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 JPA(18)-a1709 [phase space methods];
news APS(18)apr [applications, search];
Arodz APPB-a2002 [relativistic quantum mechanics];
De Vincenzo a2007 [wave equations].

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

@ __ Other dimensionalities__:
Finkelstein & Villasante PRD(85);
De Vincenzo a2007 [wave equations in 3+1 and 1+1 dimensions].

@ __Vs Dirac spinors__:
Semikoz NPB(97);
Dvoeglazov IC(00)phy;
Cahill & Cahill EJP(06)ht/05 [pedagogical].

@ __Realizations in the lab__:
Alicea PRB(10)
+ Franz Phy(10),
Stoudenmire et al PRB(11) [proposal];
Kraus & Stern NJP(11) [on a disordered triangular lattice];
Deng et al PRL(12)-a1108
+ news sn(12)aug,
Leijnse & Flensberg SST(12)-a1206 [topological superconductors];
news nat(12)feb,
PhysOrg(12)mar [and quantum computers];
Mourik et al Sci(12)apr
+ news at(12)apr,
Rokhinson et al nPhys(12)sep [as quasiparticles in nanowire devices];
Karzig et al PRX(13) [and qubit manipulation];
Tsvelik Phy(14)
[re signature in response of quantum spin liquids to an oscillating magnetic field];
Lepori et al NJP(18)-a1708 [in condensed matter systems];
Zhang et al Nat(18)mar [in semiconductor nanowires];
Manousakis et al PRL(20) [proposed test];
> s.a. graphene;
Josephson Effect.

@ __Related topics__: Jeannerot & Postma JHEP(04)hp [zero modes in cosmic string background];
Tamburini & Laveder PS(12)-a1109 [superluminal Majorana neutrinos at OPERA and apparent Lorentz violation];
Noh et al PRA(13)-a1210 ["Majoranon" and realization as qubit + continuous degree of freedom];
Pedro a1212;
Ohm & Hassler NJP(14) [coupled to electromagnetic radiation];
Li et al sRep-a1409 [non-locality].

**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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