Bispinor Space

Spinors are used extensively in physics; it is widely accepted that they are more foundamental than tensors. The foundations of the concept of spinors are groups; spinors appear as representations of groups. As is well known, both the spinors and the representations are widely used in the theory of elementary particles. Mathematicans have realized the relations between spinors and differential forms. Differential forms is a tensor field covariant at the action of SO(2n), which is a kind of tool in classcial physics. Spinors is covariant at the action of Spin group,which is regarded as an effective tool in quantum mechanics. The relations between spinors and differential forms can be regarded as equal to the quantizations of classical mathematics. Consequently, it is necessary to represent their relations. In 1994, Seiberg and Witten discovered their monopole equations:where Ψ is a spinor, F_A⁺ is a differential 2-form. These equations are nonlinear. When mathematicans have research on these equations, they get some Seiberg-Witten invariants, which are equivalent to Donaldson invariants. But Seiberg-Witten invariants are more easy to compute than Donaldson invariants. The latter equation is a self-dual differential 2-forms. We think that if we represent the equation by bispinors, it is possible to get some surprise results.Firstly, based on the concept of augmented spinor space, we have the definition of bispinor space. We can identify some spaces. Thus, we can see the spaces S S|-,Hom(5),C_2n(-1)(?)C as bispinor spaces. And they satisfy the following commutative diagramHom_c(S) → Hom_cIn order to guarantee the commutativity of the above diagram, we construct a new map...