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),

ξA = εBA ξB ,   ψA = εAB ψB ,   implying   εACεBC = δAB ,

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: XAB' \(\mapsto\)C X* A'B = XBA' (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:= {XAB', self-conjugate}; The Infeld-Van der Waerden symbol, or soldering form, is the isomorphism

va = σaAA' vAA',   and   vAA' = σaAA' va ;

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) = gab va wb = εAB ε*A'B' vAA' wBB' ;

In particular, ξa = σaAA' ξA ξA' is a null vector for all ξA.
@ 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].
@ 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].

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 GAA' , used to convert primed indices to unprimed ones; It is the natural one in 3D.
* Euclidean 3-space: Can be obtained as E:= {vAB, symmetric}; The isomorphism and the metric are given now by the Pauli matrices,

vi = σiAB vAB,   vAB = σi AB vi ;
qij = εAB εCD σiAC σjBD.

* 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

λA = 21/2 tAA' λ*A' ,   where   tAA':= σaAA' t a .

@ References: Polychronakos & Sfetsos a1609 [decomposition of the product of many spins].

And Physics
* Weyl spinors: Weyl proposed them as a way to model 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.
@ 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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