2-Spinors

SL(2, C) or Weyl Spinors > s.a. Mandelstam Identities; Soldering Form.
* Idea: An SL(2, \(\mathbb C\)) spinor space is a pair (V, ε), with V a 2D complex vector space, ε a non-degenerate antisymmetric tensor; Spinors are elements of V and its dual V*, with ε used to raise and lower indices (note the index position),

ξ_A = ε_BA ξ^B ,   ψ^A = ε^AB ψ_B ,   implying   ε^AC ε_BC = δ^A_B ,

and similarly (primed) elements of the conjugate space to V, and its dual.
* Remark: There is a natural antilinear isomorphism C from V to V*: C(λξ) = λ* C(ξ).
* Spinorial tensors: Multilinear mappings, belonging to direct products of the above four vector spaces.
* Special tensors: The rank-two self-conjugate ones: X^AB' \(\mapsto\)_C X* ^A'B = X^BA' (only among indices of the same kind is the order important).
* Automorphism group: SL(2, \(\mathbb C\)), the double covering of SO(3) (the matrices have to be unimodular to preserve the ε tensor); Spinors are the fundamental irreducible representation of the connected component to e of the Lorentz group.
* Minkowski space: Can be obtained from the above as M:= {X^AB', self-conjugate}; The Infeld-Van der Waerden symbol, or soldering form, is the isomorphism

v^a = σ^a_AA' v^AA',   and   v^AA' = σ_a^AA' v^a ;

These σs can be thought of as one matrix for each value of a; In that case, they correspond to the Pauli matrices and the 2 × 2 identity; The metric on Minkowski space is recovered from the distinguished ε by setting

(v, w) = g_ab v^a w^b = ε_AB ε*_A'B' v^AA' w^BB' ;

In particular, ξ^a = σ^a_AA' ξ^A ξ^A' is a null vector for all ξ^A.
@ General references: Weyl ZP(29); Vassiliev PRD(07)gq/06 [teleparallel model]; Gattringer & Pak NPB(08) [on the lattice].
@ Related topics: Chervova & Vassiliev a1003-MG12 [modeling in terms of Cosserat theory of elasticity]; Budinich a2003 [null vectors and O(n)].
@ In physical systems: Volovik & Zubkov NPB(14)-a1402 [emergent Weyl spinors in multi-fermion systems]; Xu et al Sci(15)aug, Lv et al PRX(15) + news SciNews(15)jul, pw(15)jul, Vishwanath Phy(15) [observation of emergent massless Weyl fermions as electron excitations in TaAs]; Lu et al Sci(15)aug [in a photonic crystal]; Cianci et al EPJC(15)-a1507 [with gravity, exact solutions]; Isaev & Podoinitsyn NPB(18)-a1712 [free massive particles with arbitrary spin].

SU(2) Spinors > s.a. SU(2) group.
* Idea: An SU(2) spinor space is a complex vector space V with a non-degenerate antisymmetric tensor ε as above, and a Hermitian inner product \(G_{AA'}\), used to convert primed indices to unprimed ones; It is the natural one in 3D.
* Euclidean 3-space: It can be obtained as E:= {v^AB, symmetric}; The isomorphism and the metric are given now by the Pauli matrices,

vⁱ = σⁱ_AB v^AB,   v^AB = σ_i ^AB vⁱ ;
q_ij = ε_AB ε_CD σ_i^AC σ_j^BD.

* Relationship with SL(2, C) spinors in spacetime: If we have a 3-surface with associated SU(2) spinors and unit normal t ^a in a spacetime (M, g), then

λ_A^† = 2^1/2 t_A^A' λ*_A' ,   where   t^AA':= σ_a^AA' t ^a .

@ References: Polychronakos & Sfetsos NPB(16)-a1609 [decomposition of the product of many spins]; Gyamfi & Barone a1706 [composition of an arbitrary collection of SU(2) spins]; > s.a. types of spinors [decomposition of products].

And Physics
* Weyl spinors: Weyl proposed them as a way to model massless elementary particles; So far, no candidate Weyl fermions have been observed in high-energy experiments, but Zahid Hasan predicted that topological effects inside crystals of tantalum arsenide should create massless quasiparticles that act like Weyl fermions, that were experimentally reported in 2015, and similar quasiparticles were later observed in electromagnetic waves.
@ Weyl spinors: Xu et al Sci(15)aug [as collective excitations in Weyl semimetals]; news pt(19)oct [in magnetic materials].
@ Field theory: Dreiner et al PRP(10) [rev, and complete set of Feynman rules for fermions]; Dvornikov & Gitman PRD(12)-a1211 [quantum massive Weyl neutrinos in external fields]; Cardoso EPJP(15)-a1401 [in spacetimes with torsion]; Canarutto IJGMP(14)-a1404-proc [and gauge freedom].
@ General relativity: O'Donnell 03; Cardoso a1004 [review of Schouten's theory].

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