Homework 3

PHYS-541: QUANTUM FIELD THEORY
HOMEWORK III
(Due November 26 , 2012)
”It is is in the peculiar confluence of special relativity and quantum mechanics that a new
set of phenomena arises: Particles can be born and particles can die. It is this matter of
birth life and death that requires the development of a new subject in physics, that of
quantum field theory.” Zee
Q1:Let us recapitulate the Lagrangian and the Hamiltonian formulation of classical mechanics.
Given the following Lagrangian
L(q, ˙q) = 1 2 G ab(⃗q) ˙q a ˙q b + h a (⃗q) ˙q a − V (⃗q),
where, q a are N coordinates, G ab is a symmetric invertible N × N matrix.
a: Find the Hamiltonian H(⃗p, ⃗q). [Please note that H is a function of the coordinates and
the momenta.]
b: Find the Euler–Lagrange equations.
c: Find the Hamilton’s equations.
d: Check that Hamilton’s equations reproduce Euler–Lagrange equations.
Q2: Higher derivative theories are now widely used. This exercise is intended to study
non-relativistic limit of such theories.
a: Find the Euler–Lagrange (or Ostrogradski) equations for the following action
∫
S = dt L(q, ˙q, ¨q).
b Also, generalize your answer to the case for which the Lagrangian depends up to the N th
t-derivative of the coordinates q(t).
Q3: Too much relativistic motion can make your head dizzy. Let us do something nonrelativistic.
Let ψ α (x) be the non-relativistic two-component electron field where α = (↑, ↓)
denote the spin up and down components respectively. The Lagrangian density for the free
electrons is given by
L = i¯hψ † ˙ψ − ¯h2
2m ⃗ ∇ψ † ⃗ ∇ψ.
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a )Find the total momentum ⃗ P and the total angular momentum ⃗ J.
In the presence of a uniform magnetic field in the x 3 direction, the Lagrangian density
becomes
L = i¯hψ † ˙ψ − 1
2m (i¯h⃗ ∇ + e c ⃗ A)ψ † (−i¯h ⃗ ∇ + e c ⃗ A)ψ − e¯h
2mc ⃗ Bψ † ⃗σψ.
In the symmetric gauge, which I happen to like, the field reads
⃗A = (− 1 2 Bx 2, 1 2 Bx 1, 0),
⃗ B = (0, 0, B).
Show that the action is invariant under a rotation around the x 3 -axis and find the third
component of the angular momentum (J 3 ).
b) The action is not explicitly invariant under translations along the x 1 or the x 2 axis.
However, show that it is invariant under the following transformation:
⃗x → ⃗x ′ = ⃗x + ⃗a
ψ(t, ⃗x) → ψ ′ (t, ⃗x ′ ) = e ieB/2¯hc(a 2x 1 −a 1 x 2 ) ψ(t, ⃗x).
c) According to our good old lady Noether, there are three associated conserved quantities
( which are called the generalized momenta ⃗ Π ). Find these conserved quantities.
Q3: Answer the following questions.
a: In D + 1 dimensions, how many generators does the Poincare group have What
do they correspond to
b : If p µ is the energy-momentum vector of a particle in 5 dimensions, how does p µ p ν
transform under the Lorentz group Find the parts which transform irreducibly
c :Why do we label particles/fields with their masses and spins
d To construct a Dirac spinor ψ D , why do we take a 2-dimensional complex left-handed
Weyl spinor and a right-handed Weyl spinor and combine them into a 4-spinor
e Pauli-Lubanski vector is defined as 2W µ ≡ ɛ µνρσ M νρ P σ
Compute the following
a) P µ W µ b) W µ if (M σρ ) αβ = i(g αρ g σβ − g ασ g βρ )
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Q4 : a) Given a left-handed Weyl spinor ψ L ∈ (1/2, 0), show that σ 2 ψL ∗ is a righthanded
spinor. ( Here σ 2 is the usual second Pauli-matrix with purely imaginary entries.
b) Given a right-handed Weyl spinor ψ R ∈ (0, 1/2), show that σ 2 ψ ∗ R is a left-handed spinor.
( Here σ 2 is the usual second Pauli-matrix with purely imaginary entries.
c) Charge conjugation operation transforms a left-handed Weyl spinor into a right handed
one vice versa. Define charge conjugation on Weyl spinors as
ψ c L ≡ iσ 2 ψ ∗ L,
ψ c R ≡ −iσ 2 ψ ∗ R
Using these definitions, construct a bispinor , call it the Dirac spinor ψ D which satifies
(ψ c D) c = ψ D .
d) Again, using these definitions, construct a bispinor , call it the Majorana spinor ψ M
which satisfies ψ c M = ψ M .
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