Spinor

Spinor is a mathematical object used in physics to describe particles with half-integer spin and, more broadly, the projective representations of the rotation group. Unlike ordinary tensors, spinors transform under a double-valued (or more generally spin) representation of the rotation group, so that a full 360-degree rotation can multiply a spinor by -1.

Mathematically, spinors arise from the Spin(n) group, the double cover of SO(n). They can be constructed as modules of the Clifford algebra Cl_n. Complex spinor representations have dimension 2^{⌊n/2⌋}. In even dimensions there are two inequivalent chiral spinor representations (Weyl spinors) of dimension 2^{n/2−1}; in odd dimensions there is a single irreducible complex spinor representation of dimension 2^{(n−1)/2}.

In three spatial dimensions, the spinor representation is two-dimensional and is realized by the fundamental representation of SU(2), with Pauli matrices serving as generators of rotations. This underpins the quantum description of electron spin, which is a spin-1/2 degree of freedom.

In relativistic quantum physics, several types of spinors are used: Dirac spinors are four-component objects that accommodate both left- and right-handed chiralities and satisfy the Dirac equation with gamma matrices; Weyl (two-component) spinors describe massless fermions; Majorana spinors are real representations possible in certain spacetime dimensions. When coupling spinors to curved spacetime, one uses spinor fields defined via a spin structure and vierbeins.

Spinors are central to the description of fermions, angular momentum, and fundamental symmetries in quantum mechanics and quantum field theory; historically they were introduced by Elie Cartan and later integrated into the Pauli and Dirac formalisms.