4-Spinors

Majorana Spinors
* Idea: A Majorana spinor space is a 4D real vector space, V_m = {^A}, with an up to sign, _AB, and a complex structure such that J = –J; It carries an irreducible real representation of the Lorentz group.
* Remark: There is no natural isomorphism between V_M and its dual.
* And Minkowski space: M can be obtained as

M = {v^A_B V_m V_m* | v^AB = –v^BA, or Jv = –vJ} ,

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

v^a = ^a_AB v^AB,    v^AB = _a^AB v^a,    and metric    (v,w) = _ab v^a w^b := tr(vw) .

* And Lorentz group: One gets a representation by ^ab:= (^{a b} – ^{b a}).
* Advantage: Simpler to generalize to n-dim than 2-spinors, and 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.
@ In 3+1 dimensions: Heß JMP(94); Ahluwalia hp/02-in; Aste 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].
@ Related topics: Borstnik et al ht/00 [mass terms]; Jeannerot & Postma JHEP(04)hp [zero modes in cosmic string background]; Semenoff & Sodano cm/06-in [zero modes].

Dirac Spinors > s.a. dirac field theory.
* Idea: Essentially pairs of an SL(2,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, · p/(E+m)) .

@ And spacetime: Bugajska JMP(86); Agostini et al CQG(04)gq/02 [and DSR].

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