4-Spinors  

Majorana Spinors > s.a. dirac field theory.
* Idea: A Majorana spinor space is a 4D real vector space, Vm = {ψA}, with an ε up to sign, ± εAB, and a complex structure such that = –εJ; It carries an irreducible real representation of the Lorentz group; Majorana fermions are their own antiparticles.
* Motivation: The search for Majorana fermions has become an important one for condensed-matter physicists; The pursuit of Majorana fermions is driven by their potential to encode quantum information in a way that solves a problem dogging quantum computing, because Majorana fermions could carry information that would be immune to environmental noise.
* Remark: There is no natural isomorphism between Vm and its dual.
* And Minkowski space: M can be obtained as

M = {vABVmVm* | vAB = –vBA, or Jv = –vJ} ,

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

va = γaAB vAB,    vAB = γaAB va,    and metric    (v,w) = ηab va wb := \(1\over4\)tr(vw) .

* And Lorentz group: One gets a representation by γab:= \(1\over2\)(γa γb – γb γa).
* Advantage: They are simpler to generalize to n dimensions than 2-spinors, and they become 2int(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 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 a1612-proc [in particle physics, solid state and quantum information]; Backens et al a1703 [and Ising spin chains].
@ In 3+1 dimensions: Heί JMP(94); Ahluwalia hp/02-proc; Aste Sym(10)-a0806 [rev].
@ In n-dimensions: Finkelstein & Villasante PRD(85).
@ 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 a1708 [in condensed matter systems]; > 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 SR-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)/2E]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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