4-Spinors |

**Majorana Spinors** > s.a. dirac
field theory.

* __Idea__: A Majorana
spinor space is a 4D real vector space, *V _{m}*
= {

*

*

*

*M* = {*v*^{A}* _{B}*
∈

with isomorphism (soldering form) given by the Dirac gamma matrices:

*v*^{a} = *γ*^{a}_{AB}*
v*^{AB}, *v*^{AB}
= *γ*_{a}^{AB}
*v*^{a}, and
metric (*v*,*w*) = *η*_{ab}
*v*^{a} *w*^{b}
:= \(1\over4\)tr(*vw*) .

* __And Lorentz group__:
One gets a representation by *γ*^{ab}:=
\(1\over2\)(*γ*^{a} *γ*^{b}
*γ*^{b} *γ*^{a}).

* __Advantage__: They
are simpler to generalize to *n* dimensions than 2-spinors, and
they become 2^{int(n/2)}-spinors.

* __Dynamics__:
Majorana spinors satisfy a wave equation different from the Dirac
equation, a result originally due to M Kirchbach.

* __Applications__:
2011, Majorana fermions are considered ideal building blocks for logic
gates in a quantum computer because of their non-commutative exchange
statistics; In addition, these particles emerge as low-energy excitations
of topological phases, which are robust against perturbations that can
lead to decoherence and would therefore be a stable platform for quantum
computation; Majorana fermions have yet to be realized experimentally.

@ __General references__: Mankoč Bortnik et al ht/00
[mass terms]; Semenoff & Sodano EJTP-cm/06-ch
[zero modes]; Wilczek nPhys(09)
[rev]; Cheng et al a1412
[re their non-Abelian statistics]; Greco JPA-a1602
[path-integral representation]; Borsten & Duff a1612-proc [in particle physics, solid state and quantum information]; Backens et al a1703 [and Ising spin chains]; Joseph et al a1709 [phase space methods].

@ __In 3+1 dimensions__: Heί JMP(94);
Ahluwalia hp/02-proc;
Aste Sym(10)-a0806
[rev].

@ __In n-__

@

@

@

**Dirac Spinors** > s.a. dirac
field theory.

* __Idea__: Essentially
pairs of an SL(2,\(\mathbb C\)) spinor together with a complex conjugate
one, that can be defined in time-orientable but non-orientable manifolds,

**u**(**p**, *m*) = [(*E*+*m*)/2*E*]^{1/2}
(1, *σ* · ** p**/(*E*+*m*)) *χ*
.

@ __General references__: Papaioannou a1707 [physical interpretation].

@ __And spacetime__: Bugajska JMP(86);
Agostini et al CQG(04)gq/02
[and DSR]; Dappiaggi et al RVMP(09)-a0904
[on a globally hyperbolic spacetime]; Antonuccio a1206
[projection onto 3+1 spacetime].

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sep 2017