Spinorska upodobitev Lorentzove grupe

V članku je obravnavana spinorska upodobitev Lorentzove grupe rotacij in potiskov. Sprva so navadene osnove matričnih Liejevih grup in teorije upodobitev, nato pa se podrobneje obravnava Lorentzova grupa in pripadajoča Liejeva algebra. Podano je zgodovinsko ozadje Diracove enačbe, ki pravzaprav motivira vpeljavo spinorske upodobitve. Natančneje so opisane transformacije Diracovih spinorjev, njihove nenavadne transformacijske lastnosti pa so pojasnjene z obstojem t.i. spin homomorfizma \(\mathrm{SL}(2,\mathbb{C}) \to \mathrm{SO}^+(1,3)\). Na koncu so navedene še vse nerazcepne upodobitve spin grupe Lorentzove grupe.

Spinor representation of the Lorentz group

This paper discusses the spinor representation of the Lorentz group of rotations and boosts. It begins with a review of the basics of matrix Lie groups and representation theory. Then the Lorentz group and its associated Lie algebra are examined in detail, offering historical context for the Dirac equation, which serves as the primary motivation for introducing the spinor representation. The transformations of Dirac spinors are analyzed in detail, and their unusual transformation properties are explained through the existence of the so-called spin homomorphism \(\mathrm{SL}(2,\mathbb{C}) \to \mathrm{SO}^+(1,3)\). The article concludes with a description of all irreducible representations of the spin group of the Lorentz group.