Relativistic Quantum Mechanics
F755, academic year 2011 — Prof. M. Kastner
Problem sheet 1 Submission deadline: August 01, 2011
Problem 1: The Lorentz group and its generators (written) 6 points
A Lorentz boost along some velocity vectorv∈R3 withv=|v|<1 is given by Λ(v) =
γ(v) γ(v)vT
γ(v)v 13+γ(v)−1v2 v⊗vT
withγ(v) = 1
√ 1−v2.
(a) Compute the transformation matrix for a Lorentz boost with velocityuinx-direction, followed by a boost with velocityvinz-direction. Show that the result can be expressed in the form Λ(φ)Λ(w), where Λ(φ) denotes a rotation around the rotational axis φ/|φ| with rotational angle φ = |φ|, and Λ(w) is a Lorentz boost with velocity vectorw. Confirm that|w|=√
u2+v2−u2v2. Is the result of the transformation invariant under the reversal of the order of thez- andx-boosts?
(b) Why does it follow from (a) that the set of all Lorentz boosts isnot a subgroup ofL+↑? (c) Compute the generators
Bk= i ∂Λ(v)
∂vk
v=0
of general Lorentz boosts. Hint: Start by expanding the matrix elements of Λ(v) to linear order inv.
(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) Use the Lie group property of the Lorentz group to argue that every element ofL+↑ can be written as a product Λ(v)Λ(ϕ) of a rotation and a Lorentz boost. Hint: Use the Zassenhaus formula for exponentials of operators!
Problem 2: Lorentz transformations as complex 2 × 2 matrices 4 points
Consider the coordinates (t, x, y, z) of a space-time point in (3 + 1) dimensions and arrange them into a complex Hermitian 2×2 matrix of the form
X =
t+z x+ iy x−iy t−z
.
Consider a transformation ofX,
AXA† =X0=
t0+z0 x0+ iy0 x0−iy0 t0−z0
,
whereA is a complex 2×2 matrix.
(a) Derive the most general condition on A that guarantees the transformation to preserve the Lorentz metric.
(b) Now assume detA= 1 [which isnotthe general result of (a)!]. How manyindependentparameters are there left in the space of transformation matricesAwith the constraint detA= 1?
(c) Determine the 2×2 complex transformation matrix for a boost of velocityv inz direction.