Relativistic Quantum Mechanics
F755, academic year 2009 — Prof. M. Kastner
Problem sheet 1 Submission deadline: July 27, 2009
Problem 1: The Lorentz group
(a) Use the definition of the Lorentz groupL to show that its dimension is six, i.e., only six of the 16 elements of a transformation matrix Λ∈L are independent.
(b) Show that the set of all Lorentz boosts
Λ(v) =
γ(v) γ(v)vT γ(v)v 13+γ(v)−1v2 v⊗vT
withγ(v) = 1
√1−v2
wherev∈R3andv=|v|<1 is not a subgroup of L+↑. (c) Compute the generators
Bk = i ∂Λ(v)
∂vk
v=0
of Lorentz boosts.
(d) Recall from your previous courses the generators of spatial rotations inR3. Write down (without proof) the generators of spatial rotations in Minkowski space.
(e) Show that the generators of Lorentz boosts and spatial rotations are linearly independent.
(f) Argue from the above findings that every element ofL+↑ can be written as a product Λ(v)Λ(ϕ) of a rotation and a Lorentz boost. Hint: Use a Baker-Campbell-Hausdorff formula!
Problem 2: Schr¨ odinger formulation of the Klein-Gordon equation
Let the two-component wave function of a particle of rest mass m0 and chargee in an electrostatic potentialϕbe defined asξ=
ϑ
χ
, where
ϑ= 1 2
ψ+ 1
m0c2(i∂t−eϕ)ψ
, χ= 1 2
ψ− 1
m0c2(i∂t−eϕ)ψ
.
Assume that ψ is a solution of the Klein-Gordon equation with electromagnetic potential ϕ. Derive from this assumption the Schr¨odinger-type differential equation governing the time evolution ofξ.
