4-Spinors| 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 Jε = −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 = {vAB ∈ Vm ⊕ Vm* | 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; 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)/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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 19 jan 2021