Download QFT I, Standardmodell der Teilchenphysik – WS 2016/17 – O

QFT I, Standardmodell der Teilchenphysik – WS 2016/17 – O. Philipsen
Exam
06. March, 2017
Problem 1 [Bose Statistics, 10 points ]:
The normal-ordered Hamiltonian of a real scalar field reads
Z
d3 p
E(p)a†(p)a(p) .
:H:=
(2π)3 2E(p)
(1)
For notational convenience, we omit the double dots in the following expressions.
(i) Show that for β ∈ R
e−βH a†(p) = a†(p)e−β(H+E(p)) .
(2)
Hint: Start by showing Ha†(p) = a†(p)(H + E(p)) .
For a thermal state with temperature T = β1 expectation values of the operator O are
calculated via
tr(Oe−βH )
,
(3)
hOiβ =
tr(e−βH )
where the trace is over all quantum mechanical states of the system.
(ii) Using the result from the previous part, show that
a†(p)a(q)
β
= (2π)3 2E(p)δ 3 (p − q)
1
eβE(p)
−1
.
(4)
Problem 2 [Mott scattering, 9 points ]:
Consider the polarized differential cross section for Mott scattering, which is the scattering
of an e− with incoming momentum p and spin index r, and outgoing momentum p0 and
spin index s, by a heavy nucleus of charge (−ze) treated as a static, spinless point charge,
2
4z 2 α2 dσ(r, s)
=
us (p0 )γ 0 ur (p) .
4
dΩ
|q|
(5)
Here α = e2 /4π is the fine structure constant, |q| = |p0 − p| and θ is the scattering angle
between p and p0 .
(i) Compute the unpolarized cross section, where no spin is observed,
dσ
8z 2 α2 0
0
2
=
EE
+
p
·
p
+
m
.
dΩ N.P.
|q|4
(6)
(ii) Show that in the non-relativistic limit one obtains the Rutherford scattering formula
(use the trigonometric identity cos θ = 1 − 2 sin2 (θ/2))
dσ
z 2 α 2 m2
=
.
(7)
dΩ N.P. |p|4 sin4 (θ/2)
Problem 3 [Chiral transformations of Dirac spinors, 5 points ]:
5
Consider the chiral transformation for a Dirac spinor ψ → ψ 0 = eiαγ ψ
(i) Find out the corresponding transformation law for the conjugate spinor ψ and for
the vector current V µ = ψγ µ ψ .
(ii) Show that the Dirac Lagrangian LDirac = ψ(iγ µ ∂µ − m)ψ is invariant under the
given chiral transformation in the massless case, while this is not the case if m 6= 0.
Problem 4 [Eigenvalues of Dirac-operator, 2 points ]:
Given an SU (N )-gauge field Aµ = Aaµ T a , the covariant Dirac-operator D[A] is defined as
D[A] = iγ µ (∂µ − igAµ ).
(8)
Its eigenfunctions φi (x) satisfy
D[A]φi (x) = λi φi (x) ,
where λ ∈ C .
(9)
Show that its eigenvalues are gauge invariant, i.e. show that given φi transforms as φ0 (x) =
U (x)φ(x) with U (x) ∈ SU (N ), then
D[A0 ]φ0i (x) = λi φ0i (x).
(10)
Problem 5 [Varia, 25 points ]:
Answer the following questions by a few brief words or formulae.
(i) Which quantity do we need to calculate in a quantum field theory in order to describe
2 → n scattering processes?
(ii) In which case is it possible to find a non-trivial Dirac-spinor that is an eigenvector
for the spin and simultaneously the helicity operator?
(iii) Discuss the differences between photons and massless quanta of the complex KleinGordon field by comparing the corresponding:
(a) equations of motion (free propagation);
(b) transformation properties under Lorentz transformation;
(c) transformation properties under parity conjugation P;
(d) transformation properties under charge conjugation C;
(e) number of dynamical degrees of freedom described by the equations of motion.
(iv) What is the difference between the “Feynman propagator” and the “2-point function” for a given field?
(v) Provide two reasons why relativistic quantum mechanics is not sufficient for a full
description of particle physics.
(vi) Which of the following processes are physically allowed? Give a reason for the
forbidden ones.
(a) e+ + e− → γ;
(b) e+ + e− → 2γ;
(c) e+ + e− → 3γ;
(d) p + p → p + p + p + p;
(e) π − + p → K − + p
(f) p → π + + π 0 ;
(g) ∆++ → p + π + ;
in 1) QCD and 2.) the Standard Model;