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];
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,

*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 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
***

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