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Problems 04

The document is a problem set for Quantum Field Theory I at ETH Zurich, focusing on the properties and representations of γ-matrices, including their algebraic identities and trace properties. It also covers spinors, completeness relations, and the Gordon identity, providing exercises to demonstrate these concepts. The document includes specific tasks to prove various mathematical relations involving γ-matrices and spinors.

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17 views2 pages

Problems 04

The document is a problem set for Quantum Field Theory I at ETH Zurich, focusing on the properties and representations of γ-matrices, including their algebraic identities and trace properties. It also covers spinors, completeness relations, and the Gordon identity, providing exercises to demonstrate these concepts. The document includes specific tasks to prove various mathematical relations involving γ-matrices and spinors.

Uploaded by

mahrusdastegir
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Quantum Field Theory I Problem Set 4

ETH Zurich, HS12 G. Abelof, J. Cancino, F. Dulat, B. Mistlberger, Prof. N. Beisert

1. Properties of γ-matrices
The γ-matrices satisfy a Clifford algebra,1

{γ µ , γ ν } = −2η µν 1. (1)

a) Show the following contraction identities using (1):

1. γ µ γµ = −4 · 1.

2. γ µ γ ν γµ = 2γ ν .

3. γ µ γ ν γ ρ γµ = 4η νρ 1.

4. γ µ γ ν γ ρ γ σ γµ = 2γ σ γ ρ γ ν .

b) Show the following trace properties using (1):

1. tr γ µ1 · · · γ µn = 0 if n is odd.

2. tr γ µ γ ν = −4η µν .

3. tr γ µ γ ν γ ρ γ σ = 4(η µν η ρσ − η µρ η νσ + η µσ η νρ ).

2. Dirac and Weyl representations of the γ-matrices


Using the Pauli matrices together with the identity,
       
0 1 0 1 0 1 2 0 −i 3 1 0
σ ≡ , σ ≡ , σ ≡ , σ ≡ , (2)
0 1 1 0 i 0 0 −1

we can realize the Dirac representation of the γ-matrices,

γD0 ≡ σ 0 ⊗ σ 3 , γDj ≡ σ j ⊗ iσ 2 (j = 1, 2, 3), (3)

where  
b11 A b12 A
A⊗B ≡ . (4)
b21 A b22 A
Denoting the Pauli matrices collectively by σ µ and defining (σ̄ 0 , σ̄ i ) = (σ 0 , −σ i ). we can
then define the γ-matrices in the Weyl representation:
 
µ 0 σµ
γW ≡ . (5)
σ̄ µ 0

Show that both representations satisfy the Clifford algebra (1). Can you show their
µ
equivalence, i.e. γW = T γDµ T −1 for some matrix T ?
−→
1
The minus sign is due to our choice of metric η µν = diag(−1, +1, +1, +1)! Alternatively, we might
use a plus sign (as in the opposite signature) and instead multiply all γ-matrices by a factor of i.

1
3. Spinors, spin sums and completeness relations
In this exercise we will use the Weyl representation (5) defined in the previous exercise.
a) Show that (p · σ)(p · σ̄) = −p2 .
b) Prove that the below 4-spinor us (~p) solves Dirac’s equation (pµ γ µ − m1)us (~p) = 0
√ 
p · σ ξs
us (~p) = √ , (6)
p · σ̄ ξs

where ξ± form a basis of 2-spinors.


c) Suppose, the 2-spinors ξ+ and ξ− are orthonormal. What does it imply for ξs† ξs and
X
ξs ξs† ? (7)
s∈{+,−}

d) Show that ūs (~p)us (~p) = 2m for s ∈ {+, −}.


e) Show the completeness relation:
X
us (~p)ūs (~p) = pµ γ µ + m1. (8)
s∈{+,−}

4. Gordon identity
Prove the Gordon identity,
1
ūt (~q)γ µ us (~p) = ūt (~q) −(q + p)µ − 21 [γ µ , γ ν ](q − p)ν us (~p).
 
(9)
2m
Hint: You can do this using just (1).

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