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; 2018, Fundamental Majorana fermions have yet to be seen
experimentally, but Majorana quasiparticles have been observed as
coordinated patterns of atoms and electrons in particular superconductors.

@ __General references__: Mankoč Borštnik 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 proc(17)-a1612 [in particle physics, solid state and quantum information];
Backens et al PRB(17)-a1703 [and Ising spin chains];
Joseph et al a1709 [phase space methods];
news APS(18)apr [applications, search].

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

@ __In n dimensions__:
Finkelstein & Villasante PRD(85).

@

@

@

**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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send feedback and suggestions to bombelli at olemiss.edu – modified 29 apr 2018