ps06soln - PHYS651 Problem Set 6 Solutions 1 S μν,S ρσ...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PHYS651: Problem Set 6 Solutions 1 . [ S μν ,S ρσ ] = i 4 2 [ γ μ γ ν ,γ ρ γ σ ]- [ γ μ γ ν ,γ σ γ ρ ]- [ γ ν γ μ ,γ ρ γ σ ] + [ γ ν γ μ ,γ σ γ ρ ] We can use that γ ν γ μ = 2 g μν 1- γ ν γ μ and the fact that 1 commutes with everything. Therefore = i 4 2 2 [ γ μ γ ν ,γ ρ γ σ ]- ( ρ ↔ σ ) Evaluate γ μ γ ν γ ρ γ σ = 2 g νρ γ μ γ σ- γ μ γ ρ γ ν γ σ = 2 g νρ γ μ γ σ- 2 g μρ γ ν γ σ + γ ρ γ μ γ ν γ σ = 2 g νρ γ μ γ σ- 2 g μρ γ ν γ σ + 2 g νσ γ ρ γ μ- γ ρ γ μ γ σ γ ν = 2 g νρ γ μ γ σ- 2 g μρ γ ν γ σ + 2 g νσ γ ρ γ μ- 2 g μσ γ ρ γ ν + γ ρ γ σ γ μ γ ν so the first 4 terms is the commutator that we need. Plugging it in yields [ S μν ,S ρσ ] = i 4 i g νρ γ μ γ σ- g μρ γ ν γ σ + g νσ γ ρ γ μ- g μσ γ ρ γ ν- g νσ γ μ γ ρ + g μσ γ ν γ ρ- g νρ γ σ γ μ + g μρ γ σ γ ν = ig μρ S νσ- ig μρ S νσ- ig νσ S μρ + ig μσ S νρ Now, you should by now know pretty well what is the algebra of the lorentz group, lets just hope you don’t have any nightmares about it and that this is the last question about it. 2 . ( a ) Using the definitions given in the problem we obtain L 1 = J 2 , 3 , L 2 = J 3 , 1 , L 3 = J 1 , 2 and K 1 = J , 1 , K 2 = J , 2 , K 3 = J , 3 , Together with the commutation relation for J μν from problem 1 we get for example L 1 ,L 2 = J 23 ,J 31 = i ( g 33 J 21- g 23 J 31- g 31 J 23 + g 21 J 33 ) =- iJ 21 = iL 3 K 1 ,K 2 = J 01 ,J 02 = i ( g 10 J 02- g 00 J 12- g 12 J 00 + g 02 J 10 ) =- iJ 12 =- iL 3 L i ,K j = 1 2 ilm J lm ,J j = i 1 2 ilm ( g m J lj- g l J mj- g mj J l + g lj J m ) = i ilm J m = i ilm K m or again in generalized notation L i ,L j = i ijk L k K i ,K j =- i...
View Full Document

Page1 / 6

ps06soln - PHYS651 Problem Set 6 Solutions 1 S μν,S ρσ...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online