0

I noticed the followings:

  1. $SL(2,\mathbb{C})$ is a double cover of $SO(1,3)$.

  2. $SL(2,\mathbb{C})$ is a double cover of $PSL(2,\mathbb{C})$

  3. $PSL(2,\mathbb{C})$ is isomorphic to $SO^+(1,3)$ which is the restricted Lorentz group.

I would expect that $PSL(2,\mathbb{C}) \cong SO(1,3)$, not $PSL(2,\mathbb{C}) \cong SO^+(1,3)$ since $SO(1,3)$ is not isomorphic to $SO^+(1,3)$.

Actually I think $SO(1,3)$ should be a 4-fold cover of $SO^+(1,3)$.

Can someone shed some light on this?

Nihar Karve
  • 8,445
Nugi
  • 541
  • 3
    Claim (1) is wrong. Where did you see it? – Qmechanic May 05 '21 at 06:30
  • I have seen it in the section 'Homomorphisms and Isomorphisms' in the wikipedia page of Lorentz group (https://en.wikipedia.org/wiki/Lorentz_group). – Nugi May 05 '21 at 06:51
  • Do not trust anything you see on wikipedia. How about you get hold of a good book? Try B. Thaller's "The Dirac equation" or the 2nd volume of Cornwell's group theory treatise. They treat coverings of the Lorentz groups in 1+3. – DanielC May 05 '21 at 07:07

1 Answers1

2

$SL(2,\mathbb{C})$ is the double cover of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$, not the proper Lorentz group $SO(1,3;\mathbb{R})$. See also e.g. this Phys.SE post. Be aware that notation and terminology may differ for different authors.

Qmechanic
  • 201,751