Problems to Standard Model - Theoretische Physik 1 ...

Problems to Standard Model SoSe 2010
Mannel, Huber, Turczyk
Sheet 4 — Handout: 04.05.2010 — Due: Monday, 17.05.2010
Problem 10: Trace Formulae of Gamma Matrices
Use the known properties of the gamma matrices to show the following trace properties
Tr [ ]
γ µ = 0
Tr [ ]
γ µ1 · . . . · γ µ2n+1 = 0 n ∈ N
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Hint:
• Make use of (γ 5 ) 2 = 1.
Tr [ γ µ γ ν
]
= 4 gµν
Tr [ γ µ γ ν γ ρ γ σ
]
= 4 (gµσ g νρ − g µρ g νσ + g µν g ρσ )
Tr [ γ 5
]
= 0
Tr [ γ µ γ 5
]
= 0
Tr [ γ µ γ ν γ 5
]
= 0
Tr [ γ µ γ ν γ ρ γ 5
]
= 0
• Show that γ µ γ µ = 4 and use it.
Tr [ γ µ γ ν γ ρ γ σ γ 5
]
= −4iɛ µνρσ .
• For the last relation show that it is is completely antisymmetric. The only totally antisymmetric
combination is given by the epsilon tensor. Therefore you can make the
ansatz
Tr [ γ µ γ ν γ ρ γ σ γ 5
]
= c ɛ µνρσ .
Problem 11: Projectors
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Consider the Dirac matrices as given in the lecture with
a) Show that the quantities
fulfil the conditions
γ 5 = iγ 0 γ 1 γ 2 γ 3 .
P L/R = 1 2 (1 ∓ γ 5)
P L + P R = 1 P L P R = P R P L = 0
(P L ) 2 = P L (P R ) 2 = P R
in general. From this you can conclude that P L/R are projectors.
b) Calculate for the given γ µ from the lecture the explicit representation of γ 5 .
c) Which component of a Dirac spinor are projected out by P L/R in the given representation
of γ 5 ? Give a physical interpretation!
please turn over

Problem 12: Spinors
In this problem we calculate the four-component spinors u(p, s) and v(p, s) with p = (E, ⃗p)
from the ones in the rest-frame, i.e. from u(k, s) and v(k, s) with k = (m,⃗0).
a) Express the energy E and the momentum ⃗p of a moving particle in terms of its velocity
⃗v. Next, establish the relation between the components of the rapidity ⃗ φ and those of ⃗v
and ⃗p.
b) Use these relations to show that, with p 0 = √ ⃗p 2 + m 2 , one has
D R/L (L(p)) ≡ e ± 1 2 φi σ i = p0 + m ± ⃗p · ⃗σ
√
2m(p 0 + m) .
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c) Finally, show that for the transformation of the Dirac spinors, one obtains
u(p, s) =
/p + m
√
2m(p 0 + m) u(k, s) and v(p, s) = − /p − m
√ v(k, s) .
2m(p 0 + m)
Hint:
• The relations cosh(x/2) = √ (cosh(x) + 1)/2 and sinh(x/2) = √ (cosh(x) − 1)/2 will be
useful.
Problem 13: Properties of the Dirac Equation
Consider the Dirac equation
(
i /∂ − m ) ψ = 0 .
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a) First prove for an arbitrary four vector a µ the relation
/a/a = a 2 1 .
Then demonstrate that each component of a Dirac spinor also satisfies the Klein-Gordon
equation.
b) Show that the adjoint Dirac equation is given by
¯ψ ( i ←− /∂ + m ) = 0 .
c) Demonstrate that the current ¯ψγ µ ψ is conserved.