Problem 1: Wave equation under space-time transformations 3 points

Relativistic Quantum Mechanics

F755, academic year 2010 — Prof. M. Kastner

Problem sheet 1 Submission deadline: August 04, 2010

Problem 1: Wave equation under space-time transformations 3 points

Consider the Galilean transformation

t= ˜t, x= ˜x+vt, y= ˜y

in (2 + 1) dimensions, where (t, x, y) are space-time coordinates in some reference frame S, (˜t,x,˜ y)˜ are space-time coordinates in frame ˜S, andv is a velocity.

(a) For some differentiable functionψ(t, x, y), express the partial derivatives

∂ψ

∂˜t,^∂ψ_∂x˜,^∂ψ_∂y˜

in terms of_∂ψ

∂t,^∂ψ_∂x,^∂ψ_∂y .

(b) Use the result from (a) to show that the wave equation

∂²ψ

∂x˜² +∂²ψ

∂y˜² −∂²ψ

∂˜t² = 0 (1)

transforms into

∂²ψ

∂x²(1−v²) +∂²ψ

∂y² −2v∂²ψ

∂x∂t −∂²ψ

∂t² = 0.

(c) Repeat (a) for the rotation transformation

t= ˜t, x= ˜xcosφ+ ˜ysinφ, y=−˜xsinφ+ ˜ycosφ.

How does the wave equation (1) change under this transformation?

(d) Repeat (a) for the Lorentz boost

t=γ(˜t−vx),˜ x=γ(˜x−v˜t), y= ˜y in (2 + 1) dimensions, whereγ= 1/√

1−v². How does the wave equation (1) change under this transformation?

Problem 2: Lorentz transformations as complex 2 × 2 matrices (written) 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.

Problem 3: The Lorentz group and its generators 3 points

(a) Show that the set of all Lorentz boosts Λ(v) =

γ(v) γ(v)v^T

γ(v)v 13+^γ(v)−1_v2 v⊗v^T

withγ(v) = 1

√ 1−v² wherev∈R³andv=|v|<1 is not a subgroup of L+^↑.

(b) Compute the generators

Bk = i ∂Λ(v)

∂vk

_v=0

of Lorentz boosts.

(c) Recall from your previous courses the generators of spatial rotations inR³. Write down (without proof) the generators of spatial rotations in Minkowski space.

(d) Show that the generators of Lorentz boosts and spatial rotations are linearly independent.