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 *σ*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 *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].

@ __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 *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,

*v*^{i} = *σ*^{i}_{AB}* v*^{AB}, *v*^{AB}
= *σ*_{i}
^{AB}* v*^{i} ;

*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

*λ*_{A}^{†} =
2^{1/2}* t*_{A}^{A'} *λ**_{A'} , where *t*^{AA'}:=
*σ*_{a}^{AA'}* t*^{ a} .

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

**And Physics
***

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