2-Spinors

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

_A = _BA ^B ,   ^A = ^AB _B ,   implying   ^AC_BC = ^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() = * C().
* Spinorial tensors: Multilinear mappings, belonging to direct products of the above four vector spaces.
* Special tensors: The rank two self-conjugate ones: X^AB' _C X* ^A'B = X^BA' (only among indices of the same kind is the order important).
* Automorphism group: SL(2,C), the double covering of SO(3) (the matrices have to be unimodular to preserve the 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 = ^a_AA' v^AA',   and   v^AA' = _a^AA' v^a ;

These 's 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 by setting

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

In particular, ^a = ^a_AA' ^{A A'} is a null vector for all ^A.
@ References: Vassiliev PRD(07)gq/06 [teleparallel model]; Gattringer & Pak NPB(08) [on the lattice].

SU(2) Spinors
* Idea: An SU(2) spinor space is a V, an 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: Can be obtained as E:= {v^AB, symmetric}; The isomorphism and the metric are given now by the Pauli matrices,

vⁱ = ⁱ_AB v^AB,   v^AB = _i ^AB vⁱ ;
q_ij = _{AB CD i}^AC _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

_A = 2^1/2 t_A^A' *_A' ,   where   t^AA':= _a^AA' t^a .

References
@ General: O'Donnell 03 [in general relativity].

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