Download FIELD THEORY 1. Consider the following lagrangian1

�
�
�
�2
V (σ, π) = − 12 µ 2 σ 2 + π 2 + 14 λ σ 2 + π 2
0.1
Thursday 3 June 2010
(µ ∈ R , λ > 0)
1. Find
Summer
sheet all
QFTthe
I
symmetries of the lagrangian and obtain all the conserved
charges corresponding FIELD
to the continuous
symmetries
THEORY 1.
Problem 1
2. Find the extremal configurations of the hamiltonian and discuss
Consider the following lagrangian1 involving two real scalar fields σ(t, x) and
Suppose
the vacuum
of the quantum theory is such that
π(t, x)
on the that
Minkowski
spacetime
1 x) | 0 µ� = υ1
�L
0 |=σ(t,
x) π)
|0� = 0
∂ σ ∂ σ + 2 ∂ µ π ∂ �µ0π |−π(t,
V (σ,
2 µ
whereand set
σ
�(t, x)
υ�
� 0� | σ
�(t, x) | 0 � = 0
� 2≡ σ(t,
�x) −
2
2
2 2
1 2
1
V (σ, π) = − 2 µ σ + π + 4 λ σ + π
(µ ∈ R , λ > 0)
3. Write the Feynman rules for the quantum theory of the real scalar fields
1. σ
theπ(t,
symmetries
of the lagrangian
�Find
(t, x)all
and
x) and comment
the result.and obtain all the conserved
charges corresponding to the continuous symmetries
FIELD THEORY 2.
2. Find the extremal configurations of the hamiltonian and discuss
Suppose
that theselfenergy
vacuum ofatthe
theory is such that
Calculate
the electron
onequantum
loop
�| 0 � =
� 0 | σ(t, x)
� 0 | π(t, x) | 0 � =
0
d4 kυ µ
2
ν
Σ2 (p/) = i(ie)
γ
D
(k)
S(p
+
k)
γ
µν
(2π)4
and set
σ
�(t, x) ≡ σ(t,
x) dimensional
−υ
� 0regularisation,
|σ
�(t, x) | 0 � = where
0
in quantum electrodynamics
using
D µν (k)
is the photon propagator in the Feynman gauge, while S(p) is the Dirac
3. Write the Feynman rules for the quantum theory of the real scalar fields
propagator. Segregate the structure of the divergent and finite parts.
σ
�(t, x) and π(t, x) and comment the result.
FIELD THEORY 2.
1
Calculate
the electron
selfenergy
one loop
Jeoffrey Goldstone
(1961)
Il Nuovo at
Cimento,
Volume 9, p.154
�
4
HINT: The extrema of the hamiltonian
correspond to space-time
dfunctional
k µ
2
ν
34 γthat
Σ
(p
/)
=
i(ie)
Dyou
S(pto
+find
k) γthe
µν (k)
independent σ- and2 π-configurations such
have
extrema of the
(2π)
potential functional V(σ, π).
in quantum electrodynamics using dimensional regularisation, where D µν (k)
is the photon propagator in the Feynman gauge, while S(p) is the Dirac
propagator. Segregate the structure of the divergent and finite parts.
1
Jeoffrey Goldstone (1961) Il Nuovo Cimento, Volume 9, p.154
3
0.2
Thursday 18 February 2010
Summer sheet QFT I
FIELD THEORY 1.
Problem 2
Consider the following classical lagrangian involving a Dirac bispinor field
L = ψ̄ (x) ( i ∂/ − M − a
/ − b/ γ5 ) ψ (x)
where, as usual, ψ̄ (x) = ψ † (x) γ 0 while a
/ = γ µ a µ , b/ = γ µ b µ with the
constant and fixed real quantities given by
a µ = (a, − a)
b0 = b ,
b1 = b2 = b3 = 0
the same for all the inertial observers 2 .
1. Show that the term ψ̄ (x) a
/ ψ (x) in the lagrangian can be eliminated
by means of a gauge transformation on the bispinor field
ψ (x)
�→
ψ � (x) = exp{− i f (x)} ψ (x)
with a suitable choice of the real and dimensionless function f (x) .
2. Find all the continuous symmetry groups of invariance for the above
classical Lagrange density and verify if the Action is invariant under
the discrete parity tranformation
(t, x)
�→
(t, − x)
ψ (x)
�→
ψ � (x � ) = γ 0 ψ (t, − x)
3. Find the explicit form of the Feynman propagator in the momentum
space, that is, the opposite of the inverse of the kinetic term
S F ( p, b ) = [ − i (p/ − M − b/γ5 ) ] −1
4. Verify that in the limit b → 0 one recovers the customary well known
Feynman propagator of the Dirac field: namely,
S F(p) =
2
i ( p/ + M )
p2 − M 2 + i ε
For example one can realistically assume b < 3 × 10 −17 eV.
11
0.4
Thursday 10 September 2009
Summer sheet QFT I
Problem 3
FIELD THEORY 1.
The minimal Goldstone model : consider the classical Lagrangian for a real
scalar field
L = 12 ∂ µ φ (x) ∂ µ φ (x) + 12 µ 2 φ 2 (x) − 14 λ φ 4 (x)
with µ ∈ R and λ > 0
1. Find all the symmetries of the above field theoretic model
2. Write the energy momentum tensor as well as the total energy and
total momentum for this model and comment about it
