1. No category

Relativity, Particles and Fields
Summer Term 2015
Prof. Dr. Nora Brambilla
Exercise Sheet No. 3
Discussion May 19th, May 20th, May 22th 2015
Dr. J. Tarr´
us, H. Martinez, D. Meindl, V. Shtabovenko
Exercise 3.1: Conserved currents
a) Consider the Lagrangian density for a complex 2-component scalar field φ
L = ∂µ φ† ∂ µ φ − m2 φ† φ,
where φ =
φ1
φ2
(1)
is an SU(2) doublet.
i) Show that L is invariant under a global SU(2) symmetry.
ii) Find the corresponding Noether currents and charges.
(Hint: Under a global SU(2) symmetry the field transforms as φ → exp (iα · τ ) φ,
where α is a real 3-component vector and τ = σ/2 are the generators of SU(2), where
σ are the Pauli matrices. Use infinitesimal transformations.)
b) Consider the Lagrangian density for a real 3-component scalar field φ = (φ1 , φ2 , φ3 )T
1
m2 T
L = ∂µ φ T ∂ µ φ −
φ φ.
2
2
(2)
i) Find the equation of motion for φ.
ii) Show that L is invariant under a global SO(3) symmetry.
iii) Find the corresponding Noether currents.
(Hint: Under a global SO(3) symmetry the field transforms as φ → exp (iθ · J )φ, where
θ is a real 3-component vector and J are the generators of SO(3), with (J k )lm = −iεklm .
Use infinitesimal transformations.)
c) Consider the Lagrangian density of the σ-model
m2
1
¯ µ ∂µ ψ+g ψ(σ+2iτ
¯
∂µ σ∂ µ σ + ∂µ π T ∂ µ π +iψγ
·πγ5 )ψ− (σ 2 +π 2 )+λ(σ 2 +π 2 )2 ,
2
2
(3)
where σ is a scalar field, π is a 3-component scalar field (π 2 ≡ π T π), ψ is a doublet of
Dirac spinor fields, τ is defined as in a) and γ 5 = iγ 0 γ 1 γ 2 γ 3 .
L=
1
i) Show that L is symmetric under:
σ → σ,
π → π−α×π,
ψ → ψ + iα · τ ψ ,
(4)
(5)
(6)
where α is an infinitesimal constant vector.
ii) Find the corresponding Noether currents.
Exercise 3.2: Dilatation transformation of the Klein-Gordon field
Consider the action for a massless real scalar field (Klein-Gordon field)
Z
1
S = d4 x (∂µ φ∂ µ φ) ,
2
(7)
where the equation of motion for φ is given by
∂µ ∂ µ φ = 0.
(8)
This is the Klein-Gordon equation with the mass set equal to zero. Under a dilation transformation with parameter α, spacetime coordinates and the field φ(x) transform as
xµ → xµ0 = eα xµ ,
φ(x) → φ0 (x0 ) = e−dφ α φ(x) .
(9)
(10)
a) Show that this transformation is a global symmetry of the action for an appropriate
choice of dφ .
(Hint: Note that the measure d4 x is not invariant under this transformation.)
b) Find the corresponding Noether current and use the equation of motion for φ to verify
that this current is conserved.
c) Show that if the action has a non-vanishing mass term
Z
1
− d4 x m2 φ2
2
(11)
then the above transformation is not a symmetry, i.e. it doesn’t leave S invariant.
d) Show that if the Klein-Gordon field remains massless but S contains an interaction
term
Z
Z
4
d x V (φ) = d4 x λφ4 ,
(12)
then the dilation symmetry is preserved.
e) Explain, why the symmetry is spoiled by a mass term but not by the particular interaction term. Will every possible interaction term preserve the symmetry?
(Hint: Compare mass dimensions of m and the coupling constant λ.)
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Exercise 3.3: Maxwell theory and energy-momentum tensor
The Lagrangian density for classical electromagnetism in the absence of charges (free Maxwell
theory) contains only kinetic term for a massless vector field and is given by
1
L = − Fµν F µν ,
4
(13)
~ is the photon field, φ is
where Fµν = ∂µ Aν − ∂ν Aµ is the field strength tensor, Aµ = (φ, A)
~ is the vector potential. E
~ and B
~ are defined as
the electric potential and A
E i = F i0 ,
1
B i = εijk F kj .
2
(14)
(15)
a) Derive the equation of motion for the photon field. Examine the components of this
equation to recover two of the four Maxwell’s equations.
b) The remaining two Maxwell’s equations follow automatically from the properties of
Fµν . To convince yourself, show that
∂µ Fλν + ∂ν Fµλ + ∂λ Fνµ = 0.
(16)
Examine the components of this equation to recover the remaining Maxwell’s equations.
c) Derive the energy-momentum tensor T µν of the free Maxwell theory.
d) In the usual understanding, the energy-momentum tensor Θµν is defined to fulfill the
following two properties
Θµν = Θνµ
∂µ Θµν = 0
(17)
(18)
Show that T µν satisfies satisfies Eq. 18, but is not symmetric in the indices.
e) Add ∂ρ (Aν F µρ ) to T µν in order to obtain a symmetric energy-momentum tensor
T µν → Θµν = T µν + ∂ρ (Aν F µρ ) .
(19)
Show that Θµν still satisfies Eq. 18.
f) Show that Θµν is traceless, i.e. Θµµ = 0.
g) Examine the components of Θµν and identify three familiar quantities: Maxwell stress
tensor, Poynting vector and electromagnetic energy density.
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Exercise groups: time and locations
Group
Tutor
1
Dominik Meindl
2
Vladyslav Shtabovenko
3
Hector Martinez
Time
Room
Tuesday May 19th , 8-10
C.3203
Wednesday May 20th, 10-12 C.3203
Friday May 22th, 10-12
PH 1141
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