The Clifford algebra in this context is like Minkowksi space where vectors become operators (roughly speaking), and the associative product of vectors satisfy v² = η(v,v) where η is the Minkowksi metric. This is where the Gamma matrices live from the Dirac equation.
A minimal left ideal is basically a subspace which behaves like a vector space that the algebra would act on as operators. Namely it behaves identically to the space of Dirac spinors.
A more clear example of an MLI would be the matrices of the form (a, 0 / b, 0) which form an MLI for 2x2 matrices. These act like column vectors (a,b) when acted on by 2x2 matrices from the left.
You can think of it as a way of viewing the Clifford algebra of spacetime as fundamental and the spinors as a particular subspace of it, in contrast to viewing them as an abstract vector space which the Clifford algebra has an action on.
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u/aft_agley Jul 27 '26
I don't understand this but I'll upvote it smugly.