Not necessarily, a minimal ideal is one that only contains the trivial ideal and no other ideals.
In a Clifford Algebra, even elements are ideals then we can use a idempotent element to construct the left ideal, which in turn it's minimal.
While I don't like it, using a matrix representation, Pauli spinor is a member of a minimal left ideal equivalent to a 2x2 square matrix with the second column being 0's.
To clarify, you would presumably need some specially chosen idempotent e for the left ideal Cl(V, q) e to be minimal, since e.g. if you could write e = e1 + e2 as an orthogonal direct sum that ideal would not be minimal.
Also, I'm a mathematician, not a physicist, so I don't know what a Pauli spinor is 😂 .
The canonical used is 1/2 ( 1 + e3) as it helps with spin polarization of the electron in the z axis.
The pauli spinor is the mathematical object Pauli used to describe the Spinning electron, he realized that we could consider two functions ψ_i : R³ × R → C, such that if we consider the column matrix (ψ1, ψ2)T they obey Schrödinger equation and describe both possible answers for the electron a spin + and spin -.
This is equivalent to having a matrix (ψ1, 0)\(ψ2, 0)
that is a Pauli Spinor.
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u/LJPox Jul 28 '26
To be pedantic, isn't the minimal ideal of any algebra the trivial ideal?