Spinors II: Isotropic Subspaces

This is a sequel.

Definition

Consider V a quadratic space. Consider W a subspace. W is called isotropic when given v, w in W arbitrary, Q(v, w) = 0.

For instance, when k = R and V is positive definite (that is, sn V = dim V) the only isotropic subspace is {0}. If V is Lorentzian (that is, sn V = dim V - 1) we have
1-dimensional isotropic subspaces: null or lightlike lines in physics talk. On the other hand, for
k = C we have:

Proposition

• Suppose V is a complex quadratic space of dimension 2n. Then the maximal dimension of an isotropic subspace is n.
• Suppose V is a complex quadratic space of dimension 2n + 1. Then the maximal dimension of an isotropic subspace is n.

It will be useful to introduce the following

Definition

Consider V a quadratic space over an arbitrary field k. Consider W a subspace of V. Define W^ to be {v in V | for any w in W: Q(v, w) = 0}. W^ is called the subspace orthogonal to W.

Several facts about orthogonal subspace will be handly. Given V, W as above:

• dim W + dim W^ = dim V
• W^^ = W
• Suppose W is isotropic. Then W^ contains W.

Consider V a quadratic space over an arbitrary field k. Consider W an isotropic subspace of V. Denote n = dim V, m = dim W. We have dim W^ = n - m but W^ contains W hence
dim W^ >= dim W i.e. n - m >= m and thus n >= 2m. It is therefore obvious that the maximal dimension of an isotropic subspace can be at most as in the proposition, even for k different from C. It remains to show that for k = C the bound can always be saturated.

Proof of Proposition

• Suppose V is a complex quadratic space of dimension 2n.
  Consider e_1 ... e_2n a basis such that Q(e_i, e_j) = delta_ij. Here delta_ij is the Kronecker symbol, that is delta_ij = 1 for i = j and delta_ij = 0 for i =/= j. Define
  f_1 = e_1 + i e_2, f_2 = e_3 + i e_4 ... f_n = e_2n-1 + i e_2n. Here i is the imaginary unit, that is i = sqrt(-1). It is easily seen that {f_j} span an n-dimensional isotropic subspace.
• Suppose V is a complex quadratic space of dimension 2n + 1.
  Consider e_1 ... e_2n, e_2n + 1 a basis such that Q(e_i, e_j) = delta_ij. Define
  f_1 = e_1 + i e_2, f_2 = e_3 + i e_4 ... f_n = e_2n-1 + i e_2n. It is easily seen that {f_j} span an n-dimensional isotropic subspace.

Definition

Consider V a complex quadratic subspace. A subspace W of V is called a maximal isotropic subspace when it is of maximal possible dimension. That is:

• For dim V = 2n, we require dim W = n.
• For dim V = 2n + 1, we require dim W = n.

Definition

Consider V a quadratic space over the field k. Consider R: V -> V an operator. V is called orthogonal when for any v, w in V we have Q(Rv, Rw) = Q(v, w). The orthogonal operators are the automorphisms of V, that is, isomorphisms of V with itself. The set of all orthogonal operators is denoted O(V).

Proposition

Consider V a complex quadratic space, U and W two maximal isotropic subspaces. Then, there exists R in O(V) s.t. R(U) = W.

This means that all maximal isotropic subspaces of a given space are essentially the same. However, we'll see in the sequels that for dim V = 2n + 1, the space of maximal isotropic subspaces has 1 connected component, whereas for dim V = 2n there are 2 connected components. For now, I won't explain what I mean by "space" and what are "connected components". These are some basic concepts of topology which I hope to explain in the sequels.