On Projections of Spinor Spaces onto Minkowski Space

Abstract

The problem of projecting spinor spaces on Minkowski space is rather old. The first formula of this type is

EquationSource% MathType!MTEF!2!1!+- % feaagGart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiEamaaBa % aaleaaiiaacqWF8oqBaeqaaOGaeyypa0Jae8NVdG3aa0baaSqaaGqa % aiaa+fgaaeaacaGGQaaaaOGae83Wdm3aa0baaSqaaiaa+X7aaeaace % GFHbGbaiaacaGFIbaaaOGae8NVdG3aaSbaaSqaaiaa+jgaaeqaaOGa % aiilaiaabccacaqGGaGaaeiiaiaabggacaGGSaGaaeOyaiabg2da9i % aaigdacaGGSaGaaGOmaiaacYcacaqGGaGaaeiiaiaabccacqWF8oqB % cqWF9aqpcaaIWaGaaiilaiaaigdacaGGSaGaaGOmaiaacYcacaaIZa % aaaa!57D0!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${{\rm{x}}_\mu } = \xi _a^*\sigma _\mu ^{\dot ab}{\xi _b},{\rm{ a}},{\rm{b}} = 1,2,{\rm{ }}\mu = 0,1,2,3$$

(1.1)

projecting the two-dirensional complex spinor space C² onto the light cone in M₄. One can consider (1.1) also as a projection on E₃. These projection are consistent with the group in the sense that SU(2) transformations of the ξ_a induce SO(3) transformations of x_l, x₂, x₃ and leave x₀ invariant. Transformations of SL(2,C) induce SO(3.1) transformations of x_μ.