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What is the difference between the $Spin(3,1)$ group and the $SO(3,1)$ group?

Qmechanic
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1 Answers1

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  1. The spin group $Spin^+(3,1;\mathbb{R})\cong SL(2,\mathbb{C})$ is the double cover of the restricted Lorentz group $SO^+(3,1;\mathbb{R})$, cf. e.g. this Phys.SE post and links therein.

  2. The spin group $Spin(3,1;\mathbb{R})$ is the double cover of the special Lorentz group $SO(3,1;\mathbb{R})$.

  3. The pin group $Pin(3,1;\mathbb{R})$ is the double cover of the Lorentz group $O(3,1;\mathbb{R})$.

References:

  1. M. Rausch de Traubenberg, Clifford Algebras in Physics, arXiv:hep-th/0506011; p. 9.
Qmechanic
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    One might want to add that $Spin(3,1)$ is the universal cover of $SO^+(3,1)$ by definition, and is isomorphic to $SL(2,{\mathbb C})$ only by accident. – WillO Jan 28 '16 at 03:46
  • @WillO: Right . – Qmechanic Jan 28 '16 at 04:37
  • @WillO one question: so the spin group is defined as the universal cover of $SO^+(3,1)$ instead of as the double cover? This means it is defined as what we get when we exponentiate the Lie algebra of $SO(3,1)$? I thought it was defined as the double cover instead. – Gold Oct 01 '17 at 23:31
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    @user1620696 : The fundamental group of $SO^+(3,1)$ is ${\bf Z}/2{\bf Z}$, so the double cover and the universal cover are the same. You can either first prove that the two covers are the same and then define the Spin group to be this common value, or you can first define the Spin group to be one or the other cover and then prove that both covers are the same. Nothing depends on this. – WillO Oct 01 '17 at 23:56