Lie Groups and Lie Algebras: Winter-2023 HW 6 : Due Wednesday 3/1
The Clifford algebra $Cl(3,1)$
S0 4614S
Write down four real gamma matrices for \(\mathfrak{so}(3,1)\). That is, find four matrices \(\gamma_m\), for \(m=t,x,y,z\), such that \(\gamma_t^2=-1\), \(\gamma_k^2=+1\) for \(k=x,y,z\), and distinct matrices anticommute, so \(\{\gamma_m,\gamma_n\}=0\) for \(m\ne n\).
The Clifford algebra \(Cl(3,1)\) consists of linear combinations of all possible products of these gamma matrices. How many such matrices are there, that is, what is the dimension of \(Cl(3,1)\) as a vector space?
The \(N\) matrices in \(Cl(3,1)\) are manifestly closed under multiplication, and therefore form a Lie algebra, which clearly has a subalgebra of dimension \(N-1\). (Why?) This subalgebra turns out to be simple. (You do NOT need to show this.) Using the Killing form, determine the number of boosts and rotations in this Lie subalgebra.
Which simple Lie algebra have you constructed?
Briefly justify your answer
Bonus: What, if anything, would change if your gamma matrices were complex rather than real?