Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 6894.13.2.5 Symplectic Structure on the SpinorsThere is a canonical symplectic structure on the space of positive spinorsS + given by the symplectic form { · , · }, which takes the value{s + , t + } = s + 1 t+ 2 − s+ 2 t+ 1on the spinorss + =⎛⎝s + 1⎞⎠ and t + =⎛⎝t + 1⎞⎠ ∈ S + .s + 2t + 2The above symplectic form is Spin(4)−invariant.There is a symplectic Riesz representationwith the following identification⎛ ⎞⎝s + 1S + ∼=−→ S +∗⎛⎠ ↦→ { ⎝s + 1⎞⎠ , ·}.s + 2s + 2The symplectic Riesz representationS + −→ S +∗is given by⎛⎝s + 1s + 2⎞⎠ ↦→⎛⎝−s + 2s + 1⎞∗⎠ .That means⎛⎝s + 1⎞⎠∗⎛= { ⎝s + 2⎞⎠ , ·}.s + 2−s + 1Just like the Hermitian case, we also have the following.By using the symplectic Riesz representation, we haveS + ⊗ S + ∼ = End( S + ),S + ⊗ S + ∼ = Λ0C ⊕ Λ +C .