
i
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[γ µ , γ ν ] with boost
and rotation generators
S ij
1
= − ijk
2
σk 0
0 σk
!
1
= ijk Σk .
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Some useful identitiesand trace identities involving γ matrices are
γ µ γµ = 41 ,
γ µ γ ν γµ = −2γ ν ,
γ µ γ ν γ ρ γµ = 4g νρ 1 ,
and
Tr γ µ = 0 ,
Tr γ µ γ ν = 4g µν ,
Tr γ µ γ ν γ ρ = 0
Tr γ µ γ ν γ ρ γ σ = 4g µν g ρσ − 4g µρ g νσ + 4g µσ g νρ .
Dirac Spinors
ψ(x) is a 4-component field that transforms under S. To define Lorentz scalars involving ψ we have
to introduce
ψ̄(x) = ψ † γ 0 ,
such that ψ̄ψ transforms as a scalar. The simpler combination ψ † ψ does not transform as a Lorentz
scalar (Lorentz boosts are not Hermitian in this representation). Lorentz vectors can be constructed as
ψ̄γ µ ψ .
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Quantum Electrodynamics (QED), Winter 2015/16
Fermions - Reference
The bar notation will be generalised such that eg. Ō ≡ O† γ 0 . It is useful to keep track of spinor
indices if you are not familiar with the spinor algebra so eg.
µ
z µ = ψ̄a γab
ψb .
Sometimes we will use this convention when highlighting the spinor indices in products - ie. a, b etc
with no raising or lowering convention (they will appear up or down, wherever convenient).
Having defined a scalar quantity we can define the Dirac action
µ
LD = ψ̄a (iγab
∂µ − m × 1ab )ψb ,
µ
which is normally written simply as LD = ψ̄(i∂/ − m)ψ with notation ∂/ab = γab
∂µ .
Varying with respect to ψ̄ we get the Dirac equation
(i∂/ − m)ψ = 0 .
The Dirac equation reduces to the Klein–Gordon equation if we consider just a scalar field (ie. field
that carries only linear momentum). This indicates that the solution (in momentum frame) to the Dirac
equation can be split into a plane wave part satisfying the Klein–Gordon equation and a second part
that carries all the angular momentum information
ψa (p) ∝ ua (p) e−i p·x .
Considering the u(p) part of the solution we can obtain the properly normalised function
!
√
s
p
·
σ
ξ
us (p) = √
,
p · σ̄ ξ s
with s = 1, 2 labeling the two independent solutions. The general solution is then
X
ψ(p) =
us (p) e−i p·x + v s (p) ei p·x ,
s
with
v s (p) =
!
√
p · σ ηs
,
√
− p · σ̄ η s
ie. positive and negative frequency (plane wave direction) modes under the assumption that both
modes have positive energy p0 > 0. Here σ = (1, σ) and σ̄ = (1, −σ) with a useful identity being
(p · σ)(p · σ̄) = p2 = m2 .
The two component spinors ξ s and η s define a spin basis. For our purposes we can always consider
them as eigenstates of eg. σ 3 such that eg.


!
!
p
p
1
0
 E − p3
 E − p3





0
1!
2
1


,

!
u (p) = 
 , u (p) = p

p
1
0




E + p3
E + p3
0
1
2
Quantum Electrodynamics (QED), Winter 2015/16
Fermions - Reference
are spin “up” and “down”.
Orthonormal properties of the full solutions are given by
ūr (p)us (p) = 2mδ rs ,
ur† (p)us (p) = 2Ep δ rs ,
v̄ r (p)v s (p) = −2mδ rs ,
v r† (p)v s (p) = 2Ep δ rs ,
ūr (p)v s (p) = v̄ r (p)us (p) = 0 .
Completeness relations will be very useful
X
ξ s ξ s† = 1 ,
s
X
s
u (p)ū(p)s = γ · p + m = p/ + m ,
s
X
v s (p)v̄(p)s = γ · p − m = p/ − m .
s
Quantisation
Introduce ladder expansion, now we need to ladder operators for both modes
Z
i
Xh
d3 p
1
s s
−i p·x
s† s
i p·x
p
ψ(x) =
.
a
u
(p)
e
+
b
v
(p)
e
p
p
(2π)3 2 Ep s
For fermions we must impose anti-commutations as the equal-time, canonical commutation relations.
This will ensure the correct statistics and ensure consistency with microcausality with respect to commutations of operators
o
n
{ψa (x), ψb (y)} = ψa† (x), ψb† (y) = 0 ,
n
o
ψa (x), ψb† (y) = δab δ (3) (x − y) ,
or, in terms of ladder operators,
n
o n
o
arp , as†
= brp , bs†
= (2π)3 δ rs δ (3) (p − q) ,
q
q
r s r s
ap , aq = bp , bq = ... = 0 .
The Hamiltonian density can be obtained by defining the momentum conjugate
π=
∂LD
= i ψ̄γ 0 = iψ † ,
∂ ψ̇
which is not the velocity because the Lagrangian is first order in ψ. Then H = π ψ̇ − LD and we can
write
Z
i
Xh
d3 p
s† s
s s†
H=
E
a
a
−
b
b
p
p p
p p ,
(2π)3
s
Z
i
Xh
d3 p
s† s
s† s
3 (3)
=
E
a
a
+
b
b
+
(2π)
δ
(0)
,
p
p
p
p
p
(2π)3
s
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Quantum Electrodynamics (QED), Winter 2015/16
Fermions - Reference
giving
h
i
s†
H, as†
p = Ep a ,
etc.. The vacuum is then defined as asp |0i = bsp |0i = 0 and we can interpret as†
p as creating a particle
s†
(positive frequency) with spin s, momentum p, and energy Ep and bp as creating an anti-particle
(negative frequency) with spin s, momentum p, and energy Ep and bs†
p.
One particle and anti-particle states are labeled as eg.
|p, s, −i =
p
2Ep as†
p |0i ,
|p, s, +i =
p
2Ep bs†
p |0i ,
and two particle states as
|p, s, −; q, r, −i =
p
p
r†
2Ep 2Eq as†
p aq |0i ,
etc. Notice that under interchange of fermions the state is equal to the negative of itself giving rise to
Fermi–Dirac statistics and the Pauli exclusion principle for the case where p = q and s = r.
Feynman Propagator
The propagator defined using the same choice of contours as the scalar case is (in momentum expansion)
i(p/ + m)
i
= 2
.
S F (p) =
p/ − m + i
p − m2 + i
The first form is defines the Green’s functions of the Dirac equation and the second form can be
obtained by inserting the completeness relation p/ +m in the numerator and using the identity {p, q} =
2 p · q in the denominator. The second form helps in keeping the matrix nature of the propagator
implicit but you can always keep the explicit notation eg.
F
Sab
(x − y) = h0| T ψa (x)ψ̄b (y) |0i .
Note that time ordering is modified for fermions to account for the statistics
(
h0| ψa (x)ψ̄b (y) |0i for x0 > y 0 ,
h0| T ψa (x)ψ̄b (y) |0i =
− h0| ψ̄b (y)ψa (x) |0i for y 0 > x0 .
This ensures that ψ(x), ψ̄(y) = 0 outside the light-cone when (x − y)2 < 0
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