3. Determine all the degenerate classical field configurations of minimal
energy and specify their symmetry
4. Choose one of the degenerate minimum field configurations φ min for
the classical Hamiltonian of the minimal Goldstone model and expand
the Lagrangian around it : which are the symmetries of the resulting
new classical Action? Comment about the classical Hamiltonian as a
function of the shifted field ϕ = φ − φ min
FIELD THEORY
Consider the Compton scattering e− γ → e− γ . The initial state contains
a photon of momentum k, energy k = | k |, and an electron at rest of
mass me . In the final state we have the scattered photon of momentum k � ,
energy k � = | k � |, and an electron which has received the recoil momentum
p � = k − k � and with the energy
�
�
p0� = E � = ( k − k � )2 + m2e = k2 + k � 2 − 2k · k � cos θ + m2e
where θ is the angle between the vectors k and k � . The electron spin states
�
are u r and ū r � (p � ) while the photon linear polarization vectors are ε µA (k � )
and ε νA (k) respectively, with r, r � , A, A � = 1, 2 .
�
�
1. Write the probability amplitude i MAA
rr � (k, k ) for this process to the
lowest order in the fine structure constant
2. Find the square modulus of the probability amplitude averaging over
the initial and summing over the final electron polarizations
20
new classical Action? Comment about the classical Hamiltonian as a
function of the shifted field ϕ = φ − φ min
Summer sheet QFT I
Problem 4
FIELD THEORY
Consider the Compton scattering e− γ → e− γ . The initial state contains
a photon of momentum k, energy k = | k |, and an electron at rest of
mass me . In the final state we have the scattered photon of momentum k � ,
energy k � = | k � |, and an electron which has received the recoil momentum
p � = k − k � and with the energy
�
�
p0� = E � = ( k − k � )2 + m2e = k2 + k � 2 − 2k · k � cos θ + m2e
where θ is the angle between the vectors k and k � . The electron spin states
�
are u r and ū r � (p � ) while the photon linear polarization vectors are ε µA (k � )
and ε νA (k) respectively, with r, r � , A, A � = 1, 2 .
�
�
1. Write the probability amplitude i MAA
rr � (k, k ) for this process to the
lowest order in the fine structure constant
2. Find the square modulus of the probability amplitude averaging over
the initial and summing over the final electron polarizations
3. Perform the photon polarization sums (averaging over the initial ones and
summing over the final ones), calculate the
20 traces and determine the unpolarized
differential cross section dσ/dΩ.
HINT: how many diagrams contribute to the amplitude at tree level?
0.5
Thursday 9 July 2009
Summer sheet QFT I
FIELD THEORY 1.
Problem 5
Consider the classical Lagrangian for the real scalar field
L=
1
1
λ
∂ µ φ (x) ∂ µ φ (x) − m 2 φ 2 (x) − φ 4 (x)
2
2
4!
and the dilatation trasformation abelian Lie group which acts on spacetime
coordinates and field according to
φ (x) → φ � (y) = e − α φ (x)
xµ → y µ = eα xµ
1. Calculate the canonical energy momentum symmetric tensor T µν
2. Calculate the Noether current J µ (x) associated to the Lie group of the
dilatation transformations and discuss its continuity equation
3. Define the new improved energy momentum symmetric tensor
Θ µν = T µν + a ( ∂ µ ∂ ν − g µν � ) φ 2 (x)
(a ∈ R)
and determine the constant a in such a manner that
g µν Θ µν = 0
for m = 0 (traceless condition)
4. Define a new dilatation current
j µ (x) ≡ x ρ Θ µρ
and show that
∂ µ j µ (x) = g λν Θ λν (x)
FIELD THEORY
Evaluate the differential cross section to the lowest order O(α2 ), where α is
the fine structure constant, for the electron-proton collision e− p −→ e− p in
the proton rest frame, disregarding the electron mass in respect to the proton
mass and the proton recoil after the collision. (Nota Bene : the proton is
assumed to be point-like)
25
0.7
Friday 29 May 2009
Summer sheet QFT I
FIELD THEORY 1.
Problem 6
Consider the planar Chern-Simons spinor electrodynamics, which is the field
theory defined on the 2+1 dimensional Minkowski spacetime with
x µ = (x0 , x1 , x2 ) = (ct, x, y)
metric tensor
g µν = g µν

