Homework 3

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 )
2