Problem 17: Spinrotations in 2 dimensions

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Friedrich-Schiller-University Jena Summer term 2016 Prof. Andreas Wipf

M.Sc. Tobias Hellwig Will be discussed: 19th week of year

Problems in Supersymmetry

Sheet 5

Problem 17: Spinrotations in 2 dimensions

Take an irreducible representation for the γ -matrices in 2 dimensions and calculate

Σ

^µν

= 1 2i γ

^µν

Also calculate the group elements S = exp(iω

µν

Σ

^µν

/2) generated by Σ. Prove explicitly the identity

S

⁻¹

γ

^ρ

S = Λ

^ρ_σ

γ

^σ

, Λ = e

^ω

,

which was already introduced in problem 13. How does a spinor transform under spin transformation.

Take a chiral representation, for example γ

⁰

= σ

₁

and γ

¹

= iσ

₂

. Let ψ be a chiral spinor, γ

∗

ψ =

±ψ.

Are these constraints compatible with Lorentz invariance?

How does a chiral spinor transform under spin transformations?

Which lorentz-invariant tensor field can you build out of bilinears of ψ?

Problem 18: Spinrotations in 4 dimensions We can choose the chiral representation

γ

µ

=

0 σ

µ

˜ σ

µ

0 , σ

µ

= (σ

0

,

−σ_i

), σ ˜

µ

= (σ

0

, σ

i

) for which the infinitesimal spin-rotations have the block-diagonal form

γ

µν

=

σ

µν

0 0 σ ˜

_µν

.

Calculate the matrices σ

_µν

and σ ˜

_µν

. What is γ

∗

=

−iγ₀

γ

₁

γ

₂

γ

₃

.