1 0
0



0 −1 0
=


0 0 −1







and described by the classical the Lagrangian density
L = 14 κ ε λµν A λ (x) F µν (x) + ψ̄(x) [ i∂/ + q A/(x) − M ] ψ(x)
where a
/ ≡ γ µ a µ and ε 012 = 1 , which involves the real vector potential
A λ (x) , the field strength F µν (x) = ∂ µ A ν (x) − ∂ ν A µ (x) , the two component
spinor


ψ
(x)


1

ψ(x) = 


ψ2 (x)
q being the charge of spinor particle, κ a real number known as the ChernSimoms coefficient and, finally, the 2 × 2 gamma matrices
γ 0 = σ1
γ 1 = i σ2
γ 2 = i σ3
where σı ( ı = 1, 2, 3 ) are the Pauli matrices, in such a manner that the
Clifford algebra {γ µ , γ ν } = 2g µν I holds true.
1. Determine the canonical dimensions of the gauge potential and spinor
field in natural units as well as in physical units
2. Find the Euler-Lagrange field equations and make some comments
3. Specify the conditions under which the Action is gauge invariant
4. Find the normal mode decomposition of the quantized free spinor field,
i.e. for q = 0 , in the massless case M = 0
Score : 1. pt. 2, 2. pt. 3, 3. pt. 10 ; 4. pt 15
34
Summer sheet QFT I
Problem 7
Write the expressions for the QED Feynman amplitudes corresponding to the diagrams
given in the figure. Specify the momentum routing. Of course, you are NOT supposed to
perform the loop integrals.
Summer sheet QFT I
Problem 8: Path integral for a free particle
Summer sheet QFT I
Problem 9: Generating functional for fermion fields.
Using the generating functional derive:
1. the two-point Green function (i.e. the Feynman propagator) for the field ψ(x);
2. the photon-fermion vertex in QED.
Problem 10: The propagator for a massive vector field (using the generating
functional)
Use the generating functional.
Problem 11: CP properties of kaon non-leptonic decays
HINT: the pion is a pseudo-scalar particle.
Summer sheet QFT I
Problem 12: Properties of Majorana fermions
Summer sheet QFT I