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The double cover of $SO(4)$ is isomorphic to $SU(2)\times SU(2)$, which might be related to $SO(3,1)$.

Also, I have seen that $so(3,1)=Cl^2(3,1)$. However, the notation confused me a bit, does the super index of 2 means $Cl(3,1)\times Cl(3,1)$? If so, how does one prove that $so(3,1)=Cl^2(3,1)$?

What other isomophisms or equalities does $SO(3,1)$ have?

Qmechanic
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JavaGamesJAR
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1 Answers1

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  1. More generally, for a Clifford algebra $Cl(V,Q)$ over a $d$-dimensional vector space $V$ with a quadratic form $Q$ of signature $(s,t)$ there exists a linear isomorphism $$Cl(V,Q)~\cong~ \bigwedge\! V\tag{1}$$ with the exterior algebra $\bigwedge\! V$. Both sides of eq. (1) are $2^d$-dimensional. In particular for quadratic elements $$Cl^2(V,Q)~\cong~ \bigwedge\!{}^2 V~\cong~ so(V,Q),\tag{2}$$ where $so(V,Q)$ is the $\frac{d(d-1)}{2}$-dimensional special orthogonal Lie algebra associated with $(V,Q)$.

    In OP's case $(V,Q)$ is spacetime endowed with the Minkowski metric.

  2. Concerning group isomorphisms: The restricted Lorentz group $$SO^+(3,1;\mathbb{R})~\cong~ PSL(2,\mathbb{C}),$$ cf .e.g. this Phys.SE post.

Qmechanic
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