Spin Structure and Dirac Operator

Abstract

Let V be a finite dimensional vector space over ℝ with a symmetric bilinear form 〈 , 〉. The associated quadratic form is Q(v) = 〈v, v〉, v ∈ V, We have Q(rv) = r²Q(v), r ∈ ℝ, and

$$Q\left( {v + w} \right) = Q\left( v \right) + 2\left\langle {v,w} \right\rangle + Q\left( w \right)$$

.