Editing Spinor (section)

=== Abstract spinors ===
There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of {{math|Cℓ(''V'', ''g'')}} on itself. These are subspaces of the Clifford algebra of the form {{math|Cℓ(''V'', ''g'')''ω''}}, admitting the evident action of {{math|Cℓ(''V'', ''g'')}} by left-multiplication: {{math|''c'' : ''xω'' → ''cxω''}}. There are two variations on this theme: one can either find a primitive element {{math|''ω''}} that is a [[nilpotent]] element of the Clifford algebra, or one that is an [[idempotent]]. The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it.<ref>This construction is due to Cartan (1913). The treatment here is based on {{Harvtxt|Chevalley|1996}}.</ref> In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of {{math|''V''}}, and then specify the action of the Clifford algebra ''externally'' to that vector space.

In either approach, the fundamental notion is that of an [[isotropic subspace]] {{math|''W''}}. Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of {{math|''V''}} is given.

As above, we let {{math|(''V'', ''g'')}} be an {{math|''n''}}-dimensional complex vector space equipped with a nondegenerate bilinear form. If {{math|''V''}} is a real vector space, then we replace {{math|''V''}} by its [[complexification]] <math>V \otimes_\Reals \Complex</math> and let {{math|''g''}} denote the induced bilinear form on <math>V \otimes_\Reals \Complex</math>. Let {{math|''W''}} be a maximal isotropic subspace, i.e. a maximal subspace of {{math|''V''}} such that {{math|1=''g''{{!}}<sub>''W''</sub> = 0}}. If {{math|1=''n'' =  2''k''}} is even, then let {{math|''W''{{′}}}} be an isotropic subspace complementary to {{math|''W''}}. If {{math|1=''n'' =  2''k'' + 1}} is odd, let {{math|''W''{{′}}}} be a maximal isotropic subspace with {{math|1=''W'' ∩ ''W''{{′}} = 0}}, and let {{math|''U''}} be the orthogonal complement of {{math|''W'' ⊕ ''W''{{′}}}}. In both the even- and odd-dimensional cases {{math|''W''}} and {{math|''W''{{′}}}} have dimension {{math|''k''}}. In the odd-dimensional case, {{math|''U''}} is one-dimensional, spanned by a unit vector {{math|''u''}}.